The Perfect Binary One-Error-Correcting Codes of Length 15: Part I--Classification

1 The Perfect Binary One-Error -Correcti ng Codes of Length 15 : Par t I—Classificati on Patric R. J. ¨ Osterg ˚ ard, Olli Pottonen Abstract A complete class ification o f the perfect binary one-er ror-correcting codes of length 15 as well as their e xten sions of length 16 is presented. There are 5 983 such inequiv alent perfect codes and 2 165 e xte nded perfect codes. Efficient generation of these codes relies on the recent classification of Stein er qu adrup le systems of order 16 . Utilizin g a result o f Blackmore, the optimal b inary o ne-err or-correcting codes of leng th 14 and the (15 , 1 02 4 , 4) code s are also classified; there are 38 408 and 5 9 83 such codes, respectively . Index T erms classification, Hamming code, perfect binary code, Steiner system I . I N T RO D U C T I O N Consider the space F n 2 of dimensi on n over t he Galois field F 2 = { 0 , 1 } . A binary code of length n is a subs et of F n 2 . The (Hamming) dist ance d ( x , y ) between two codew ords x , y is the numb er of coordinates in which they diffe r , and the (Hamming) weight wt( x ) is the num ber of nonzero coordinates. The support of a codew ord is the set of non zero coordinates, supp( x ) = { i : x i 6 = 0 } . Accordingly , d ( x , y ) = wt( x − y ) = | supp( x − y ) | . A code has minimum dista n ce d if d is the lar gest int eger such that the d istance between any dis- tinct cod e words is at least d . Then t he balls of radiu s ⌊ ( d − 1) / 2 ⌋ centered around the code words are nonintersecting, and the code is said to be a ⌊ ( d − 1) / 2 ⌋ -error -correcting code. If these ball s tile th e whole sp ace, t hen the code is called perfect . The parameters of perfect codes over an alphabet of prime order are well known [1], and perfect binary codes exist with d = 1 ; d = n ; d = ( n − 1) / 2 for odd n ; d = 3 , n = 2 m − 1 for m ≥ 2 ; and d = 7 , n = 23 . The first t hree types of codes are called trivial, the fourth has the p arameters of Hamm ing codes, and the last one is the binary Golay code. A perfect code with min imum d istance d is also called a ⌊ ( d − 1) / 2 ⌋ -perfect code. A binary code with l ength n , m inimum distance d , and M codewords is called a ( n, M , d ) code. In this notation a binary 1 -perfect code is a (2 m − 1 , 2 2 m − m − 1 , 3) code. T wo related families are the extended and shortened 1 -perfect codes, which have parameters (2 m , 2 2 m − m − 1 , 4) and (2 m − 2 , 2 2 m − m − 2 , 3) , respectiv ely . Existence of b inary 1 -perfect codes foll ows from the existence of Hamm ing codes, whi ch are the uniq ue linear 1 -perfect codes. Still constructi ng all 1 -perfect codes i s a long standing open problem. It makes sense to approach this issue by consid ering the number of in equ i valent codes (or more formally the numb er of equiv alence classes). T wo codes are said to be equiv alent if one is obtai ned from the other by permuting coordinates and adding a constant vector; a formal definition appears in Section II. There is trivially a uni que 1 -perfect code of length 3 . Zaremba [2] showed that also th e 1 -perfect code of leng th 7 i s u nique. Howe ver , already the next case of length 15 has until this work wit hstood all attempts of compl ete classification, although s e veral constructions of such codes ha ve been published; see the surve ys [3], [4]. It turns out that these result s were not far from a complete classi fication as for the number of codes found. The growth of th e number of 1 -perfect bi nary cod es is doub le exponential in th e This work was supported in part by the Graduate School in Electronics, T elecommunication and Automation and by the Academy of Finland, Grant Numbers 107493 and 110196. P . R. J. ¨ Osterg ˚ ard and O. Pottonen are with the Department of Communications and Net working, Helsinki Univ ersity of T echnology TKK, P .O.Box 3000, F I-02015 T KK, F inland (e-mail: patric.ostergard@tkk.fi, olli.pottonen@tkk.fi) 2 length of the code, see [5] for a lower bound on this number . For an in-depth t reatment of the topic of classifying comb inatorial objects, see [6]. The aim of t he current work is to obtain a com plete classification of inequ iv alent 1 -perfect binary codes of length 15 . By comp uter s earch it is here shown t hat t heir number is 5 983 . Al so the codes obtained b y extending, shortening or extending and shortening are classified; the nu mbers of (1 6 , 2 048 , 4) , (14 , 1 024 , 3) and ( 15 , 1 024 , 4) codes t urn out to be 2 165 , 3 8 408 and 5 983 respectively . In the rest of the p aper we document the classification of the extended 1-perfect codes of length 16 , which yields cl assifications of the 1-perfect codes of length 15 and the shortened 1-perfect codes of length 14 . In Section II we define som e concepts and consider construction of extended 1-perfect codes via Steiner s ystems. In Section III we present algorithms for d etecting and rejecting equiv alent codes, and in Section IV, we take a brief l ook at the results; a separate, more d etailed st udy of the classi fied codes wil l appear in a separate paper [7]. Finally , in Section V we giv e a consi stency check for gaining confidence in the comput ational results. I I . P R E L I M I N A R I E S A N D C O N S T RU C T I O N A permutation π of the set { 1 , 2 , . . . , n } acts on code words by permuting the coordinates: π (( c 1 , c 2 , . . . , c n )) = ( c π − 1 (1) , c π − 1 (2) , . . . , c π − 1 ( n ) ) . P airs ( π , x ) form the wr eat h product S 2 ≀ S n , which acts on codes as ( π, x )( C ) = π ( C + x ) = π ( C ) + π ( x ) . T w o codes, C 1 and C 2 , are isomorp h ic if C 1 = π ( C 2 ) for some π and equivalent if C 1 = π ( C 2 + x ) for some π , x . The auto morphism g roup o f a code C , Aut( C ) , is the group of all pairs ( π , x ) such that C = π ( C + x ) . T w o important subg roups of Aut( C ) are the gr oup of symmetri es , Sym( C ) = { π : π ( C ) = C } and the kernel Ker( C ) = { x : C + x = C } . If the code cont ains the all-zero w ord, 0 , then the elements of the kernel are code words. A Steiner system S ( t, k , v ) can be viewed as a code S ⊂ F v 2 with t he p roperty that each codew ord o f S has weight k , and for any y ∈ F v 2 with wt( y ) = t , there is a un ique x ∈ S such that supp( y ) ⊆ supp( x ) . Usually Steiner syst ems are defined as set systems rather than codes, b ut our definit ion i s more directly applicable for t his work. The parameter v is the or der o f the sys tem. Steiner systems S (2 , 3 , v ) and S (3 , 4 , v ) , which are called Steiner t riple systems and Steiner quadr u ple systems , respectively , are related to 1-perfect codes in t he following way . If C is a 1-perfe ct binary code of length v and x ∈ C , then th e code words of C + x with weight 3 form a Stein er triple system of order v . Similarly , i f C is an extended 1-perfect bi nary code and x ∈ C , then t he code words of C + x with weight 4 form a Steiner quadruple system. These s ystems are, respectively , the neighborhood tri ple system and nei g hborhood quadruple system associated with t he code and the codew ord. As a starting point for th e classification of the e xtend ed 1-perfect binary codes of lengt h 16, we hav e the classification [8] of Steiner quadrupl e sy stems o f o rder 16; th ere are 1 05 4 1 63 such desi gns. W e want to find, for each S (3 , 4 , 16) , all extended 1-perfect binary cod es in which it occurs. This can be done by pu ncturing any coordinate, aug menting t he result ing code to 1-perfect codes in all possi ble ways, and finally extending eve ry result ing code with a parity bit . When augmenting a set of code words to a 1-perfect code, we con sider a 1-perfect code as a set of ball s with radius one that form a partition of the ambient s pace. Accordingly , finding a code (with specified code words) is a special case of the exact cover pr oblem , where we are give n a set S and a collection U of its subs ets, and the task is to form a partition of S by using sets in U . Let the set Q cont ain the code words obtai ned by puncturing the all-zero code word and its neig hborhood quadruple system. In this case we h a ve S = F 15 2 \ B ( Q ) , and U = { B ( x ) : B ( x ) ∩ B ( Q ) = ∅} , where B ( x ) = { y : d ( x , y ) ≤ 1 } and B ( C ) = { B ( x ) : x ∈ C } . W e use the libexact software [9] for solv ing such instances of the exact cove r problems. In th e search we could i n fact have m ade use of the f act that all 1-perfect binary codes 3 are self-comp lementary—in ot her words, th e all-one word is alw ays in the kernel—but this w oul d not hav e had an y practical significance as the search was rather fast. Since t he all-zero word and its neighborho od quadruple system contain 141 of the 2 048 codewords, 1907 new codew ords are needed. Searching for these was a remarkably easy computational task; on a verage the search trees i n which codes were found had 1978 nodes and those in wh ich no codes were found had 3 n odes. I I I . I S O M O R P H R E J E C T I O N The general framework by McKay [10] was used to carry out i somorph rejection, altho ugh a less sophisticated method w ould ha ve sufficed in this w ork. Recall th at we augment a Steiner q uadruple system Q to a n e xtend ed 1-perfect code C . W e accept C if it passes the following two tests; otherwis e i t is rejected. First we require that C shall be the mini mum (with respect to s ome practically computable total order of cod es) u nder the action of Aut( Q ) . Second, we compute the canonical equiv alence class representativ e c E ( C ) , consider π , x for which π ( C + x ) = c E ( C ) and require that x and 0 a re on the same Au t( C ) orbi t (we define c E so that x ∈ C always hol ds). When the e xtended 1-perf ect codes have been classified, classifyi ng the 1-perfe ct codes is straightfor- ward. A ll 1-perfect codes are obtained by puncturing the extended codes, and the resulting 1-perfect codes are equiv alent if and only if they are obtained by puncturin g the same extended code at coordinates which are in the same orbit o f the automorphism group. A compl ete classification of the (14 , 1 024 , 3 ) codes is obtained sim ilarly , since each such code is obtained by shortening a uni que (up to equivalence) 1-perfect code of lengt h 15 ; this resul t w as proved by Blackmore [11]. Although a code can be shortened at any coordinate in two ways, by selecting the code words w ith 0 or 1 in a certain coo rdinate, both selections lead to equivalent codes. T his follows from the fact that e very 1-perfect binary code is self-complementary . Furthermore we note that any (15 , 1 024 , 4) code is obt ained by e xtending a (14 , 1 024 , 3 ) code with a parity b it. Hence all such codes are obt ained by shortening and extending a perfect cod e, or equiv alently removing all words of chosen parity . As th e perfect cod es are self-complem entary , we get (up t o equiva- lence) same code by chosing either odd o r e ven par it y . As this mapping is reversible, we conclude that there is on e-to-one correspondence between equivalence classes of (15 , 2 048 , 3) codes and equiv alence classes of (1 5 , 1 024 , 4) codes, and in both cases their number is 5 983 . Let C be a (15 , 1 02 4 , 4 ) code C and let C ′ be the corresponding perfect code. The g roup Aut( C ′ ) contains Aut( C ) as a subgroup, and Aut( C ′ ) has one more generator than Aut( C ) , namely the all-one code word. Accordingly | Aut( C ′ ) | = 2 | Aut( C ) | . W e still hav e to describe an alg orithm for canonical labeling. The most s traightforward approach o f using the general purpose isomorph ism nauty [12] is rather slow on codes as lar ge and regular as the (16 , 2 048 , 4) codes; t his was also noted by Phelps [13]. Hence a tailored approach is necessary . The method presented below has a lot in common with the one described in [13]. An alternative m ethod based on minim um distance graphs w oul d also work [14], cf. [15]. A triangle consists of 3 code words with mutual distance 4 . T riangles constitute an easily computable and rath er sensi tive in variant of Steiner quadrupl e system s. Distingu ishing the isomorphis m class es of the neighborhood qu adruple sys tems of a code also constitu tes an in variant of the extended 1-perfec t codes. These two in variants turned out to be useful for speeding up our computations. A canonical isomorphism class representative c I ( C ) for a code C can be computed by us ing nauty to label a correspondi ng graph canonically . Moreover , nau ty comp utes generators of the group Sym( C ) . Canonical equiv alence class representati ve can be defined as c E ( C ) = min { c I ( C + x ) : x ∈ C } , where the mi nimum is again taken wi th respect to some practically com putable total order of codes. Note that t wo codew ords, x and y , are in the same orbi t of Aut ( C ) if and only if c I ( C + x ) = c I ( C + y ) . Because of th is, if we know that x and y are in the sam e orbit, and c I ( C + x ) h as been computed, then there is no need to comp ute c I ( C + y ) . A lso if c I ( C + x ) = c I ( C ) , then nauty yields a permutation π 4 such that π ( C + x ) = C . Th e pairs ( π , x ) are coset representativ es of Aut( C ) with respect to Sym( C ) , so we get generators of t he group Aut( C ) . I V . R E S U L T S There are exactly 2 165 inequiv alent extended 1-perfect codes o f length 16 , 5 98 3 inequivalent 1-perfect codes of lengt h 15 , 38 4 08 shortened 1-perfect codes of length 14 and 5 983 (15 , 1 024 , 4) codes. The orders of t he automo rphism groups of the codes are p resented in T ables I, II and III. As noted in Section III, the order of the aut omorphism group of a (15 , 1 024 , 4) code is half of the order of the autom orphism group of the correspond ing perfect code. The codes hav e been made ava ilabl e in electroni c form by i ncluding them in the so urce of the arXiv ver - sion of this paper . Downloading http: //arxiv.o rg/e- print/0806.2513v3 and uncomp ressing it with g unzip and tar yields the files perfect15 and ex tended16 . Only 15 590 of the 1 054 163 nonis omorphic S (3 , 4 , 16) can be augmented to a 1-perfect code, and the total number of extensions is 22 814 . The computati onally intensiv e part of this result was the earlier classification of S (3 , 4 , 16) , which requi red severa l y ears of CPU time, while all searches describ ed in this paper took onl y a coupl e of hou rs of CPU tim e. A detailed study o f t he properties of t he classified codes will appear in a second part of t his arti cle [7]. T ABLE I A U T O M O R PH I S M G RO U P S O F (16 , 2 048 , 4) C O D E S | Aut( C ) | # | Aut( C ) | # | Aut( C ) | # 128 11 5 376 1 196 608 6 192 5 6 144 23 262 144 3 256 105 8 192 174 344 064 1 384 9 10 752 2 393 216 3 512 377 12 288 22 524 288 2 672 2 16 384 103 688 128 1 768 19 2 4 576 12 786 432 2 1 024 416 32 768 47 1 572 864 3 1 344 1 43 008 2 2 359 296 1 1 536 21 49 152 18 2 752 512 1 1 920 1 61 440 1 3 145 728 1 2 048 394 65 536 33 5 505 024 2 2 688 1 86 016 3 6 291 456 1 3 072 18 98 304 12 660 602 880 1 4 096 298 131 072 6 V . C O N S I S T E N C Y C H E C K T o g et confidence in t he result s, we performed a consi stency check si milar t o the one used, for example, in [8]. In this check we count the total number of codes in t wo different ways and ensure that the resul ts agree. First we consider the set C of equiv alence class representatives obtained in the classification. By t he orbit-stabilizer theorem, the total number of extended 1-perfect cod es is X C ∈C 16! · 2 16 Aut( C ) , where 16! · 2 16 is t he order of the wreath product group acting on t he codes. Let Q consist of t he representative Steiner q uadruple systems, and let E ( Q ) be the number of all extended 1-perfect codes obtain ed by augmentin g Q . Applying the orbit-stabilizer t heorem, we get t he expression 1 2 0 48 X Q ∈Q 16! · 2 16 · E ( Q ) Aut( Q ) , 5 T ABLE II A U T O M O R PH I S M G RO U P S O F (15 , 2 048 , 3) C O D E S | Aut( C ) | # | Aut( C ) | # | Aut( C ) | # 8 3 512 1 017 24 576 7 12 3 672 3 32 768 8 16 5 768 32 43 008 4 24 10 1 024 697 49 152 10 32 138 1 536 17 65 536 5 42 2 2 048 406 98 304 1 48 12 2 688 1 131 072 1 64 542 3 072 37 172 032 1 96 22 3 840 1 196 608 5 120 1 4 096 202 344 064 2 128 1 230 5 376 4 393 216 2 192 18 6 144 35 589 824 1 256 1 319 8 192 94 41 287 680 1 336 3 12 288 7 384 30 16 384 44 T ABLE III A U T O M O R PH I S M G RO U P S O F (14 , 1 024 , 3) C O D E S | Aut( C ) | # | Aut( C ) | # | Aut( C ) | # 1 5 168 1 8 192 80 2 75 192 80 12 288 18 3 8 256 4 392 16 384 14 4 425 336 5 21 504 1 6 39 384 114 24 576 15 8 1 162 512 2 469 32 768 14 12 56 768 30 49 152 1 16 3 465 1 024 1 346 65 536 1 21 4 1 344 1 86 016 1 24 39 1 536 54 98 304 2 32 7 311 2 048 527 172 032 1 48 59 2 688 6 196 608 2 64 9 068 3 072 55 1 376 256 1 96 49 4 096 222 128 7 172 6 144 18 where the division by 2 048 is necessary since each code is counted once for each code word. Both formulas yield the same re sul t, 2 795 493 027 033 907 200 . Sim ilarly we also counted t he 1-perfe ct codes and short ened 1-perfect cod es in two diffe rent ways; their number is 1 397 746 513 516 9 53 600 . Indeed, there are tw ice as many extended 1-perfect codes as there are 1-perfec t codes, si nce each 1-perfect code admits two extensions: one with even parity bi t and one wit h od d. Sim ilarly , we get a bijectio n from the 1-perfect codes t o shortened 1 -perfect codes if we s horten each code by t aking, for i nstance, the cod e words with value 0 in coordinate 15 and removing that coordinate. Thus there are equall y many 1-perfect codes and short ened 1-perfect codes. R E F E R E N C E S [1] F . J. MacW illiams and N. J. A. Sloane, The Theory of Err or-Corr ecting Codes . Amsterdam: North-Holland, 1977. [2] S. K. Zaremba, “Cov ering problems concerning Abelian groups, ” J. L ondon Math. Soc. , vol. 27, pp. 242–246, 1952. [3] T . E tzion and A. 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