Coding Theory and Algebraic Combinatorics

Chapter 1 Co ding theory and algebraic com binatorics Michael Huber ∗ Institut f¨ ur Mathematik, T e chnische Universit¨ at Berlin, Str aße des 17. Jun i 136, D- 10623 Berlin, Germany, mhub er@math.tu-b erlin.de This c hapter introduces and elaborates on the fruitful interpla y of coding th eory and algebraic com b in atorics, with most of the fo cus on the in teraction of cod es with com binatorial designs, ﬁnite geometries, simple groups, sph ere packings, kissing num b ers, lattices, and asso ciation schemes. In particular, sp ecial interest is devoted to the relationship b etw een codes and combinatoria l d esigns. W e describe and recapitulate important results in the development of the state of the art. In addition, we giv e illustrative examples and constructions, and highlight recent adv ances. Finally , we provide a collection of signiﬁcant op en problems and chal lenges concerning fut u re researc h. 1.1. In tro duction The classical publications “A mathematical theory of comm unicatio n” by C. E . Shannon [1] and “Err or detecting and erro r c orrecting co des” by R. W. Ham- ming [2 ] gave birth to the twin disc iplines of informa tion theory and co ding theory . Since their inceptions the interactions of information and co ding theory with many mathematical branches hav e contin ually deep ened. This is in particula r true for the close connection b et ween co ding theory and algebraic co m binato rics. This c hapter in tro duces and elab orates on this fruitful in terplay of coding the- ory and alg ebraic combinatorics, with most of the focus on t he interaction of codes with combinatorial designs, ﬁnite g eometries, simple gr o ups, s phere pa c kings, kiss- ing n um be r s, la ttices, and asso ciation schemes. In particular, sp ecial interest is devoted to the rela tionship betw een co des and co m binato rial des ig ns. Since we do not ass ume the reader is familiar with the theor y of combinatorial desig ns, an accessible and r e asonably s e lf-con tained exp osition is provided. Subsequently , we describ e a nd rec a pitulate imp ortant results in the dev elopment of the state of the art, provide illustrative examples and c o nstructions, a nd highlight recent adv ances . F urther more, we g iv e a collectio n of sig niﬁcan t o pen problems and challenges con- cerning future resea rc h. ∗ The author gratefully ac kno wledges supp ort by the Deutsch e F orsc hungsgemeinsc haft (DFG ). 1 2 Michael Hub er The chapter is org anized as follows. In Sec. 1.2, we give a brief account of basic notions of a lgebraic c o ding theo ry . Section 1.3 co nsists o f the main part of the chapter: After an introduction to ﬁnite pro jective planes and c o m bina torial designs, a subsection on ba sic connections b etw een co des and combinatorial desig ns follows. The next subsectio n is on p erfect c odes and designs, and address es further related concepts. Subsection 1.3.4 deals with the classical Assmus-Mattson Theore m and v arious ana logues. A subs ection on co des and ﬁnite geo metries follows the discussion on the non-existence o f a pro jectiv e plane of or der 10. In Subsection 1.3.6, int errelatio ns betw een the Golay co des, the Mathieu-Witt designs, and the Mathieu groups are studied. Subsection 1.3 .7 deals with the Golay co des and the Leech lattice, a s well as rec e n t milestones concer ning kis s ing num ber s and sphere packings. The last topic of this section consider s co des and ass o ciatio n schemes. The chapter concludes with sections on directions for further r esearch as well as conclusions and exercises. 1.2. Bac kground F or o ur further purp oses, we give a shor t account of basic notions of algebraic co ding theory . F or additional informa tion on the sub ject of algebra ic co ding theory , the reader is r eferred to [3–13]. F o r some his to rical no tes on its orig ins , see [14 ] a nd [6, Chap. 1], as well as [15 ] for a historical survey on co ding theory a nd information theory . W e denote by F n the set o f a ll n -tuples fro m a q -symbol a lphabet. If q is a prime p ow er , we take the ﬁnite ﬁeld F = F q with q elements, and in terpret F n as an n -dimensiona l vector spa ce F n q ov er F q . The elemen ts of F n are called ve ct ors (or wor ds ) a nd will b e deno ted by b old symbols. The (Hamming) distanc e b et ween tw o co dewords x , y ∈ F n is deﬁned by the nu mber o f co ordinate p o sitions in which they diﬀer, i.e. d ( x , y ) : = |{ i | 1 ≤ i ≤ n, x i 6 = y i }| . The weight w ( x ) of a co dew ord x is deﬁned by w ( x ) : = d ( x , 0 ) , whenever 0 is a n element of F . A subset C ⊆ F n is called a ( q -ary ) c o de of length n ( binary if q = 2 , tern ary if q = 3). The elements of C are called c o dewor ds . A line ar c o de (or [ n, k ] c o de ) ov er the ﬁeld F q is a k -dimensional linear subspac e C of the vector s pace F n q . W e note that large pa rts of co ding theo ry ar e concerned with linea r co des. In particular, as many combinatorial conﬁgurations can b e des c ribed by their incidence matrices , co ding theorists have sta rted in the ea r ly 1960 ’s to co nsider as co des the vector spaces spa nned by the r o ws of the resp ectiv e incidence matrices ov er some g iv en ﬁeld. Co ding t he ory and algebr aic c ombinatorics 3 The minimum distanc e d of a co de C is deﬁned as d : = min { d ( x , y ) | x , y ∈ C, x 6 = y } . Clearly , the minimum dista nce o f a linea r co de is equa l to its minimum weight , i.e. the minim um of the weigh ts of a ll no n-zero co dewords. An [ n, k , d ] c o de is an [ n , k ] co de with minimum distance d . The minimum distance of a (not necessar ily linear) co de C determines the e rror- correcting capability o f C : If d = 2 e + 1, then C is called an e - err or-c orr e cting c o de . Deﬁning by S e ( x ) : = { y ∈ F n | d ( x , y ) ≤ e } the spher e (or b al l ) of ra dius e around a co deword x of C , this implies that the spheres of r adius e aro und distinct co dew ords a re disjoint. Counting the num b er of co dew ords in a spher e o f r adius e yields to the subse- quent spher e p acking (or Hamming ) Bound . Theorem 1.1. L et C b e a q -ary c o de of length n and minimum distanc e d = 2 e + 1 . Then | C | · e X i =0  n i  ( q − 1 ) i ≤ q n . If equality holds, then C is called a p erfe ct c o de . Eq uiv alen tly , C is p erfect if the s pheres of radius e around all co dew ords cover the whole space F n . Certainly , per fect co des are comb inatorially interesting ob jects, how ever, they are extremely rare. W e will call tw o co des (p ermutation) e quivalent if one is obtained fro m the other by applying a ﬁxed p ermu tation to the co ordina te p ositions for all co de words. A gener ator matrix G for an [ n, k ] c ode C is a ( k × n )-matrix for which the rows are a basis of C . W e say that G is in standar d form if G = ( I k , P ), where I k is the ( k × k ) identit y matrix. F or a n [ n, k ] co de C , let C ⊥ : = { x ∈ F n q | ∀ y ∈ C [ h x , y i = 0 ] } denote the dual c o de of C , where h x , y i is the standard inner (or dot) pro duct in F n q . The co de C ⊥ is an [ n, n − k ] co de. If H is a gener ator matrix for C ⊥ , then clearly C = { x ∈ F n q | x H T = 0 } , and H is called a p arity che ck matrix for the co de C . If G = ( I k , P ) is a ge ne r ator matrix of C , then H = ( − P T , I n − k ) is a parity c heck matr ix of C . A c ode C is called s elf-dual if C = C ⊥ . If C ⊂ C ⊥ , then C is ca lled self-ortho gonal . 4 Michael Hub er If C is a linea r co de o f length n ov er F q , then C : = { ( c 1 , . . . , c n , c n +1 ) | ( c 1 , . . . , c n ) ∈ C, n +1 X i =1 c i = 0 } deﬁnes the exten de d c o de cor resp onding to C . The s ym b ol c n +1 is called the over al l p arity che ck symb ol . Conversely , C is the pu n ctur e d (or shortene d ) co de of C . The weight distribution o f a linear co de C of leng th n is the se quence { A i } n i =0 , where A i denotes the n umber o f co dewords in C o f weigh t i . The po lynomial A ( x ) : = n X i =0 A i x i is the weight enu mer ator o f C . The weigh t enumerators of a liner co de a nd its dual co de a re rela ted, as shown by the following theorem, which is one of the most imp ortant results in the theor y of erro r -correcting co des. Theorem 1.2. (MacWilliams [16 ]). L et C b e an [ n , k ] c o de over F q with weight enumer ator A ( x ) and let A ⊥ ( x ) b e the weight enumer ator of the dual c o de C ⊥ . Then A ⊥ ( x ) = q − k (1 + ( q − 1) x ) n A  1 − x 1 + ( q − 1) x  . W e note tha t the concept of the weight enumerator can b e gener alized to no n- linear co des (so- called distanc e enum er ator , cf. [17 , 18] and Subsection 1.3.8). An [ n , k ] co de C ov er F q is called cyclic if ∀ ( c 0 ,c 1 ,...,c n − 1 ) ∈ C [( c n − 1 , c 0 , . . . , c n − 2 ) ∈ C ] , i.e. any cyclic shift of a co deword is ag a in a co deword. W e adopt the us ual conv en- tion for cyclic co des that n a nd q ar e co prime. Using the isomo rphism ( a 0 , a 1 , . . . , a n − 1 ) ⇄ a 0 + a 1 x + . . . + a n − 1 x n − 1 betw een F n q and the residue class ring F q [ x ] / ( x n − 1 ), it follows tha t a cyclic co de corres p onds to an ideal in F q [ x ] / ( x n − 1). 1.3. Though ts for Practitioners In the following, we introduce and elab orate on the fruitful in terplay o f co ding theory and algebra ic combinatorics, with mo st of the fo cus on the int eraction of co des with combinatorial desig ns, ﬁnite g eometries, simple gro ups, spher e packings, kissing num ber s, lattices, a nd ass ociatio n schemes. In par ticular, sp ecial interest is devoted to the re la tionship b etw een co des and combinatorial designs. W e give an accessible and rea sonably self-contained exp osition in the ﬁrst subs e ction a s we do not a ssume the rea der is familiar with the theory of combinatorial designs. In what Co ding t he ory and algebr aic c ombinatorics 5 follows, w e de s cribe and reca pitula te imp ortant results in the developmen t of the state of the art. In addition, we give illustra tiv e e xamples a nd constructions, a nd highlight r ecen t achievemen ts. 1.3.1. Intr o du ction to ﬁni te pr oje cti ve pl anes and c ombinatori al de- signs Combinatorial desig n theory is a sub ject of consider able interest in dis c rete ma th- ematics. W e give in this subsection an intro duction to the topic , with emphasis on the construction of some impo rtan t designs. F or a more gener al tr eatmen t of com- binatorial designs, the rea der is referr e d to [19–24]. In pa rticular, [1 9, 2 1] provide encyclop edias o n key results. Besides co ding theory, there a re many interesting connectio ns of design theory to other ﬁelds. W e mention in our co n text esp ecially its links to ﬁnite g eome- tries [25], incidence geo metry [26], group theory [2 7 – 30], graph theory [4 , 31], cryp- tography [32–34], as well as classiﬁcatio n a lgorithms [35]. In addition to tha t, we recommend [22 , 36–39] for the re ader interested in the br oad area of c om bina torics in general. W e sta rt by int ro ducing se veral no tions. Deﬁnition 1.1. A pr oje ctive plane of or der n is a pair of p oints a nd lines such that the following pr oper ties hold: (i) any tw o distinct p oints ar e on a unique line, (ii) any tw o distinct lines int ersect in a unique p oin t, (iii) there exists a qu adr angle , i.e. four po ints no three of which are on a common line, (iv) there ar e n + 1 points on each line, n + 1 lines thro ugh each p oint and the total nu mber o f p oin ts, resp ectively lines , is n 2 + n + 1. It follows easily from (i), (ii), and (iii) that the num ber of p oints o n a line is a constant. When s etting this co ns tan t e qual to n + 1, then (iv) is a consequence o f (i) a nd (iii). Combinatorial desig ns can b e r e garded as genera lizations of pro jective planes: Deﬁnition 1.2. F or p ositive integers t ≤ k ≤ v and λ , w e deﬁne a t - design , or more precis ely a t - ( v , k , λ ) design , to b e a pair D = ( X , B ), w he r e X is a ﬁnite set of p oints , a nd B a set of k -element subsets of X ca lled blo cks , with the pro perty that any t p oints ar e contained in precisely λ blo cks. W e will denote p oin ts b y lower-case and blo cks by upper -case Latin letters. Via conv ention, we set v : = | X | and b : = |B | . Througho ut this chapter, ‘rep eated blo c k s ’ are not allowed, that is, the same k -element subset of p oin ts may not o ccur twice as a blo ck. If t < k < v holds, then we sp eak of a non- trivial t -design. 6 Michael Hub er Designs may b e represented alge br aically in ter ms o f incidence matrices: Let D = ( X , B ) b e a t -design, a nd let the p oin ts b e lab eled { x 1 , . . . , x v } and the blo c ks be lab eled { B 1 , . . . , B b } . Then, the ( b × v )-matrix A = ( a ij ) (1 ≤ i ≤ b , 1 ≤ j ≤ v ) deﬁned by a ij : =  1 , if x j ∈ B i 0 , otherwise is called an incidenc e matrix of D . Clearly , A depe nds on the resp ectiv e la b eling, how ever, it is unique up to column and row p ermu tation. If D 1 = ( X 1 , B 1 ) and D 2 = ( X 2 , B 2 ) are t wo t -desig ns, then a bijective map α : X 1 − → X 2 is called an isomorphism of D 1 onto D 2 , if B ∈ B 1 ⇐ ⇒ α ( B ) ∈ B 2 . In this case, the designs D 1 and D 2 are isomorphic . An iso mo rphism of a desig n D onto itself is called an aut omorph ism of D . Ev iden tly , the set of all automo r phisms of a design D fo rm a gr oup under comp osition, the ful l automorphism gr oup of D . An y subgro up of it will b e called an automorphism gr oup o f D . If D = ( X , B ) is a t -( v , k , λ ) des ign with t ≥ 2, a nd x ∈ X a rbitrary , then the derive d design with r esp e ct to x is D x = ( X x , B x ), where X x = X \{ x } , B x = { B \{ x } | x ∈ B ∈ B } . In this ca se, D is a ls o called an extens ion o f D x . Ob- viously , D x is a ( t − 1)-( v − 1 , k − 1 , λ ) design. The c omplementary design D is obtained by replacing each blo c k o f D by its co mplemen t. F or his to rical reas ons, a t -( v , k , λ ) design with λ = 1 is called a St einer t - design . Sometimes this is a lso known as a Steiner system if the pa r ameter t is cle arly given from the c o n tex t. The sp ecial case of a Steiner desig n with parameters t = 2 and k = 3 is called a Steiner tr iple system of or der v (brieﬂy S T S ( v )). The question reg arding their existence was pos ed in the classical “Combinatorische Aufgab e” (185 3) of the nine- teent h century geometer Jakob Steiner [40]: “W elc he Zah l, N , von Elementen hat d ie Eigensc haft, dass sich die Elemente so zu dreien ordnen lassen, dass je zw ei in einer, ab er nur in einer V erbindung vork ommen?” How ever, there had b een ea r lier work on these pa rticular des igns go ing ba c k to, in particular , J. P l¨ uck er, W. S. B. W oo lhouse, and most no ta bly T. P . Kirkman. F or a n acco un t on the early histor y of designs, see [21, Chap. I.2] and [4 1]. A Steiner design with par ameters t = 3 and k = 4 is ca lled a St einer qu adruple system of or der v (brieﬂy S QS ( v )). If a 2- design has equally many p oints and blo c k s, i.e. v = b , then we sp eak of a squar e design (as its incidence matrix is sq uare). By tradition, square designs are often called symmet r ic designs , although her e the term do es not imply any symmetry of the design. F or more on these interesting des igns, see, e.g., [42]. Co ding t he ory and algebr aic c ombinatorics 7 W e give some illustrative examples of ﬁnite pro jective planes and combinatorial designs. W e a ssume that q is always a prime p o wer. Example 1 .1. Let us cho ose as p oin t set X = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } and a s blo c k s e t B = {{ 1 , 2 , 4 } , { 2 , 3 , 5 } , { 3 , 4 , 6 } , { 4 , 5 , 7 } , { 1 , 5 , 6 } , { 2 , 6 , 7 } , { 1 , 3 , 7 }} . This gives a 2-(7 , 3 , 1) design, the well-kno wn F ano plane , the smallest design arising from a pr o jective geo metry , which is unique up to isomorphism. W e g iv e the usua l representation of this pro jective plane of o rder 2 by the following diagra m: 7 1 3 6 2 5 4 Fig. 1.1. F ano plane Example 1 .2. W e take as p oint set X = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } and a s blo c k s e t B = {{ 1 , 2 , 3 } , { 4 , 5 , 6 } , { 7 , 8 , 9 } , { 1 , 4 , 7 } , { 2 , 5 , 8 } , { 3 , 6 , 9 } , { 1 , 5 , 9 } , { 2 , 6 , 7 } , { 3 , 4 , 8 } , { 1 , 6 , 8 } , { 2 , 4 , 9 } , { 3 , 5 , 7 }} . This gives a 2-(9 , 3 , 1) design, the smallest non-tr ivial design aris ing fr om a n aﬃne geometry , which is a g ain unique up to iso morphism. This aﬃne plane o f order 3 can b e constr ucted from the a rray 1 2 3 4 5 6 7 8 9 as shown in Figure 1.2. 8 Michael Hub er 1 4 7 2 5 8 3 6 9 Fig. 1.2. Aﬃne plane of order 3 More ge ner ally , we obtain: Example 1.3. W e choose a s p oint set X the set of 1-dimensio nal subspaces of a vector space V = V ( d, q ) of dimension d ≥ 3 ov er F q . As blo c k set B we ta k e the s et of 2- dimensional subspaces o f V . Then there ar e v = ( q d − 1) / ( q − 1) po in ts and each blo c k B ∈ B co n ta ins k = q + 1 p oints. Since obviously any t wo 1-dimensio nal subspaces s pa n a sing le 2 -dimensional subspa ce, any tw o distinct po in ts are contained in a unique blo c k. Th us, the pr oje ctive sp ac e P G ( d − 1 , q ) is an example of a 2-( q d − 1 q − 1 , q + 1 , 1) design. F or d = 3, the particular designs are pr oje ctive planes of or der q , which ar e square designs. Mo re generally , for any ﬁxed i with 1 ≤ i ≤ d − 2, the p oints and i -dimensional subspaces of P G ( d − 1 , q ) (i.e. the ( i + 1)-dimensio nal subspaces of V ) yie ld a 2 -design. Example 1.4. W e take as p oint set X the set of elements of a vector space V = V ( d, q ) o f dimension d ≥ 2 ov er F q . As blo ck s et B we choo se the set of aﬃne lines of V (i.e. the translates of 1-dimensional subspaces of V ). Then there ar e v = q d po in ts and each blo c k B ∈ B contains k = q p oint s. As clearly a n y tw o distinct p o in ts lie on exactly one line, they are contained in a unique blo c k. Hence, we obtain the aﬃne sp ac e AG ( d, q ) as an example of a 2 -( q d , q , 1) design. When d = 2, these designs are aﬃne planes of or der q . More generally , for any ﬁxed i with 1 ≤ i ≤ d − 1, the p oints and i -dimensiona l subspaces of AG ( d, q ) form a 2- design. Remark 1.1. It is well-established that b oth aﬃne a nd pro jectiv e pla ne s of o rder n e x ist whenever n is a prime p ow er. The conjecture that no such planes ex ist with or de r s other than prime p ow e r s is unre s olv ed so far. The clas sical r esult of R. H. Bruck a nd H. J. Ryser [43] s till remains the o nly g e ne r al statement: If n ≡ 1 or 2 (mo d 4) and n is not equa l to the sum of tw o squar es of integers, then n do es not o ccur a s the or der of a ﬁnite pro jective plane. The smalles t integer that is not a prime p ow e r and not cov er ed b y the Bruck-Ryser Theor em is 10. Using substantial Co ding t he ory and algebr aic c ombinatorics 9 computer analy sis, C. W. H. L a m, L. Thiel, and S. Swiercz [44] proved the non- existence of a pro jective plane of order 1 0 (cf. Remar k 1.10). The next smallest nu mber to consider is 1 2, for which neither a p ositive nor a negative answer has bee n proved. Needless to mention that — apa rt from the existence pr oblem — the questio n on the num b e r of diﬀerent isomor phis m t yp es (when exis ten t) is fundamental. Ther e are, for example, pr ecisely four non-isomor phic pro jective pla nes of o rder 9. F o r a further discussion, in par ticular of the rich history of a ﬃne and pr o jective planes, we r efer, e.g., to [25 , 45–49]. Example 1.5 . W e take as po in ts the vertices of a 3-dimensio nal c ube. As illustrated in Figure 1.3, we can cho ose three types of blo cks: (i) a fac e (six o f these), (ii) t wo opp osite edges (six of these), (iii) an insc r ibed regula r tetrahedro n (tw o of these). This gives a 3 -(8 , 4 , 1) design, which is unique up to is omorphism. Fig. 1.3. Steiner quadruple s ystem of order 8 W e have more gener ally: Example 1.6. In AG ( d, q ) a ny thre e distinct p oin ts deﬁne a plane unless they are collinear (that is , lie on the same line). If the underlying ﬁeld is F 2 , then the lines contain only tw o p oin ts and hence any thr ee p oints canno t b e co llinear. Therefore, the p oints and planes o f the aﬃne s pa ce AG ( d, 2) form a 3-(2 d , 4 , 1) design. More generally , for any ﬁxed i with 2 ≤ i ≤ d − 1 , the p oints and i -dimens ional subspaces of AG ( d, 2) for m a 3-design. Example 1.7. The unique 2-(9 , 3 , 1) des ig n whos e p oin ts and blo cks a re the po in ts and lines of the aﬃne plane AG (2 , 3) can b e extended precisely three times to the following desig ns which ar e a ls o unique up to is o morphism: the 3- (10 , 4 , 1) design which is the M¨ obius plane of order 3 with P Γ L (2 , 9 ) as full automor phism gr oup, and the tw o Mathieu-Witt designs 4-(11 , 5 , 1) and 5-(12 , 6 , 1) with the sp oradic Mathieu gro ups M 11 and M 12 as po in t 4-tr ansitiv e and p oin t 5-transitive full auto- morphism groups, resp ectiv ely . 10 Michael Hub er T o constr uc t the ‘lar ge’ Mathieu-Witt designs one starts with the 2-(21 , 5 , 1) design w ho se p oints a nd blocks are the p oints a nd lines of the pro jec- tive plane P G (2 , 4). This ca n b e e x tended a lso exac tly three times to the fo l- lowing unique des igns: the Mathieu-Witt design 3-(22 , 6 , 1) with Aut( M 22 ) as p oint 3-transitive full automo r phism gr oup as well as the Mathieu-Witt designs 4 -(23 , 7 , 1) and 5- (24 , 8 , 1) with M 23 and M 24 as p oint 4-tr a nsitiv e and po in t 5 -transitive full automorphism gro ups, r espectively . The ﬁve Mathieu groups were the ﬁrst s pora dic simple gro ups a nd were dis- cov er ed by E. Mathieu [50 , 51] ov er one hundred years ago. They a re the only ﬁnite 4- and 5- transitive per mutation groups apart from the sy mmetric or alter- nating gro ups. The Steiner des igns a sso c iated with the Mathieu g roups were ﬁrst constructed by b oth R. D. Car mic hael [2 8 ] and E . Witt [5 2], and their uniq ueness established up to is o morphism by Witt [53]. F rom the meanwhile v a rious alter- native constructio ns, we mention esp ecially those o f H. L ¨ uneburg [54] and M. As- ch bacher [55, Chap. 6]. How ever, the easiest wa y to co nstruct and prove uniqueness of the Ma thieu- Witt des igns is via co ding theory , using the related binary and ternary Golay c o des (see Subsection 1.3 .6). Remark 1.2. By cla ssifying Steiner desig ns whic h admit automorphism groups with suﬃciently strong sy mmetr y prop erties, sp eciﬁc characterizations of the Mathieu-Witt des igns with their related Mathieu gro ups w e re obtained (see, e.g., [5 6–61] and [62, Chap. 5] for a survey). Remark 1.3. W e mention tha t, in ge ne r al, for t = 2 and 3, there a re ma n y inﬁnite classes o f Steiner t -desig ns, but for t = 4 a nd 5 only a ﬁnite num b er ar e known. Although L. T eirlinck [63] has shown that non- trivial t -designs ex is t fo r a ll v alues of t , no Steiner t -designs hav e b e e n constr ucted for t ≥ 6 so far . In what follows, we need some helpful co m bina torial to ols: A standar d combinatorial double co un ting argument gives the following as s er- tions. Lemma 1.1 . L et D = ( X , B ) b e a t - ( v , k , λ ) design. F or a p ositive int e ger s ≤ t , let S ⊆ X with | S | = s . Then the total nu mb er λ s of blo cks c ontaining al l the p oints of S is given by λ s = λ  v − s t − s   k − s t − s  . In p articular, for t ≥ 2 , a t - ( v , k , λ ) design is also an s - ( v , k , λ s ) design. F or histor ical reasons , it is custo mary to set r : = λ 1 to b e the total num be r o f blo c ks cont aining a given po in t (referring to the ‘replication num b er’ from statistical design of exp eriment s, o ne of the orig ins o f designs theory). Co ding the ory and algebr aic co mbinatorics 11 Lemma 1.2. L et D = ( X , B ) b e a t - ( v , k , λ ) design. Then the fol lowing holds: (a) bk = v r . (b)  v t  λ = b  k t  . (c) r ( k − 1) = λ 2 ( v − 1) for t ≥ 2 . Since in Lemma 1.1 each λ s m ust b e an in teg er, we have moreov er the subsequent necessary arithmetic conditions. Lemma 1.3. L et D = ( X , B ) b e a t - ( v , k , λ ) design. Then λ  v − s t − s  ≡ 0 (mo d  k − s t − s  ) for e ach p ositive inte ger s ≤ t . The following theorem is an imp ortant result in the theor y of designs , ge nerally known as Fisher’s In e quality . Theorem 1 .3. (Fisher [64 ]). If D = ( X , B ) is a non- trivial t - ( v , k , λ ) design with t ≥ 2 , then we have b ≥ v , that is, t her e ar e at le ast as many blo cks as p oints in D . W e r emark that equality holds exactly for square desig ns when t = 2. Obviously , the equality v = b implies r = k by L emma 1.2 (a). 1.3.2. Basi c c onne ctions b etwe en c o des and c ombinatorial designs There is a rich and fruitful interpla y b et ween co ding theory and des ign theory . In particular, ma n y t - designs have b een found in the last decades by considering the co dew ords of ﬁxed weigh t in some s pecial, o ften linear co des. As we will see in the sequel, these c odes typically exhibit a high degree of regula rit y . There is an amo un t of literature [4 , 7, 13 , 31, 65–72] discussing to some extent in mo re detail v ario us relations b etw een co des and desig ns. F or a co deword x ∈ F n , the set supp( x ) : = { i | x i 6 = 0 } of all co ordinate p ositions with non- zero co or dinates is ca lled the su pp ort of x . W e shall often form a t -design o f a co de in the following wa y: Giv en a (usually linear) co de of length n , which contains the zero vector, and no n-zero weight w , we choo se as po in t s et X the set o f n co ordinate p ositions of the co de a nd as blo ck set B the suppo rts of a ll co dewords of weigh t w . Since we do not allow r epeated blo cks, clear ly only distinct repr esen tatives o f suppo rts for co dewords with the same supp orts ar e taken in the non-binar y c ase. 12 Michael Hub er W e g ive some elementary exa mples. Example 1 .8. The ma trix G =     1 1 0 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 1     is a gene r ator matrix of a binar y [7 , 4 , 3] Hamming co de, w hich is the smallest non-trivial Hamming co de (see also Example 1 .12). This co de is a p erfect single- error -correcting co de with weigh t distributio n A 0 = A 7 = 1, A 3 = A 4 = 7. The seven co dew ords of weigh t 3 are precis ely the seven rows of the incidence matrix            1 1 0 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 1 0 1 0 0 0 1            of the F a no pla ne P G (2 , 2 ) of Fig. 1.1. The suppo rts o f the seven co dewords of weigh t 4 yield the complementary 2-(7 , 4 , 2) design, i.e. the bipla ne of order 2. Example 1.9. The matr ix ( I 4 , J 4 − I 4 ), where J 4 denotes the (4 × 4 ) all-one matrix, generates the extended binary [8 , 4 , 4] Hamming co de. This c ode is self-dual and has weigh t distribution A 0 = A 8 = 1, A 4 = 14. As any tw o co dew ords of weight 4 hav e dista nce at lea st 4, they have at most t wo 1’s in common, and hence no co dew ord of weight 3 can app ear as a subw ord of mor e than one co deword. On the other hand, there are  8 3  = 5 6 words of weigh t 3 and ea c h c odeword o f weigh t 4 has four subw or ds of weigh t 3. Hence each co deword of weight 3 is a subw ord of exactly one co dew ord of weight 4 . Therefor e, the s upports o f the fourteen co dewords of weigh t 4 form a 3-(8 , 4 , 1) design, which is the unique S QS (8) (cf. Exa mple 1.5). W e give also a basic example of a non-line a r co de co nstructed from design theory . Example 1. 1 0. W e take the rows o f an incidence matrix of the (unique) Hadamar d 2-(11 , 5 , 2) design, and adjo in the all-one co deword. Then, the tw elve co dew ords hav e mutual distance 6, and if we delete a co ordinate, we get a binar y non-line a r co de of length 1 0 and minimum distance 5. F or a detaile d description of the connection be t ween non-linea r co des and design theory as well as the application of design theor y in the area of (ma jo rit y -logic) deco ding, the rea der is referr e d, e.g., to [1 3, 71, 72]. Co ding the ory and algebr aic co mbinatorics 13 Using highly tra nsitiv e per m utatio n gr oups, a further construction o f des igns from co des can b e describ ed (see , e.g., [31]). Theorem 1.4. L et C b e a c o de which admits an automorph ism gr oup acting t -homo gene ously (in p articular, t -tr ansitively) on the set of c o or dinates. Then the supp orts of the c o dewor ds of any non- zer o weight form a t -design. Example 1.11. The r -th order R e e d-Mul ler (RM) c o de RM( r , m ) of length 2 m is a binary [2 m , P r i =0  m i  , 2 m − r ] co de with its co dew ords the v alue- v ectors of all Bo olean functions in m v a riables of de g ree at most r . These c odes were ﬁr st co nsidered by D. E. Muller [7 3] and I. S. Reed [7 4] in 1954 . The dual of the Reed- Muller co de RM( r , m ) is RM( m − r − 1 , m ). Clearly , the extended binary [8 , 4 , 4] Hamming co de in Example 1.9 is RM(1 , 3). Alternatively , a co dew o rd in RM( r , m ) ca n b e view ed as the sum of characteristic functions of subspaces of dimensio n at least m − r of the aﬃne space AG ( m, 2). Thus, the full automorphism g r oup of RM( r, m ) contains the 3 -transitive gr o up AGL ( m, 2 ) of all aﬃne transfo rmations, and hence the co dew ords of any ﬁxed non-zer o weigh t yield a 3-design. 1.3.3. Perfe ct c o des and designs The in terplay between co ding theo ry and combinatorial des igns is most evidently seen in the r elationship b etw een p erfect c o des a nd t -designs. Theorem 1.5. (Assm us a nd Mattson [75]). A line ar e -err or-c orr e cting c o de of length n over F q is p erfe ct if and only if the supp orts of the c o dewor ds of minimum weight d = 2 e + 1 form an ( e + 1) - ( n, d, ( q − 1) e ) design. The questio n “Does every Steiner triple system on n points extend to a Steiner q uadruple system on n + 1 p oin ts?” which go es also back to Jakob Steiner [40 ], is s till unreso lv e d in general. Ho wev er, in terms of binary e -er r or-corr ecting co des, there is a p ositive a nsw er. Theorem 1 .6. (Assmus and Mattson [7 5]). L et C b e a (not ne c essarily line ar) binary e -err or c orr e cting c o de of length n , which c ontains the zer o ve ctor. Then C is p erfe ct if and only if the su pp orts of the c o dewor ds of minimum weight d = 2 e + 1 form a Steiner ( e + 1) - ( n, d, 1) design. Mor e over, t he su pp orts of the minimum c o dewor ds in the ext ende d c o de C form a Steiner ( e + 2) - ( n + 1 , d + 1 , 1 ) design. W e hav e seen in Example 1.8 a nd Example 1.9 that the supp orts o f the s ev en co dew ords of weight 3 in the binary [7 , 4 , 3] Hamming co de form a S T S (7), while the supp orts of the four teen co de words of weight 4 in the extended [8 , 4 , 4] Hamming 14 Michael Hub er co de yield a S QS (8). In view of the ab ov e theo rems, we get mor e genera lly : Example 1.12. Let n : = ( q m − 1) / ( q − 1). W e cons ide r a ( m × n )-matr ix H ov er F q such that no t wo columns of H ar e linea rly dep enden t. Then H clearly is a parity chec k matrix of a n [ n, n − m, 3] co de, which is the Hamming c o de ov e r F q . The n um ber of its co dewords is q n − m , and for any co deword x , we hav e S 1 ( x ) = 1 + n ( q − 1 ) = q m . Hence, by the Sphere Pac king Bound (Theo r em 1.1), this co de is per fect, and the supp orts of co dew ords of minimum weigh t 3 form a 2-( n, 3 , q − 1) design. F ur thermore, in a binary [2 m − 1 , 2 m − 1 − m, 3] Hamming co de the supp orts of co dewords of weigh t 3 for m a S T S (2 m − 1 ), a nd the supp orts of the co dewords of weigh t 4 in the ex tended co de yield a S QS (2 m ). Note. The Hamming co des were develop ed by R. W. Hamming [2 ] in the mid 1940’s , who was employ ed at Bell Lab oratories , and addr essed a need fo r error correctio n in his work o n the primitive computers o f the time. W e remark that the extended binar y [2 m , 2 m − m − 1 , 4] Hamming co de is the Reed-Muller c ode RM( m − 2 , m ). Example 1.13. The binary Golay c o de is a [23 , 1 2 , 7] co de, while the ternary Golay c o de is a [11 , 6 , 5] co de. F or b oth co des, the par ameters imply e q ualit y in the Sphere Pac king Bound, and hence these co des are p erfect. W e will discuss later v arious constr uctions of these s ome of the most famous co des (see Example 1 .1 4 a nd Construction 1.12). By the ab ov e theorems, the supp orts of co dewords of minim um weigh t 7 in the binary [23 , 12 , 7] Golay co de form a Steiner 4-(2 3 , 7 , 1) design, and the s upports o f the co dewords of weigh t 8 in the extended binar y [24 , 12 , 8 ] Golay co de give a Steiner 5-(2 4 , 8 , 1) desig n. T he supp orts of co dewords of minimum weigh t 5 in the ter nary [11 , 6 , 5] Golay co de yield a 3-(11 , 5 , 4) design. It can b e shown (e.g., via Theorem 1 .4) that this is indeed a Steiner 4 -(11 , 5 , 1) design. W e will see in Example 1.15 that the supp o rts o f the co dewords of weight 6 in the extended ter nary [12 , 6 , 6 ] Golay co de give a Steiner 5-(12 , 6 , 1) desig n; thus the ab o ve results ar e not b est p ossible. Note. The Go la y co des were discov ered b y M. J. E. Go la y [76] in 1949 in the pro cess o f extending Ha mming’s co nstruction. They hav e numerous pra ctical rea l- world applica tions, e.g ., the use of the extended binary Golay co de in the V oyager spacecra ft progra m during the early 1980’s or in contemporar y standar d Automatic Link Establishment (ALE) in High F r equency (HF) data communication for F or- ward Er ror Cor rection (FEC). Remark 1.4. It is easily seen from their co ns truction that the Hamming co des are unique (up to equiv alence). It was shown by V. Pless [77 ] that this is also tr ue for the Golay co des. Moreov er, the binary and ternary Go la y co des are the only no n-trivial per fect e -error -correcting co des with e > 1 ov er a n y ﬁeld F q . Using integral ro ots of the Lloyd po lynomial, this r emark able fa c t was prov e n by A. Tiet¨ av¨ a inen [7 8 ] a nd Co ding the ory and algebr aic co mbinatorics 15 J. H. v an Lint [79], and indep enden tly b y V. A. Z ino v’ev and V. K. Leo n t’ev [80 ]. M. R. Best [8 1] and Y. Hong [82] extended this result to arbitrar y alpha b ets for e > 2 . F or a thoroug h acc o un t of p erfect co des, we refer to [83] and [84, Chap. 11]. As trivial per fect co des can only form trivial designs, we have (up to equiv alence) a co mplete list o f non-trivial linear p erfect co des with their ass ocia ted designs: Code Code parameters Design parameters Hamming co de [ q m − 1 q − 1 , q m − 1 q − 1 − m, 3] q any pri me p o w er 2-( q m − 1 q − 1 , 3 , q − 1) binary Golay co de [23 , 12 , 7] q = 2 4-(23 , 7 , 1) ternary Gola y co de [11 , 6 , 5] q = 3 4-(11 , 5 , 1) There are v a rious constructions of no n-linear single-err o r-correc ting p erfect co des. F or more details, see, e.g., [9, 13, 71, 72, 85] and refere nces therein. How ever, a classiﬁcation of these co des seems out o f rea c h at present, although some progr ess has b een made recently , see, for instance [86–88 ]. Remark 1.5 . The long-standing question whether every Steiner triple system of order 2 m − 1 oc c urs in a per fect co de has b een answered r ecen tly in the neg ativ e. Relying on the class iﬁcation [8 9] of the Steiner qua druple systems of order 16, it was shown in [90 ] that the unique anti-Pasc h Steiner triple system of o rder 15 provides a co un ter example. Remark 1.6. Due to the close rela tionship betw een p erfect co des and some of the most interesting des igns, several natural extensions o f p erfect co des hav e b een examined in this resp ect: Ne arly p erfe ct c o des [91], and the mor e genera l cla ss of uniformly p acke d c o des [92 , 93], were s tudied extensively and even tually lead to t -designs. H. C. A. v an Tilb org [9 4] show ed that e -e rror corr e c ting uniformly pack ed co des do not exist for e > 3 , and class iﬁed thos e for e ≤ 3 . F or more details, see [4, 10 , 1 3 , 94]. The concept of diameter p erfe ct c o des [95, 96] is r elated particularly to Steiner designs. F or fur ther generalizatio ns of p erfect co des, see e.g., [8 4, Chap. 11] and [13, Chap. 6]. 1.3.4. The Assmus-Mattson The or em and anal o gues W e co nsider in this s ubsection one of the mo st fundamental res ults in the interplay of co ding theor y a nd design theor y . W e sta rt by introducing tw o impo rtan t classes of co des. Let q b e an o dd prime p o wer. W e deﬁne a function χ (the so- c alled L e gendr e- symb ol ) on F q by χ ( x ) : =    0 , if x = 0 1 , if x is a no n-zero squar e − 1 , o therwise . 16 Michael Hub er W e note that χ is a character o n the m ultiplicative g roup of F q . Using the elements of F q as row a nd column la bels a i and a j (0 ≤ i, j < q ), resp ectively , a ma trix Q = ( q ij ) of order q can b e deﬁned by q ij : = χ ( a j − a i ) . (1.1) If q is a pr ime, then Q is a circula n t matrix. W e ca ll a matrix C q +1 : =      0 1 · · · 1 χ ( − 1) . . . Q χ ( − 1)      of or der q + 1 a Paley matrix . Thes e matr ic es were constructed by R. A. Paley in 1933 and are a sp eciﬁc type o f confere nc e matrice s, which hav e their o r igin in the application to c o nference telephone c ir cuits. Construction 1.7. Let n b e an o dd prime, and q b e a quadr atic r esidue (mo d n ), i.e. q ( n − 1) / 2 ≡ 1 (mo d n ). The quadr atic r esidue c o de (or QR c o de ) of length n ov er F q is a [ n, ( n + 1 ) / 2] c ode with minim um weight d ≥ √ n (so-ca lled Squar e R o ot Boun d ). It can b e g enerated b y the (0 , 1)-cir culan t ma trix of order n with top row the incidence vector o f the non-zer o q uadratic residues (mo d n ). These co des are a specia l clas s of cyclic co des a nd were ﬁrst constr ucted by A. M. Gleason in 1964. F or n ≡ 3 (mo d 4 ), the extended quadra tic re sidue co de is self-dua l. W e note for the imp ortant binary ca se that q is a quadratic residue (mod n ) if and only if n ≡ ± 1 (mo d 8). Note. By a theorem o f A. M. Gleason and E. Prang e, the full automorphism g roup of an extended qua dratic res idue co de of length n contains the g roup P S L (2 , n ) o f all linea r fractio na l transfor mations whos e determina n t is a non-zer o square. Example 1.14. The bina ry [7 , 4 , 3] Hamming co de is a quadr atic r esidue co de of length 7 over F 2 . The binar y [23 , 12 , 7 ] Golay co de is a quadratic residue co de of length 23 over F 2 , while the terna ry [1 1 , 6 , 5] Golay co de is a quadr atic r esidue co de of length 1 1 ov er F 3 . Construction 1.8 . F or q ≡ − 1 (mo d 6 ) a prime p o wer, the Pless symmetry c o de Sym 2( q +1) of dimension q + 1 is a ternary [2( q + 1) , q + 1] co de with g enerator matr ix G 2( q +1) : = ( I q +1 , C q +1 ), where C q +1 is a Paley ma trix. Since C q +1 C T q +1 = − I q +1 (ov er F 3 ) for q ≡ − 1 (mod 3), the co de Sym 2( q +1) is self-dual. This inﬁnite family of cyclic co des were intro duced by V. Pless [97, 98] in 197 2. W e note that the ﬁr st symmetry co de S 12 is equiv alent to the extended [1 2 , 6 , 6] Golay co de. The celebrated Assmus-Mattson Theorem gives a suﬃcient condition for the co dew ords of co nstan t weigh t in a linear co de to form a t -des ign. Co ding the ory and algebr aic co mbinatorics 17 Theorem 1.9. (Assmus and Mattson [9 9]). L et C b e an [ n, k , d ] c o de over F q and C ⊥ b e the [ n, n − k, e ] dual c o de. L et n 0 b e the lar gest int e ger such that n 0 − n 0 + q − 2 q − 1 < d , and deﬁne m 0 similarly for t he dual c o de C ⊥ , wher e as if q = 2 , we assu m e t hat n 0 = m 0 = n . F or some inte ger t with 0 < t < d , let us su pp ose that ther e ar e at most d − t non-zer o weights w in C ⊥ with w ≤ n − t . Then, for any weight v with d ≤ v ≤ n 0 , the supp orts of c o dewor ds of weight v in C form a t -design. F urthermor e, for any weight w with e ≤ w ≤ min { n − t, m 0 } , t he su pp ort of the c o dewor ds w in C ⊥ also form a t -design. The pr oo f of the theorem inv olves a clever use o f the MacWilliams relatio ns (Theorem 1.2). Along with these, Lemma 1 .1 and the immediate o bserv ation that co dew ords of weight less than n 0 with the same supp ort must b e scalar multiples of each o ther, form the basis of the pro of (for a detailed pro of, see, e.g., [4, Chap. 14 ]). Remark 1.7. Un til this re sult by E. F. Assmus, Jr. and H. F. Mattson, Jr . in 1969, only very few 5-designs were known: the Mathieu-Witt desig ns 5 -(12 , 6 , 1) and 5- (24 , 8 , 1) , the 5- (24 , 8 , 48) design formed by the co dewords o f weight 12 (the do de c ads ) in the extended binary Go la y co de, as well a s 5-(12 , 6 , 2 ) and 5 -(24 , 8 , 2) designs which had b een found without using co ding theor y . How ever, by using the Assmus-Mattson Theorem, it was p ossible to ﬁnd a num b er of new 5-desig ns . In particular, the theore m is most useful when the dual co de has relatively few non- zero weights. Nevertheless, it has not b een po ssible to detect t -desig ns for t > 5 by the Assmus-Mattson T heo rem. W e illustr a te in the following examples some applications of the theorem. Example 1 .15. The extended binar y [24 , 12 , 8 ] Golay co de is self-dua l (cf. Con- struction 1.7) and has co dewords of weigh t 0 , 8 , 1 2 , 16, a nd 2 4 in view o f Theo - rem 1 .2. F o r t = 5, we o btain the Steiner 5-(24 , 8 , 1 ) design as in Ex a mple 1 .13. In the self-dual extended terna ry [1 2 , 6 , 6] Go la y co de all co dewords are divisible b y 3, and hence for t = 5 , the suppo rts o f the co dew ords of weight 6 for m a Steiner 5-(12 , 6 , 1) design. Example 1.16. The extended quadratic residue co de o f length 48 over F 2 is self- dual with minimum dista nc e 12. By Theorem 1.2, it has co dewords of weight 0 , 12 , 16 , 20 , 2 4 , 28 , 32 , 36, and 48. F or t = 5, each of the v a lues v = 12 , 1 6 , 20 , or 24 yields a diﬀerent 5-design and its complementary design. Example 1.17 . The P less symmetry co de Sym 36 of dimension 18 is self-dual (cf. Co nstruction 1.8) with minimum distance 12. The supp orts of c o dewords of weigh t 12 , 15 , 18 , and 2 1 yield 5-des ig ns together with their complementary desig ns. Remark 1.8. W e give an overview o f the state of k no wledge concerning co des over F q with their ass ocia ted 5- designs (cf. a lso the tables in [13, Chap. 16 ], [65 , 71 , 72 ]). In fact, thes e co des are all se lf- dua l. T rivial desig ns as w ell as co mplemen tar y designs are o mitted. 18 Michael Hub er Code Code parameters Design parameters Ref. Extended cyclic code [18 , 9 , 8] q = 4 5-(18 , 8 , 6) [100] 5-(18 , 10 , 180) Extended binary Gola y code [24 , 12 , 8] q = 2 5-(24 , 8 , 1) [101] 5-(24 , 12 , 48) Extended ternary Gola y co de [12 , 6 , 6] q = 3 5-(12 , 6 , 1) Lifted Golay co de o ve r Z 4 [24 , 12] Z 4 5-(24 , 10 , 36) [102, 103] 5-(24 , 11 , 336) [102] 5-(24 , 12 , 1584) [102] 5-(24 , 12 , 1632) [102] Extended quadric residue codes [24 , 12 , 9] q = 3 5-(24 , 9 , 6) [65, 99] 5-(24 , 12 , 576) 5-(24 , 15 , 8580) [30 , 15 , 12] q = 4 5-(30 , 12 , 220) [65, 99] 5-(30 , 14 , 5390) 5-(30 , 16 , 123000) [48 , 24 , 12] q = 2 5-(48 , 12 , 8) [65, 99] 5-(48 , 16 , 1365) 5-(48 , 20 , 36176) 5-(48 , 24 , 190680) [48 , 24 , 15] q = 3 5-(48 , 12 , 364) [65, 99] 5-(48 , 18 , 50456) 5-(48 , 21 , 2957388) 5-(48 , 24 , 71307600) 5-(48 , 27 , 749999640) [60 , 30 , 18] q = 3 5-(60 , 18 , 3060) [65, 99] 5-(60 , 21 , 449820) 5-(60 , 24 , 34337160) 5-(60 , 27 , 1271766600) 5-(60 , 30 , 24140500956) 5-(60 , 33 , 239329029060) Pless symmetry codes [24 , 12 , 9] q = 3 5-(24 , 9 , 6) [98] 5-(24 , 12 , 576) 5-(24 , 15 , 8580) [36 , 18 , 12] q = 3 5-(36 , 12 , 45) [98] 5-(36 , 15 , 5577) 5-(36 , 18 , 209685) 5-(36 , 21 , 2438973) [48 , 24 , 15] q = 3 5-(48 , 12 , 364) [98] 5-(48 , 18 , 50456) 5-(48 , 21 , 2957388) 5-(48 , 24 , 71307600) 5-(48 , 27 , 749999640) [60 , 30 , 18] q = 3 5-(60 , 18 , 3060) [97, 98] 5-(60 , 21 , 449820) 5-(60 , 24 , 34337160) 5-(60 , 27 , 1271766600) 5-(60 , 30 , 24140500956) 5-(60 , 33 , 239329029060) Note. The lifted Golay co de ov e r Z 4 is deﬁned in [104 ] as the extended Hensel lifted quadric residue co de of le ngth 24. The supp orts of the co dewords of Ha mming weigh t 10 in the lifted Golay c o de and cer tain extremal double circulant Typ e II Co ding the ory and algebr aic co mbinatorics 19 co des of length 24 yield (non-iso morphic) 5-(2 4 , 10 , 36) desig ns. W e further note that the quadr a tic residue co des and the Pless symmetry c o des listed in the ta ble with the same par ameters are not equiv ale nt as shown in [98] by insp e c ting sp eciﬁc elements o f the automo r phism gro up P S L (2 , q ). Remark 1.9. The concept of the weigh t enumerator can b e generalized to non- linear co des (so-called distanc e enumer ator ), which leads to a n ana log of the MacWilliams relations as well as to simila r results to the Ass m us- Mattson T he- orem for non-linear co des (s e e [1 7, 18, 105 ] and Subsection 1.3.8). The ques tion whether there is a n analogous result to the Ass m us- Mattson theo rem fo r co des ov er Z 4 prop osed in [10 2] was answered in the aﬃr mativ e in [106]. F urther generaliz a - tions o f the Assmus-Mattson Theor e m are known, see in particular [107–113]. 1.3.5. C o des and ﬁni te ge ometries Let A be an incidence matrix o f a pro jective pla ne P G (2 , n ) of o rder n . When we consider the subspace C of F n 2 + n +1 2 spanned b y the rows of A , we obta in for o dd n only the [ n 2 + n + 1 , n 2 + n, 2] co de co nsisting of all co dewords of even weigh t. The case for even n is more int eresting, in particular if n ≡ 2 (mo d 4). Theorem 1.1 0 . F or n ≡ 2 (mo d 4 ) , the r ows of an incidenc e matrix of a pr oje ctive plane P G (2 , n ) of or der n gener ate a binary c o de C of dimension ( n 2 + n + 2) / 2 , and t he ext ende d c o de C is self-dual. In a pro jective plane P G (2 , n ) of even or der n , there exist s e ts of n + 2 p oin ts, no three of which are collinea r, and which are ca lle d hyp er ovals (sometimes just ovals , cf. [46 ]). This gives furthermore Theorem 1.11. The c o de C has minimum weight n + 1 . Mor e over, the c o dewor ds of minimum weight c orr esp ond to the lines and those of weight n + 2 to t he hyp er ovals of P G (2 , n ) . Remark 1.10. The ab ov e tw o theorems arose in the context of the exa mina tion of the e xistence o f a pro jective pla ne of order 10 (cf. Remark 1.1; for detailed pro ofs see, e.g., [4, Chapt. 13]). Assuming the existence of such a plane, the obtained prop erties o f the corr esponding co de lead to very extensive computer s e arches. F or example, in an ea rly crucia l step, it was shown [114 ] that this co de co uld not have co dew ords of weigh t 1 5. On the v a r ious attempts to attack the problem and the ﬁnal veriﬁcation of the non- existence, we r efer to [44, 115, 116 ] a s well as [22, Chap. 17] and [3 5, Chap. 12]. Note. W e note that at present the F a no plane is the only known pro jective pla ne with order n ≡ 2 (mo d 4). 20 Michael Hub er F or further acc o un ts on co des and ﬁnite geometries , the reader is referr ed, e.g ., to [66, Cha p. 5 and 6] and [3, 4, 6 7, 11 6–119], a s well as [120 ] from a mor e gr oup- theoretical p ersp ectiv e and [121 ] with an emphasis on quadra tic for ms ov er F 2 . 1.3.6. Golay c o des, Mathieu-Witt designs, and Mathieu gr oups W e highlig h t s ome of the rema rk able and natural interrelations b et w een the Golay co des, the Mathieu-Witt designs, and the Mathieu groups . There a r e v arious diﬀerent constructions for the Golay co des b esides the de- scription as quadra tic residue co des in Exa mple 1.14. W e brieﬂy illus trate some exemplary constr uc tio ns. F or further details and mor e constr uctio ns, we r efer to [1 2 2], [13, Chap. 20], [4, Chap. 11], and [123, Chap. 11]. Construction 1.12. • Starting with the zer o vector in F 24 2 , a linear co de o f le ngth 24 can b e obtained by successively taking the lexicographica lly lea st binar y co deword which ha s not b een used and which ha s distance a t least 8 to any predecesso r. At the end of this pro cess, we have 40 96 co dew o rds whic h for m the extended binary Go la y co de. This constructio n is due to J. H. Conw ay and N. J. A. Sloane [1 24]. • Let A b e an incidence ma trix of the (unique) 2-(11 , 6 , 3) design. Then G : = ( I 12 , P ) with P : =      0 1 · · · 1 1 . . . A 1      is a (12 × 24)-matr ix in which ea c h row (exc ept the top r o w) has eig ht 1’s, and generates the extended binary Golay co de. • Let N b e an (12 × 12 )- adjacency matrix of the gra ph fo r med b y the vertices and edg e s of the r egular icosahedro n. Then G : = ( I 12 , J 12 − N ) is a gener ator matrix for the extended binary Golay co de. • W e r ecall that F 4 = { 0 , 1 , ω , ω 2 } is the ﬁeld of four elements with ω 2 = ω + 1. The hexac o de is the [6 , 3 , 4] co de over F 4 generated by the matrix G : = ( I 3 , P ) with P : =   1 ω 2 ω 1 ω ω 2 1 1 1   . The extended binary Golay co de ca n b e deﬁned by identifying e ac h co deword with a bina ry (4 × 6)-matrix M (with rows m 0 , m 1 , m 2 , m 3 ), satisfying (i) each column o f M has the sa me parity a s the ﬁrst r o w m 0 , (ii) the sum m 1 + ω m 2 + ω 2 m 3 lies in the hex aco de. Co ding the ory and algebr aic co mbinatorics 21 This description is essentially equiv alent to the computationa l too l MOG (Mir- acle Oct ad Gener ator) of R. T. Cur tis [125]. The construction via the hex a code is by Conw ay , see, e.g ., [123, Chap. 11]. • Let Q b e the c irculant matrix of order 5 deﬁned by Eq. (1.1). Then G : = ( I 6 , P ), where P is the ma trix Q bo r dered on top with a row o f 1 ’s, is a generator matrix of the terna ry Golay co de. Remark 1. 11. Referring to Ex a mple 1.7, we no te that the automo rphism g roups of the Golay co des are isomorphic to the pa rticular Mathieu groups, as was ﬁrst po in ted out in [101, 12 6]. Mo reov er, the Golay co des ar e re lated in a par ticularly deep and interesting wa y to a larg er family of s pora dic ﬁnite simple g roups (cf., e.g., [5 5]). Remark 1.12. W e hav e seen in E xample 1.1 3 that the supp orts of the co dewords of weigh t 8 in the extended bina ry [24 , 12 , 8] Go la y co de form a Steiner 5-(24 , 8 , 1) design. The uniqueness of the la rge Mathieu-Witt desig n (up to isomor phism) can be esta blis hed easily via co ding theor y (cf. Exa mple 1.7). The main par t is to s ho w that any binary co de of 4 096 c o dewords, including the zero vector, of length 24 a nd minim um distance 8, is linea r a nd ca n b e determined uniquely (up to equiv a lence). F or further details, in particula r for a uniqueness pro of of the small Mathieu-Witt designs, see, e.g., [7 0, 12 2 ] and [4, Cha p. 1 1]. 1.3.7. Golay c o des, L e e ch lattic e, kissing numb ers, and spher e p ack- ings Sphere packings clo sely c onnect mathema tics and informatio n theor y via the sam- pling theorem as observed by C. E. Shanno n [1] in his classical article of 1948. Rephrased in a more geometric languag e, this ca n b e expr essed as follows: “Nearly eq ual signals are represented by neighboring p oints, so to keep the signals distinct, Shannon represents them by n -dimensional ‘billiard balls’, an d is th ere- fore led to ask: what is the b est wa y to pack ‘billiard balls’ in n dimensions?” [127] One o f the most remark a ble lattices, the L e e ch lattic e in R 24 , plays a cr ucial role in c la ssical spher e packings. W e recall that a lattic e in R n is a discrete subg roup of R n of rank n . The ex tended binary Golay co de led to the discovery b y John Leech [1 28] of the 24 -dimensional Euclidean lattice named a fter him. There are v arious constructions b esides the us ual ones from the bina ry and ternar y Golay co des in the meant ime, see, e.g., [129], [123, Cha p. 24]. W e outline some of the fundamen tal connections b et ween s phere packings and the Leech lattice. The Kissing Numb er Pr oblem deals with the maximal num b er τ n of equal size non-ov erlapping spher e s in the n -dimensio nal Euclide a n spac e R n that can touch a given sphere of the same siz e . Only a few of these num be r s are actually known. F or dimensions n = 1 , 2 , 3, the classica l solutions a re: τ 1 = 2, τ 2 = 6, τ 3 = 12 . The num b er τ 3 was the sub ject of a famous controv er sy b etw een Isaa c Newton and 22 Michael Hub er David Gr e gory in 1694, and was ﬁnally veriﬁed only in 19 53 by K. Sch¨ utte and B. L. v a n der W aerden [1 30]. Using an a pproach initia ted by P . Delsarte [18 , 131] in the early 197 0’s which gives linear progr amming upper b ounds for binary erro r - correcting co des a nd for spher ical co des [132] (cf. Subsection 1 .3.8), A. M. O dlyzk o and N. J. A. Slo ane [1 33], and indep endently V. I. Levensh tein [1 34], proved that τ 8 = 2 4 0 and τ 24 = 1 9 6560. These exact solutions are the num ber of no n-zero vectors of minimal length in the ro ot lattice E 8 and in the Le e c h lattice, r espectively . By ex tending and improving Dels a rte’s metho d, O. R. Musin [1 35] veriﬁed in 2 003 that τ 4 = 2 4, which is the num b er o f non-zero vectors of minimal length in the ro ot lattice D 4 . The Spher e Packing Pr oblem asks for the maximal density of a pa c k ing of equal size no n-o verlapping spheres in the n -dimensio na l Euclidean space R n . A sphere packing is called a lattic e p acking if the centers of the spheres form a lattice in R n . The Leech la ttice is the unique densest lattice pa c king (up to sca ling a nd is ometries) in R 24 , as was shown by H. Co hn and A. K umar [136, 1 37] r e cen tly in 2004 , ag ain by a mo diﬁcation of Delsar te’s metho d. Mor e o ver, they show ed that the density of any sphere packing in R 24 cannot exceed the one given by the Leech lattice by a factor of more than 1 + 1 . 65 · 10 − 30 (via a computer calculation). The pro of is based on the work [138] by Cohn and N. D. Elkies in 200 3 in which linear pro gramming bo unds for the Sphere Packing Problem a re introduced and new upp e r b ounds on the density o f sphere packings in R n with dimension n ≤ 3 6 a re prov e n. F or further details on the K issing Num be r Problem and the Spher e Pac king Problem, s ee [123 , Chap. 1], [12 7 , 13 9], [14], as well as the survey a rticles [140–142]. F or a n on-line databa se on lattices, see [14 3 ]. 1.3.8. C o des and asso ciation schemes An y ﬁnite nonempty subs e t o f the unit sphere S n − 1 in the n -dimensiona l Euclide a n space R n is called a spheric al c o de . These co des hav e many practical a pplications, e.g., in the desig n of signals for data transmission and storag e . As a sp ecial class of spherical co des, spheric al designs were introduce d by P . Delsa rte, J.-M. Go ethals and J. Seidel [132] in 19 77 as ana logs on S n − 1 of the classic al combinatorial designs. F or example, in S 2 the tetrahedr on is a spherical 2-design; the o ctahedr on a nd the cube ar e spher ical 3- designs, and the icosahedr o n and the do decahedron are spher- ical 5 -designs. In order to obtain the linear prog ramming upp er bo und mentioned in the previo us subsection, Kr a wtch ouk p olynomials were inv olved in the case o f binary err or-corr ecting co des a nd Gegenbauer po lynomials in the case o f spherical co des. How ever, Delsar te’s approa c h was indeed m uc h more general and far-reaching. He developed for asso ciation schemes, which hav e their origin in the statistical theory o f design of exp eriments, a theor y to unify many o f the ob jects w e hav e bee n addr essing in this chapter. W e g iv e a formal deﬁnition of asso ciation schemes Co ding the ory and algebr aic co mbinatorics 23 in the s ense of Delsarte [18] a s well as intro duce the Hamming and the J ohnson schemes a s imp ortant examples o f the t w o fundamental cla s ses of P -p olynomial and Q -p olynomial asso ciation schemes . Deﬁnition 1. 3. A d - class asso ciation scheme is a ﬁnite p oint set X together with d + 1 re la tions R i on X , s atisfying (i) { R 0 , R 1 , . . . , R d } is a partition of X × X , (ii) R 0 = { ( x, x ) | x ∈ X } , (iii) for e ac h i with 0 ≤ i ≤ d , there exists a j with 0 ≤ j ≤ d such that ( x, y ) ∈ R i implies ( y , x ) ∈ R j , (iv) for any ( x, y ) ∈ R k , the num b er p k ij of p oints z ∈ X with ( x, z ) ∈ R i and ( z , y ) ∈ R j depe nds only on i, j and k , (v) p k ij = p k j i for all i , j and k . The num b ers p k ij are ca lle d the interse ction numb ers of the asso ciation scheme. Two po in ts x, y ∈ X are calle d i -th asso ciates if { x, y } ∈ R i . Example 1.1 8 . The Hamming scheme H ( n, q ) has as p oint set X the set F n of all n -tuples fro m a q -s ym b ol alphab e t; tw o n -tuples a r e i -th as s ocia tes if their Hamming distance is i . The Johnson scheme J ( v , k ), with k ≤ 1 2 v , ha s as p oin t set X the set of all k -elemen t s ubsets of a set of size v ; tw o k -element s ubset S 1 , S 2 are i -th asso ciates if | S 1 ∩ S 2 | = k − i . Delsarte intro duced the Hamming and Jo hnson schemes as settings fo r the clas - sical conce pt of error -correcting co des and combinatorial t -de s igns, resp ectively . In this manner, certain results b e come for mally dual, like the Sphere Pac king Bound (Theorem 1.1) and Fisher’s Inequality (Theore m 1.3). F or a more extended trea tmen t of a sso c ia tion schemes, the rea der is refer r ed, e.g., to [144–148], [4, Cha p. 1 7], [13, Chap. 21], and in pa rticular to [14 9, 150] with an emphasis on the close connection b et ween co ding theory a nd asso ciations s c he mes . F or a sur vey on spher ical designs, see [21 , Chap. VI.54 ]. 1.4. Directions for further researc h W e pres en t in this section a collection of signiﬁca nt op en pro blems and challenges concerning future resear c h. Problem 1.1. (cf. [40]). Doe s every Steiner tr iple system on n p oints extend to a Steiner quadruple system on n + 1 p oints? Problem 1.2. Do es ther e exist any non-tr ivial Steiner 6-desig n ? Problem 1.3. (cf. [13, p. 180]). Find all no n-linear sing le-error- correcting p erfect co des ov er F q . 24 Michael Hub er Problem 1.4. (cf. [6, p. 106 ]). Cha r acterize c o des where all co dewords of the same weigh t (or of minimum weigh t) form a non-trivia l desig n. Problem 1.5 . (cf. [6, p. 116]). Find a pro of of the non-existence o f a pro jective plane of order 10 without the help o f a computer o r with a n easily r epro ducible computer progra m. Problem 1.6. Do es there exis t a n y ﬁnite pro jective pla ne of o rder 12, or of a n y other order that is neither a prime p ow er no r cov e r ed b y the Bruck-Ryser Theor em (cf. Rema rk 1.1)? Problem 1.7. Do es the r oo t lattice D 4 give the unique kissing num b er conﬁgura - tion in R 4 ? Problem 1.8. So lve the Kissing Number P roblem in n dimensions for any n > 4 apart from n = 8 a nd 24. F o r pr esen tly known lower and upp er b ounds, we refer to [151] a nd [15 2], r espectively . Also a n y improvemen ts of these b ounds w ould b e desirable. Problem 1.9. (cf. [13 8 , Conj. 8.1]). V erify the conjecture that the Lee c h lattice is the unique densest sphere packing in R 24 . 1.5. Conclusions Over the last sixty years a s ubstan tial a moun t of resear c h ha s b een ins pir ed by the v arious interactions o f co ding theory and alge braic combinatorics. The fruitful int erplay often reveals the high degr ee of r egularity of b oth the co des a nd the combinatorial structures. This has lea d to a vivid area of r esearch connecting c losely mathematics with informa tion and co ding theory . The emerging metho ds ca n b e applied so metimes surprisingly eﬀectively , e.g., in view of the recent a dv ances on kissing num b ers a nd sphere packings. A further developmen t of this b eautiful interplay a s well as its application to concrete problems w ould b e desirable, certainly also in view of the v arious s till op en and lo ng-standing problems. 1.6. T erminolo g ies/Keyw ords Erro r -correcting co des, combinatorial desig ns , p erfect co des and related concepts, Assmus-Mattson Theore m and analogues, pr o jective geometries, non- e xistence of a pr o jective plane of order 10 , Golay co des, Leech lattice, kis sing num b ers, sphere packings, s pherical co des, asso ciation s c hemes . Co ding the ory and algebr aic co mbinatorics 25 1.7. Exercises (1) V erify (n umerically) that the Steiner quadruple sy s tem S QS (8) of o rder 8 (cf. E xample 1.5) ha s 14 blo cks, a nd that the Mathieu-Witt desig n 5-(2 4 , 8 , 1) (cf. E xample 1.7) has 75 9 blo cks. (2) What are the parameters of the 2-design consisting of the p oint s and hyp e r - planes (i.e. the ( d − 2)-dimensional pr o jective subspaces) of the pro jective space P G ( d − 1 , q )? (3) Do es there exist a self-dual [8 , 4] co de over the ﬁnite ﬁeld F 2 ? (4) Show that the ter na ry [11 , 6 , 5] Go la y co de has 13 2 co dewords of weigh t 5. (5) Co mpute the weigh t distribution o f the binary [23 , 12 , 7 ] Golay co de. (6) Show that any bina ry co de of 4096 co dewords, including the zero vector, of length 2 4 a nd minim um distance 8 is linear . (7) Give a pro of for the Sphere Packing Bound (cf. Theorem 1.1). (8) Give a pro of for Fisher’s Inequality (cf. Theor em 1 .3). (9) Show that a binary co de gene r ated by the rows of a n incidence matrix of any pro jectiv e plane P G (2 , n ) of even order n has dimens io n at mos t ( n 2 + n + 2) / 2 (cf. Theo rem 1.1 0). (10) (T o dd’s Lemma). In the Mathieu-Witt design 5- (2 4 , 8 , 1) , if B 1 and B 2 are blo c ks ( o ctads ) meeting in four p oin ts, then B 1 + B 2 is also a blo ck. Solutions: ad (1): By Le mma 1.2 (b), we hav e to calculate b = 8 · 7 · 6 4 · 3 · 2 = 14 in the ca se of the Steiner quadruple sys tem S QS (8), and b = 24 · 23 · 22 · 21 · 20 8 · 7 · 6 · 5 · 4 = 759 in the case of the Mathieu-Witt design 5-(24 , 8 , 1). ad (2): Starting from Example 1 .3, we o btain via counting arguments (or by using the trans itivit y prop erties o f the genera l linear group) that the po in ts and hyper planes of the pro jectiv e space P G ( d − 1 , q ) form a 2-( q d − 1 q − 1 , q d − 1 − 1 q − 1 , q d − 2 − 1 q − 1 ) design. ad (3): Y es, the extended binary [8 , 4 , 4 ] Hamming c o de is self-dual (cf. Exa mple 1.9). ad (4): Since the ternar y [11 , 6 , 5] Golay co de is p erfect (cf. Example 1.13), every word of w eight 3 in F 11 3 has distance 2 to a co deword o f weigh t 5 . Thus A 5 = 2 3 ·  11 3  /  5 2  = 132. ad (5): The binary [23 , 12 , 7 ] Golay co de contains the zer o vector a nd is p erfect. This determines the weigh t distribution as follows A 0 = A 23 = 1 , A 7 = A 16 = 253 , A 8 = A 15 = 506, A 11 = A 12 = 1288. ad (6): Let C denote a binar y co de of 4096 co dewords, including the zer o vector, o f length 2 4 and minimum dista nce 8. Deleting a n y co ordinate leads to a co de 26 Michael Hub er which has the same weight distribution as the co de given in Exer cise (5). Hence, the co de C only has c odewords of w eight 0 , 8 , 1 2 , 16 and 24 . This is still tr ue if the co de C is translated by any co dew ord (i.e. C + x for any x ∈ C ). Thus, the distances b etw een pa irs of co dew ords are als o 0 , 8 , 1 2 , 16 and 24. Therefo re, the standar d inner pro duct h x , y i v anishes for any tw o co dewords x , y ∈ C , and hence C is self-ortho g onal. F or cardinality reasons , we co nclude that C is self-dual and hence in particular linear. ad (7): The sum P e i =0  n i  ( q − 1) i counts the n um be r of words in a sphere of ra dius e . As the s pher es of ra dius e a bout distinct co dew o rds are disjoint, we obtain | C | · P e i =0  n i  ( q − 1) i words. Clearly , this n um b e r canno t excee d the total nu mber q n of words, and the cla im follows. ad (8): As a no n-trivial t -des ign with t ≥ 2 is also a non-tr iv ial 2-desig n by Le mma 1.1, it is suﬃcient to prove the assertio n for an arbitra ry non-tr ivial 2 -( v , k , λ ) design D . Let A b e an incidence matrix of D as deﬁned in Subse ction 1.3.1. Clearly , the ( i, k )-th entry ( AA t ) ik = b X j =1 ( A ) ij ( A t ) j k = b X j =1 a ij a kj of the ( v × v )- ma trix AA t is the to ta l num b er of blo cks co n taining b oth x i and x k , and is thus eq ua l to r if i = k , and to λ if i 6 = k . Hence AA t = ( r − λ ) I + λJ, where I denotes the ( v × v )-unit matrix a nd J the ( v × v )-matr ix with all entries equal to 1 . Using e le men tar y r o w and column op erations, it follows eas ily that det( AA t ) = rk ( r − λ ) v − 1 . Thu s AA t is non-singular (i.e. its determinant is non-z e ro) as r = λ would imply v = k by Lemma 1.1, yielding that the des ign is trivial. Therefore, the matrix AA t has r ank( A ) = v . B ut, if b < v , then rank( A ) ≤ b < v , and th us rank( AA t ) < v , a c o n tr adiction. It follows that b ≥ v , pr o ving the claim. ad (9): Let C deno te a binar y co de generated by the r o ws of an incidence ma trix of P G (2 , n ). By assumption n is even, and hence the extended co de C m ust be self-orthog onal. Therefo r e, the dimens ion o f C is at most n 2 + n + 2 / 2. ad (10 ): F o r given blo c ks B 1 = { 01 , 0 2 , 03 , 04 , 05 , 06 , 07 , 08 } and B 2 = { 0 1 , 02 , 03 , 04 , 09 , 10 , 11 , 1 2 } in the Mathieu-Witt design 5 -(24 , 8 , 1), let us ass ume that B 1 + B 2 is not a blo ck. The blo ck B 3 which cont ains { 05 , 06 , 0 7 , 08 , 09 } must contain just o ne mor e p oint of B 2 , s ay B 3 = { 0 5 , 06 , 07 , 08 , 09 , 10 , 13 , 14 } . Similarly , B 4 = { 05 , 0 6 , 07 , 08 , 11 , 12 , 15 , 16 } is the blo ck co n ta ining { 0 5 , 06 , 07 , 08 , 11 } . But hence, it is impos s ible to ﬁnd a blo ck whic h contains { 05 , 06 , 07 , 0 9 , 11 } and intersects with B i , 1 ≤ i ≤ 4, in 0, 2 or 4 p oints. There fore, w e obtain a contradiction as there must b e a blo ck containing any ﬁve p oints by Deﬁni- tion 1.2. Co ding the ory and algebr aic co mbinatorics 27 References [1] C. E. Shannon, A mathematical theory of comm unication, Bel l Syst. T e ch. J. 27 , 379–423 and 623–656, (1948). [2] R. W. H amming, Error detecting and error correcting codes, Bel l Syst. T e ch. J. 29 , 147–160 , (1950). [3] E. R. Berlek amp, Algeb r aic Co ding The ory . (McGra w-Hill, New Y ork, 1968; R evised edition: A egean Park Press 1984). [4] P . J. Cameron and J. H. v an Lint, Designs, Gr aphs, Co des and their Links . (Cam- bridge Univ. Press, Cam bridge, 1991). [5] R. Hill, A First Course in Co ding The ory . (Clarendon Press, Oxford, 1986). [6] W. C. Huﬀman and V. Pless, Eds., Handb o ok of Co di ng The ory . vol. I and I I, (North- Holland, Amsterdam, New Y ork, Ox ford, 1998). [7] W. C. Huﬀman and V. Pless, F undamentals of Err or-Corr e cting Co des . (Cambridge Univ. Press, Cam bridge, 2003). [8] A. Betten, M. Braun, H . F rip ertinger, A. Kerb er, A. Kohnert, and A. W assermann, Err or-Corr e cting Line ar Co des . (Springer, Berlin, Heidelb erg, N ew Y ork, 2006). [9] J. H . van Lint, Co des , I n ed s. R . L. Graham, M. Gr¨ otsc hel, and L. Lov´ asz, Hand- b o ok of Combinatorics , vol. I, pp . 773–807 . N orth-Holland, Amsterdam, New Y ork, Oxford, (1995). [10] J. H. v an Lint, Intr o duction to Co ding The ory . (S pringer, Berlin, Heidelb erg, New Y ork, 1999), 3rd edition. [11] W. W. P eterson and E. J. W eldon, Jr., Err or-Corr e cting Co des . (MIT Press, Cam- bridge, 1972), 2nd edition. [12] R. Roth, Intr o duction to Co ding The ory . (Cambridge Univ. Press, Cambridge, 2006). [13] F. J. MacWilliams and N . J. A. Sloane, The The ory of Err or-Corr e cting Co des . (North-Holland, Amsterdam, New Y ork, Ox ford, 1977; 12. impression 2006). [14] T. M. Thompson, F r om Err or-Corr e cting Co des thr ough Spher e Packings to Simpl e Gr oups . (Carus Math. Monograph 21 , 1983). [15] A. R. Calderbank, The art of signaling: ﬁ ft y years of cod ing theory , IEEE T r ans. Inform. The ory . 44 , 2561–2595, (1998). [16] F. J. MacWilliams , A theorem on th e distribution of w eigh ts in a sy stematic co de, Bel l Syst. T e ch. J. 42 , 79–94, (1963). [17] F. J. MacWilliams, N . J. A. Sloane, and J.-M. Go ethals, The MacWilliams identities for n on - linear co des, Bel l System T e ch. J. 51 , 803–819, (1972). [18] P . Delsarte, An algebraic approach to t he association schemes of co ding t heory , Philips R es. R ep orts Suppl. 10 , (1973). [19] T. Beth, D. Jun gnic kel, and H. Lenz, Design The ory . vol. I and I I, Encyclop e dia of Math. and Its Applic ations 69/78 , ( Cambridge Univ. Press, Cambridge, 1999). [20] P . J. Cameron, Par al lel i sms of Com pl ete Designs . (Cam b ridge U n iv. Press, Cam- bridge, 1976). [21] C. J. Colbou rn and J. H. Dinitz, Eds., Handb o ok of Combinatorial Designs . (CRC Press, Boca Raton, 2006), 2nd edition. [22] M. H all, Jr., Combi natorial The ory . (J. Wiley , New Y ork, 1986), 2nd edition. [23] D. R. Hughes and F. C. Piper, Design The ory . (Cam b ridge Univ. Press, Cam bridge, 1985). [24] D. R. Stinson, Combinatorial Designs: Construc tions and Analysis . (Sp ringer, Berlin, Heidelb erg, New Y ork, 2004). [25] P . D em b ows ki, Finite Ge ometries . (Sprin ger, Berlin, H eidelberg, New Y ork, 1968; Reprint 1997). 28 Michael Hub er [26] F. Buekenhout, Ed., Handb o ok of Incidenc e Ge ometry . (North- Holland, A msterdam, New Y ork, Oxford, 1995). [27] P . J. Cameron, Permutation Gr oups . (Cambridge Univ . Press, Cam bridge, 1999). [28] R. D. Carmichael , Intr o duction to the The ory of Gr oups of Finite O r der . (Ginn , Boston, 1937; Reprint: Dov er Publications, New Y ork, 1956). [29] J. D. Dixon and B. Mortimer, Permutation Gr oups . ( S pringer, Berlin, Heidelb erg, New Y ork, 1996). [30] H. Wielandt, Fini te Permutation Gr oups . (Academic Press, New Y ork, 1964). [31] V. D. T onchev, Combinatorial Conﬁgur ations: Designs, Co des, Gr aphs . (Longman, Harlo w, 1988). [32] D. Pei, Authentic ation Co des and Combinatorial Designs . (CRC Press, Bo ca R aton, 2006). [33] D. R. St in son, Combinatorial designs and crypto gr aphy , I n ed. K. W alker, Surveys in Com bi nator ics, 1993 , pp . 257–287 . Cambridge Univ . Press, Cambridge, (1993). [34] C. J. Colbourn , J. H . Dinitz, and D. R . Stinson, Applic ations of c ombinatorial designs to c ommunic ations, crypto gr aphy, and networking , In eds. J. D. Lamb and D. A. Preece, Surveys in C ombinatorics, 1999 , pp. 37–100. Cambridge Univ. Press, Cam bridge, (1999). [35] P . K aski and P . R. J. ¨ Osterg ˚ ard, C l assiﬁc ation Algorith ms for Co des and Designs . (Springer, Berlin, Heidelb erg, N ew Y ork, 2006). [36] P . J. Cameron, Combinatorics: T opics, T e chniques, Algorithms . (Cam bridge Univ . Press, Cam bridge, 1994; Reprint 1996). [37] J. H. v an Lint and R. M. Wilson, A Course in Combinatorics . (Cambridge Un iv. Press, Cam bridge, 2001), 2nd edition. [38] H. J. Ry ser, Ed., Combinatorial M athemat ics . (Math. Assoc. Amer., Buﬀalo, NY , 1963). [39] R. L. Graham, M. Gr¨ otschel, and L. Lo v´ asz, Eds., Handb o ok of Combinatorics . vol. I and I I, (North - Holland, Amsterdam, New Y ork, O x ford, 1995). [40] J. St einer, Combinatorisc he Aufgab e, J. R eine Ange w. Math. 45 , 181–182, (1853). [41] R. J. Wilson, The early h istory of b lock designs, Re nd. Sem. Mat. Messina Ser. I I . 9 , 267–276, (2003). [42] Y. J. Ionin and M. S. Shrik hande, Combi natorics of Symmetric Designs . (Cambridge Univ. Press, Cam bridge, 2006). [43] R. H. Bruck and H. J. Ryser, The non-ex istence of certain ﬁn ite pro jectiv e p lanes, Canad. J. Math. 1 , 88–93, (1949). [44] C. W. H. Lam, L. Thiel, and S. S wiercz, The n on-existence of ﬁnite pro jective planes of order 10, Canad. J. Math. 41 , 1117–112 3, (1989). [45] A. Beutelspac her, Pr oje ctive Planes , In ed. F. Buekenhout, Handb o ok of I ncidenc e Ge ometry , pp. 101–136. N orth-Holland, Amsterdam, New Y ork, Oxford, (1995). [46] J. W. P . Hirsc hfeld, Pr oje ctive Ge ometries over Finite Fields . (Clarendon Press, Oxford, 1998), 2nd edition. [47] D. R . Hughes and F. C. Pip er, Pr oje ctive Planes . (Springer, Berlin, Heidelb erg, New Y ork, 1982), 2nd edition. [48] H. L ¨ u n eburg, T r anslation Pl anes . (S p ringer, Berlin, Heidelb erg, New Y ork, 1980). [49] G. Pick ert, Pr ojektive Eb enen . (Springer, Berlin, H eidelberg, N ew Y ork, 1975), 2nd edition. [50] E. Mathieu, M´ emoire sur l’ ´ etude des fonctions de plu sieurs quantiti ´ es, J. Math. Pur es Appl. 6 , 241–323, (1861). [51] E. Mathieu, Sur la fonction cinq fois transitive de 24 quantit ´ es, J. Math. Pur es Appl. 18 , 25–46, (1873). Co ding the ory and algebr aic co mbinatorics 29 [52] E. Witt, Die 5-fach transitive n Grupp en von Mathieu, Abh. Math. Sem. Univ. Ham- bur g . 12 , 256–264, (1938). [53] E. Witt, ¨ Ub er Steinersche Systeme, Abh. Math. Sem. Univ. Hambur g . 12 , 265–275 , (1938). [54] H. L ¨ uneburg, T r ansitive Erweiterungen end licher Permutationsgrupp en . (Sp ringer, Berlin, Heidelb erg, New Y ork, 1969). [55] M. A schbac her, Sp or adic Gr oups . (Cambridge U niv. Press, Cambridge, 1994). [56] J. Tits, S u r les syst` emes d e Steiner asso ci ´ es aux trois “grands” group es de Mathieu, R endic. Math. 23 , 166–184, (1964). [57] H. L ¨ u n eburg, F ahnenhomogene Quadru pelsysteme, Math. Z. 89 , 82–90, (1965). [58] M. Hub er, Classiﬁcati on of ﬂ ag-tran sitive Steiner q uadruple sy stems, J. Combi n. The ory, Series A . 94 , 180–190, (2001). [59] M. Hub er, The classiﬁcation of ﬂag-transitive S teiner 3-designs, Adv . Ge om. 5 , 195–221 , (2005). [60] M. Hub er, The classiﬁcation of ﬂag-transitive Steiner 4-d esigns, J. Algebr. Comb. 26 , 183–207, (2007). [61] M. Hub er, A census of highly symmetric combinatorial d esig ns, J. Algebr. Comb. 26 , 453–476, (2007). [62] M. Hub er, Flag-tr ansitive Steiner Designs . (Birkh¨ auser, Basel, Berlin, Boston, to app ear). [63] L. T eirli nck, N on-trivial t -designs without rep eated blocks exist for all t , Discr ete Math. 65 , 301–311 , (1987). [64] R. A. Fisher, An ex amination of the diﬀerent p ossible solutions of a problem in incomplete blocks, Ann. Eugenics . 10 , 52–75, (1940). [65] E. F. Assmus, Jr. and H. F. Mattson, Jr., Coding and combinatori cs, SIAM R ev. 16 , 349–388, (1974). [66] E. F. Assm us, Jr. and J. D. Key , Designs and their Co des . ( Cam bridge Un iv. Press, Cam bridge, 1993). [67] E. F. Assmus, Jr. and J. D. K ey , D esigns and co des: an up date, Des. C o des Cryp- to gr aphy . 9 , 7–27, (1996). [68] I. F. Blake, Co des and designs, Math. Mag. 52 , 81–95, (1979). [69] J. H. v an Lint, Co des and designs , In ed. M. Aigner, Higher Combi nator ics: Pr o c. NA TO A dv. Study Inst. (Berlin) , pp. 241–256. R eidel, Dordrech t, Boston, (1977). [70] J. H. v an Lint, Co des and c ombinatorial designs , In eds. D. Jungnickel and S. A. V anstone, Pr o c. Marshal l Hal l c onfer enc e on c o ding the ory, design the ory, gr oup the ory (Burlington, VT) , pp. 31–39. J. Wiley , New Y ork, (1993). [71] V. D. T onchev, Co des and designs , In eds. W. C. Huﬀman and V. Pless, Handb o ok of Co ding The ory , vol. I I, pp. 1229–1267. N orth-Holland, Amsterdam, New Y ork, Oxford, (1998). [72] V. D. T onchev, C o des , In eds. C. J. Colbourn and J. H. Dinitz, Handb o ok of Com- binatorial Designs , p p . 677–702. CRC Press, Bo ca Raton, 2nd edition, (2006). [73] D. E. Muller, Application of b oolean algebra to switching circuit design and to error correction, IEEE T r ans. Com p. 3 , 6–12, (1954). [74] I. S . Reed, A class of multiple-error-correcting codes and th e decod ing sc h eme, IEEE T r ans. I nform. The ory . 4 , 38–49, (1954). [75] E. F. A ssm us, Jr. and H. F. Mattson, Jr., On tactical conﬁgurations and error- correcting co des, J. Combin. The ory, Series A . 2 , 243–257 , (1967). [76] M. J. E. Gola y , Notes on digital co ding, Pr o c. IRE . 37 , 657, (1949). [77] V. Pless, O n th e uniq ueness of th e Golay co des, J. Combi n. The ory, Series A . 5 , 215–228 , (1968). 30 Michael Hub er [78] A. Tiet¨ av¨ ainen, On the nonexistence of p erfect co des ov er ﬁ nite ﬁelds, SIAM J. Appl. M ath. 24 , 88–96, (1973). [79] J. H. van Lint, Nonexistenc e the or ems for p erfe ct err or c orr e cting c o des , In eds. G. Birkh oﬀ and M. Hall, Jr., Computers in Algebr aic Numb er The ory , vol. IV, pp. 89–95. SIAM-A MS Proc., Providence, R I, (1971). [80] V. A. Zinov’ev and V. K. Leont’ev, The nonex istence of p erfect co des ov er Galois ﬁelds, Pr obl. Contr ol and I nform. The ory . 2 , 123–132, (1973). [81] M. R. Best. A c ontribution to the nonexistenc e of p erfe ct c o des . PhD thesis, Univer- sit y of Amsterdam, (1982). [82] Y. Hong, On the n onexistence of unknown p erfect 6- and 8-co des in Hamming sc hemes H ( n, q ) with q arbitrary , Osaka J. Math. 21 , 687–700, (1984). [83] J. H. v an Lint, A survey of p erfect codes, R o cky Mountain J. Math. 5 , 189–224, (1975). [84] G. Cohen, I . H onk ala, S. Litsyn, and A. Lobstein, Covering Co des . (N orth-Holland, Amsterdam, New Y ork, Oxford, 1997). [85] A. M. Romanov, A survey of metho ds for constructing nonlinear p erfect binary codes, Di skr etn. Anal. Issle d. Op er. Ser. 1 . 13 , 60–88, (2006). [86] S. V. Av gustino v ic h, O. H eden, and F. I. Solov’ev a, The classiﬁcation of some p erfect codes, Des. C o des Crypto gr aphy . 31 , 313–318, (2004). [87] O. Heden and M. Hessler, O n the classiﬁcation of p erfect co des: side class structu res, Des. Co des Crypto gr aphy . 40 , 319–333, (2006). [88] K. T. Phelps, J. Rif` a, and M. Villanuev a, Kernels and p -k ernels of p r -ary 1-p erfect codes, Des. C o des Crypto gr aphy . 37 , 243–261, (2005). [89] P . K aski, P . R. J. ¨ Osterg ˚ ard, and O. Pottonen, The Steiner quadrup le sy stems of order 16, J. Combin. The ory, Series A . 113 , 1764–17 70, (2006). [90] P . R. J. ¨ Osterg ˚ ard and O. Pottonen, There exist Steiner triple systems of order 15 that do not o ccur in a p erfect binary one-error-correcting code, J. Combin. Des. 15 , 465–468 , (2007). [91] J.-M. Goethals and S. Snover, N early p erfect binary co des, Discr ete Math. 2 , 65–88, (1972). [92] N. V. Semako v , V. A . Zinov’ev, and G. V. Zaitsev, Uniformly pack ed co d es, Pr obl. Per e daci Inform. 7 , 38–50, ( 1971 ). [93] J.-M. Goethals and H. C. A. v an Tilb org, Uniformly pack ed co des, Philips R es. R ep orts . 30 , 9–36, (1975). [94] H. C. A. v an Tilb org. Unif ormly p acke d c o des . PhD thesis, T ec hnology Universi ty Eindhov en, (1976). [95] R. Ahlsw ede, H. K . Ay dinian, and L. H. Khachatrian, On p erfect co des and related concepts, Des. Co des Crypto gr aphy . 22 , 221–237, (2001). [96] M. Schw artz and T. Etzion, Codes and anticodes in the Grassman graph, J. Combi n. The ory, Series A . 97 , 27–42, (2002). [97] V. Pless, The weigh t of the sy mmetry co de for p = 29 an d th e 5-designs obt ained therein, Ann. New Y ork A c ad. Sci. 175 , 310–313 , (1970). [98] V. Pless, Sy mmetry co des o ver GF (3) and n ew ﬁve-designs, J. Combin. The ory, Series A . 12 , 119–142, (1972). [99] E. F. Assm us, Jr. and H . F. Mattson, Jr., New 5-designs, J. C ombin. The ory, Series A . 6 , 122–151 , (1969). [100] F. J. MacWilliams, A. M. Odlyzko, N. J. A. Sloane, and H . N. W ard, Self-dual co des o ver GF (4), J. Combin. The ory, Series A . 25 , 288–318, (1978). [101] L. J. Paige, A note on t h e Mathieu groups, Canad. J. Math. 9 , 15–18, (1956). [102] M. Harada, New 5-d esigns constructed from the lifted Gola y co de ov er Z 4 , J. C om - Co ding the ory and algebr aic co mbinatorics 31 bin. D es. 6 , 225–229, ( 1998 ). [103] T. A . Gulliver and M. H arada, Ext remal d ouble circulant t yp e I I codes o ver Z 4 and construction of 5-(24 , 10 , 36) designs, Discr ete Math. 194 , 129–137 , (1999). [104] A. Bonnecaze, P . S ol ´ e, and A. R. Calderbank, Quaternary qu adratic residue codes and u nimodular lattices, IEEE T r ans. Inform. The ory . 41 , 366–377 , (1995). [105] P . D elsarte, F our fundamental parameters of a co de and their com binatorial signiﬁ- cance, I nformation and Contr ol . 23 , 407–438, (1973). [106] K. T anab e, A criterion for designs in Z 4 -cod es on the symmetrized weigh t enumer- ator, Des. C o des Crypto gr aphy . 30 , 169–185 , (2003). [107] A. R. Calderbank, P . Delsarte, and N. J. A. Sloane, A strengthening of the Assmus- Mattson t heorem, IEEE T r ans. Inf orm. T he ory . 37 , 1261–126 8, (1991). [108] G. T. Kennedy and V. Pless, On designs and formally self-dual co des, Des. Co des Crypto gr aphy . 4 , 43–55, (1994). [109] J. Simonis, MacWilliams identities and co ordinate partitions, Line ar Algebr a Appl. 216 , 81–91, (1995). [110] C. Bac hoc, On harmonic weigh t enumerators of b inary co des, Des. Co des Crypto g- r aphy . 18 , 11–28, (1999). [111] J.-L. Kim and V . Pless, Designs in ad d itiv e cod es ov er GF (4), Des. Co des Crypto g- r aphy . 30 , 187–199, (2003). [112] D.-J. Shin, P . V. Ku mar, and T. Helleseth, An Assmus-Ma ttson-type approac h for identif ying 3-designs from linear codes ove r Z 4 , Des. Co des Crypto gr aphy . 31 , 75–92, (2004). [113] T. Britz and K. Shiromoto, Designs from sub code supp orts of linear cod es, Des. Co des Crypto gr aphy . 46 , 175–189, (2008). [114] F. J. MacWilliams, N. J. A. Sloane, and J. G. Thompson, On th e existence of a pro jective plane of order 10, J. Combin. The ory, Series A . 14 , 66–78, (1973). [115] C. W. H . Lam, The search for a ﬁ nite pro jective plane of order 10, Amer . Math. Monthly . 98 , 305–318, (1991). [116] P . J. Cameron, Finite Ge ometries , In eds. R. L. Graham, M. Gr¨ otsc hel, and L. Lo v´ asz, Handb o ok of Combinatorics , vol . I, pp. 647–692. North- Holland, Ams- terdam, New Y ork, Oxford, (1995). [117] E. F. Assmus, Jr. and J. D. Key , Polynomi al c o des and ﬁnite ge ometries , In eds. W. C. Hu ﬀman and V. Pless, Handb o ok of Co ding The ory , vol. I I, pp. 1269–13 43. North-Holland, Amsterdam, New Y ork, Ox ford, (1998). [118] J. A. Thas, Finite geometries, vari eties and co des, Do c. Math. J. DMV, Extra V ol. ICM I II. pp. 397–408, (1998). [119] L. Storme. Pro jective geometry and co ding theory , COM 2 MAC Lect. Note Series 9 . Combin. and Comput. Math., Center Pohang U niv. of Science and Technology , (2003). [120] C. Hering. On co des and pro jective designs. T echnical R eport 344, Kyoto Univ., Math. Researc h Inst. Seminar Notes, (1979). [121] P . J. Cameron. Finite geometry and co ding t h eory , So crates Course Notes, (1999). Published electronically at http://dwispc8. vub.ac.be/ Potenza/lectnotes.html . [122] T. Beth and D. Jungnick el, Mathieu gr oups, Witt designs, and Golay c o des , I n ed s. M. Aigner and D. Jungnick el, Ge ometries and Gr oups , pp. 157–179. Springer, Berlin, Heidelb erg, New Y ork, (1981). [123] J. H. Con w a y and N. J. A. Sloane, Spher e Packings, L attic es and Gr oups . (Springer, Berlin, Heidelb erg, New Y ork, 1998), 3rd edition. [124] J. H. Con w a y and N. J. A. Sloane, Lexicographic co des: error-correcting cod es from game th eory , IEEE T r ans. Inform. The ory . 32 , 337–348 , (1986). 32 Michael Hub er [125] R. T. Curtis, A new com b inatorial approac h to M 24 , M ath. Pr o c. Cambridge Philos. So c. 79 , 25–42, (1976). [126] E. F. A ssm us, Jr. an d H. F. Mattson, Jr., P erfect cod es and th e Mathieu groups, Ar ch. Math. 17 , 121–135, (1966). [127] N. J. A. S loane, The Sphere Packing Problem, Do c. Math. J. DMV, Ext ra V ol. ICM I II. pp. 387–396, (1998). [128] J. Leech, Some sphere packings in higher space, Canad. J. Math. 16 , 657–682, (1964). [129] J. H. Con w a y and N. J. A. Sloane, Tw enty-three constructions for the Leech lattice, Pr o c. R oy. So c. L ond., Series A . 381 , 275–283, ( 1982 ). [130] K. Sch ¨ utte and B. L. v an der W aerden, Das Problem der dreizehn Ku geln, Math. Ann . 125 , 325–334, (1953). [131] P . Delsarte, Bounds for unrestricted co des, by linear programming, Philips R es. R ep orts . 27 , 272–289, (1972). [132] P . Delsarte, J.-M. Go ethals, and J. Seidel, Sph erical cod es and designs, Ge om. De d- ic ata . 6 , 363–388, (1977). [133] A. M. Od lyzk o and N . J. A. S loane, New b ounds on the num b er of unit spheres that can touch a unit sphere in n dimensions, J. Combin. The ory, Series A . 26 , 210–21 4, (1979). [134] V. I. Levensh tein, On b ounds for packings in n - dimensional Euclidean space, Sov. Math. Doklady . 20 , 417–421, (1979). [135] O. R. Musin, The k issing number in four dimensions, Ann. Math. (to app ear). Preprint published electronically at http://arxiv .org/abs/ma th.MG/0309430 . [136] H. Cohn and A. Kumar, The densest lattice in tw enty-four dimensions, El e ctr on. R es. Anno unc. Amer. Math. So c. 10 , 58–67, (2004). [137] H. Cohn and A. Kumar, Optimalit y and uniqueness of the Leec h lattice among lattices, A nn. Math. (to app ear). Preprint published electronically at http://arx iv.org/abs /math.MG/0403263 . [138] H. Cohn and N. D. Elkies, New upp er b ounds on sphere packings, A nn. Math. 157 , 689–714 , (2003). [139] O. R . Musin. An ext ension of Delsarte’s meth od. The kissing problem in three and four dimensions. In Pr o c. COE Workshop on Spher e Packings (Kyushu Univ. 2004) , pp. 1–25, (2005). [140] N. D. Elkies, Lattices, linear codes, and inv ariants, Notic es Amer. Math. So c. 47 , 1238–12 45 and 1382–139 1, (2000). [141] F. Pfender and G. M. Ziegler, Kissing num b ers, sphere packings and some unex- p ected pro ofs, Notic es Amer. Math. So c. 51 , 873–883, (2004). [142] K. Bezdek, Sph ere packings revisited, Eur op. J. Comb. 27 , 864–883 , (2006). [143] G. N ebe and N. J. A. Sloane. A catalogue of lattices. Pub lished electronically at http://www .research. att.com/ ~ njas/latti ces/ . [144] R. C. Bose and T. Shimamoto, Classiﬁcation and analysis of partially balanced incomplete blo c k designs with tw o asso ciate classes, J. A mer. Statist. Asso c. 47 , 151–184 , (1952). [145] R. C. Bose and D. M. Mesner, On linear associative algebras corresp on d ing to asso- ciation schemes of partially balanced designs, Ann. Math. Statist . 30 , 21–38, (1959). [146] E. Bannai and T. I to, Algeb r aic Combinatorics I: Asso ciation Schemes . (Benjamin, New Y ork, 1984). [147] A. E. Brouw er, A. M. Cohen, and A. Neumaier, Distanc e-R e gular Gr aphs . (Springer, Berlin, Heidelb erg, New Y ork, 1989). [148] A. E. Brouw er and W. H. Haemers, Asso ci ation Schemes , In eds. R. L. Graham, Co ding the ory and algebr aic co mbinatorics 33 M. Gr¨ otsc hel, and L. Lov´ asz, Handb o ok of Combinatorics , vol. I, pp. 747–771. North- Holland, Amsterdam, New Y ork, Ox ford, (1995). [149] P . Camion, Co des and asso ciation schemes: Basic pr op erties of asso ciation schemes r elevant to c o di ng , In ed s. W. C. Huﬀman and V . Pless, Handb o ok of Co ding The ory , vol . I I, pp. 1441–1567 . North-Holland, Amsterdam, New Y ork, Ox ford, (1998). [150] P . Delsarte and V. Levensthein, Asso ciation sc hemes and co ding theory , IEEE T r ans. Inform. The ory . 44 , 2477–2504, (1998). [151] G. Neb e and N. J. A. Sloane. T able of the highest kissing num b ers presently kn o wn. Published electronically at http://www.rese arch.att.c om/ ~ njas/latti ces/ . [152] C. Ba choc and F. V allen tin, New upp er b ounds for kissing num b ers from semideﬁnite programming. Preprin t published electronically at http://arxiv. org/abs/ma th.MG/0608426 . 34 Michael Hub er Index [ n, k ] co de, 2 t -design, 5 aﬃne geometry , 8 aﬃne plane, 8 aﬃne space, 8 algebraic com binatorics, 1 association scheme, 22 automorphism, 6 automorphism group, 6 ball, 3 binary Gola y co de, 14 block, 5 Bruck-Ryser Theorem, 8 classiﬁcation algorithm, 5 codin g t heory , 1, 5 collinear, 9 com binatorial design th eory , 5 com binatorics, 1 cryptography , 5 cub e, 9 cyclic code, 4 derived design, 6 design, 5 diameter p erfect co de, 15 dod ecad, 17 dual co d e, 3 error-correcting co d e, 3 extended co de, 4 extension, 6 F ano plane, 7 ﬁnite geometries, 5 Fisher’s Inequality , 11 full aut omorph ism group , 6 generator matrix, 3 graph th eory , 5 group th eory , 5 Hamming Bound, 3 Hamming cod e, 14 Hamming distance, 2 Hamming scheme, 23 hexacod e, 20 hyperov al, 19 incidence geometry , 5 incidence matrix, 6 information theory , 21 isomorphism, 6 Johnson sc heme, 23 Kissing N um b er Problem, 21 lattice, 21 lattice p ac king, 22 Leec h lattice, 21 linear co de, 2 Mathieu groups, 9 Mathieu-Witt d esigns, 9 minim um distance, 3 minim um w eigh t, 3 Miracle O ctad Generator, 21 nearly p erfect co de, 15 non-trivial d esign, 5 octad, 25 o v al, 19 o veral l parity chec k symbol, 4 P aley matrix, 16 35 36 Index parity c heck matrix, 3 p erfect co de, 3 p erm u tation equival ent, 3 Pless symmetry cod e, 16 p oin t, 5 pro jective geometry , 8 pro jective plane, 5, 8 pro jective space, 8 punctured cod e, 4 QR cod e, 16 quadrangle, 5 quadratic residue co de, 16 Reed-Muller (RM) co de, 13 regular tetrahedron, 9 self-dual co de, 3 self-orthogonal co de, 3 Shannon’s Sampling Theorem, 21 sphere, 3 sphere packing b ound, 3 Sphere Pa cking Problem, 22 spherical co de, 22 spherical design, 22 sp oradic simple groups, 10 square design, 6 Square Ro ot Bound, 16 standard form, 3 statistical design of exp erimen ts, 10 Steiner t -design, 6 Steiner d esign, 6 Steiner q uadruple system, 6 Steiner sy stem, 6 Steiner t rip le sy stem, 6, 7 supp ort, 11 symmetric design, 6 ternary Golay co de, 14 trivial d esign, 5 uniformly pack ed co des, 15 w eigh t, 2 w eigh t distribution, 4 w eigh t enumerator, 4 w ord, 2

Coding Theory and Algebraic Combinatorics

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment