Bounds on Codes Based on Graph Theory

Bounds on Codes Based on Graph Th eory Salim Y . El Rouayh eb ECE Departmen t T exas A&M University College Station, TX 77 843 salim@ece.tamu.ed u Costas N. Georghiades ECE Depar tment T exas A&M University College Station, TX 77 843 georghiade s@ece.tamu.edu Emina Soljanin Math. Sc. Cen ter Bell Lab s, Lucent Murray Hill, NJ 07974 emina@lucent.c om Alex Sprintson ECE Depar tment T exas A&M University College Station, TX 77 843 spalex@ece.tamu.edu Abstract — Let A q ( n, d ) be the maximum order (maximum number of codewords) of a q -ary code of length n and Hamming distance at least d . And let A ( n, d, w ) that of a b inary code of constant weight w . Buildin g on results from alg ebraic graph the- ory and Erd ˝ os-ko-Rado like theor ems in extr emal combin atorics, we show how sev eral known bounds on A q ( n, d ) and A ( n, d, w ) can be easily obtained i n a sin gle framew ork. For instance, both the Hamming and Singleton bounds can deriv ed as an application of a p roperty relating the cliq ue number and th e indep endence number of v ertex transitive graphs. Using the same t echniques, we also derive some new boun ds and present some ad ditional applications. I . I N T RO D U C T I O N Let Σ = { 0 , 1 , . . . , q − 1 } be an alph abet of order q . A q -ar y code C of length n and order | C | is a subset of Σ n containing | C | elements ( codewords). The we ight w t ( c ) o f a c odeword c is th e n umber o f its non -zero entries. A w constant weig ht code is a code wher e all the codewords have the same weight w . The Hamming d istance d ( c, c ′ ) between two codew ords c and c ′ is the numb er o f positions where they ha ve d ifferent entries. The m inimum Hamming distan ce o f a co de C is the largest integer ∆ such that ∀ c, c ′ ∈ C, d ( c, c ′ ) ≥ ∆ . Let A q ( n, d ) b e the max imal numb er of codew ords that a q -ary code of len gth n and min imum Hamming distance d can possibly con tain ([1, Chapte r 17]). A ( n, d, w ) is defin ed similarly for binary cod es with constant weight w . F inding the values o f A q ( n, d ) and A ( n, d, w ) is a basic prob lem in “classical” codin g theory [2], [1]. Finding a general exact expression for the maximal order of codes is a difficult task. I n fact, it was de scribed in [ 4], as “a h opeless task”. For this reason, mu ch of the re search do ne has foc used on boun ding these quantities. The dual pr oblem, consisting of find ing the maximal or der of a set of cod ew ords satisfying an upper bound on their pairwise Ham ming distance (anticodes), is well studied in extremal combinato rics. Su rprisingly eno ugh, it has a closed form solution [3], [4], [5]. Using tools fr om algeb raic g raph th eory , we dr aw a lin k b e- tween the maxim al order of cod es and tha t of anti-code s. Then using results like the ce lebrated Erd ˝ os-ko-Rado theorem, we rederive some known in equalities o n A q ( n, d ) and A ( n, d, w ) and other sim ilarly defined quan tities an d gi ve some ne w bound s. This paper is organized as f ollows. I n Section II we br iefly introdu ce some of the ne eded b ackgro und in gr aph the ory . In Section III we sho w how the tools intro duced can be u sed to derive upper bound s on A q ( n, d ) . In Sections IV and V we derive bou nds o n the maxima l size of constant and d oubly constant weigh t cod es, respectively . In Section VI, we show how th e described techniques can b e used to solve other problem s. W e conclud e in Section VII, where we summarize our results and present some open qu estions. I I . G R A P H T H E O RY B AC K G RO U N D W e start b y giving a brief sum mary of some graph theo ret- ical concepts and results that will be needed in this paper . For more details, we refer the interested read er to [6] an d [7]. Let G ( V , E ) be an undirected g raph, where V is its vertex set and E is its edge set ( E ⊆ V × V ). W e also use V( G ) to denote the vertex set o f G and E( G ) its edge set. If { u, v } is an ed ge in G , i.e. { u , v } ∈ E ( G ) , we say that the vertices u and v are adjac ent and write u ∼ v . The complemen t of a graph G is th e graph ¯ G defined over the same vertex set b ut where two vertices are ad jacent in ¯ G iff they are n ot in G . W e denote by ω ( G ) the clique n umber of a graph G , defined as the largest number o f vertices o f G that ar e pairwise adjacent. In contrast α ( G ) , the ind ependen ce number o f G , is the largest numb er of vertice s in G such that no two of them are adjacen t. It c an be easily seen that α ( G ) = ω ( ¯ G ) . In addition , the chr oma tic n umber χ ( G ) o f G is the minimum number of colors needed to color its vertices such that d ifferent colors are assigned to adjacent vertices. Definition 1 ( Graph Automorph ism [7]): Let G ( V , E ) be a graph an d φ a bijection fro m V to its elf. φ is called an automorp hism o f G iff ∀ u, v ∈ V , u ∼ v ⇔ φ ( u ) ∼ φ ( v ) . The set of all automo rphisms of G is a group und er co mposi- tion; it is called the automorphism group of G and it is denoted Aut ( G ) . For exam ple, the co mplete grap h on n vertices K n has S n , the symmetric g roup of ord er n , as its automorp hism group . In o ther words, Aut ( K n ) ∼ = S n . Definition 2 ( V erte x T ransitive Graph [7]): W e say that graph G ( V , E ) is vertex tr ansitiv e if f ∀ u, v ∈ V , ∃ φ ∈ Aut ( G ) s.t. φ ( u ) = v . Definition 3 ( Cayley Graphs): Let H b e a gr oup and S ⊂ H such th at S is closed under inv ersion and th e iden tity element of H 1 H / ∈ S . Th e Cayley gra ph C ( H, S ) is the graph with vertex set H and wher e for any g , h ∈ H , g ∼ h iff hg − 1 ∈ S . Next, we give with out a pr oof an imp ortant result fr om [7] (Lemma 7.2.2) that will be instrumenta l in deriving our results. Theor em 1: Let G ( V , E ) be a vertex transitive graph, then α ( G ) ω ( G ) ≤ | V ( G ) | . I I I . B O U N D S O N C O D E S Definition 4 ( Hamming Graph [2]): The Hammin g graph H q ( n, d ) , n ∈ N and 1 ≤ d ≤ n , has as vertices a ll the q -ary seq uences of length n , and two vertices are ad jacent iff their Ham ming distan ce is larger or equal to d . That is, V ( H q ( n, d )) = Σ n , w here Σ = { 0 , 1 , . . . , q − 1 } . and u ∼ v iff d ( u, v ) ≥ d . Recall that A q ( n, d ) d enotes the maximum numb er of codewords in a q-ary code of length n and minimum Hamming distance d . When the subscript is o mitted we assume q = 2 , i.e. A ( n, d ) = A 2 ( n, d ) . It can be easily seen that A q ( n, d ) = ω ( H q ( n, d )) . Let S n,d , 1 ≤ d ≤ n , be a subset of the group ( Z n q , +) , where a ddition is done modulo q , such that S n,d = { s ∈ Z n q ; w t ( s ) ≥ d } . It is easy to ch eck th at S n,d is closed under in version and does no t conta in the identity element (the all zero seq uence). T he next lemma asserts that the Hamm ing graph is in fact a Cayley g raph. Lemma 1: H q ( n, d ) = C ( Z n q , S n,d ) . Pr o of: T ake Σ = ( Z q , +) . The result then follows easily from the fact that ∀ x, y ∈ Z n q , d ( x, y ) = w t ( x − y ) . Lemma 2: Th e Hamm ing grap h H q ( n, d ) is vertex tran si- ti ve. Pr o of: Follows From Lemma 1 and the fact that Cayley Graphs are vertex transitiv e [7, Thm. 3 .1.2]. For a clearer presentation , we also g i ve h ere a direct pro of. T ake Σ = ( Z q , +) . And ∀ u, v , x ∈ Σ n , defin e the f unction φ u,v ( x ) = x + v − u . φ u,v ( x ) is an automorp hism of H q ( n, d ) . In fact, d ( φ u,v ( x ) , φ u,v ( y )) = d ( x + v − u, y + v − u ) = wt ( x + v − u − ( y + v − u )) = wt ( x − y ) = d ( x, y ) . Also, φ u,v ( x ) takes u to v . Cor olla ry 1: A q ( n, d ) α ( H q ( n, d )) ≤ q n Pr o of: Follo ws from Lemma 2 and Thm. 1. Notice that α ( H q ( n, d )) , th e independen ce number of the Hamming graph H q ( n, d ) , is a ctually the maxim um number of sequences such that the Hamming distance between any two of them is at most d − 1 . Following [ 3], we define N q ( n, s ) to be the maxim um number of q -ary sequence s of len gth n that intersect pairwise (h av e th e same entr ies) in at least s positions. It fo llows tha t α ( H q ( n, d )) = N q ( n, t ); with t = n − d + 1 (1) Lemma 3 ( Singleton Bound ): A q ( n, d ) ≤ q n − d +1 Pr o of: Consider the set T ( n, t ) of q -ary sequences of length n that all have the same element in the first t = n − d + 1 entries. By definitio n, N q ( n, t ) ≥ | T ( n, t ) | = q n − t . T hen, by (1) and Cor ollary 1, A q ( n, d ) ≤ q n q n − t = q n − d +1 . Lemma 4 ( Hamming Bound ): A q ( n, d ) ≤ q n P ⌊ d − 1 2 ⌋ i =0  n i  ( q − 1) i . Pr o of: Th e proof is similar to that of Lemma 3 and is done by finding a different lo wer boun d on N q ( n, t ) . In fact, con sider the ball B ( n, r ) = { x ∈ Σ n ; w t ( x ) ≤ r } . By th e tr iangle in equality , ∀ x, y ∈ B ( n, ⌊ d − 1 2 ⌋ ) , d ( x, y ) ≤ d − 1 . Therefore N q ( n, t ) ≥ | B ( n, ⌊ d − 1 2 ⌋ ) | , and A q ( n, d ) ≤ q n B ( n, ⌊ d − 1 2 ⌋ ) . The num ber N q ( n, t ) is well stud ied in extremal co mbi- natorics [3] [5], an d a closed for m for it is known. Thus, exact expr essions o f N q ( n, t ) can be u sed to der i ve better upper bo unds on A q ( n, d ) . For instance, if n − t is even, N 2 ( n, t ) = P n − t 2 i =0  n i  . Thus, in this case, B ( n, ⌊ d − 1 2 ⌋ ) is a maximal an ticode. Howe ver , whe n n − t is odd , N 2 ( n, t ) = 2 P n − t − 1 2 i =0  n − 1 i  [3, Thm . Kl] and [8 ]. Theref ore, we obtain the following lemma. Lemma 5: A ( n, d ) ≤ 2 n − 1 P d − 2 2 i =0  n − 1 i  , if d is even . (2) Notice that the above b ound is tighter than the Hamming bound for even d since 2 d − 2 2 X i =0  n − 1 i  − d − 2 2 X i =0  n i  =  n − 1 d − 2 2  > 0 . This new improved Hamming boun d w as r ecently proven in [9] using d ifferent techniq ues than the on e presented here. Next we g i ve a n ew up per bound on A q ( n, d ) fo r alp habets of arbitrar y size. Lemma 6: For q ≥ 3 , t = n − d + 1 and r = ⌊ min { n − t 2 , t − 1 q − 2 }⌋ , A q ( n, d ) ≤ q t +2 r P r i =0  t +2 r i  ( q − 1) i . (3) Pr o of: The proof follows from Coro llary 1 and Thm. 2 in [5] or the Diametric Theor em of [3]. Note that for q ≥ t + 1 , N q ( n, t ) = q n − t [5, Corollar y 1], i.e. a ma ximal an ticode would be the tri vial set T ( n, t ) described in th e proof of Lemm a 3. In this case, the boun d of (3) boils down to the Singleton b ound . For d e ven and n n ot much larger than t , th e next lemma provides an imp rovement on the Hamming b ound for nonbi- nary alphabe ts. Lemma 7: For d o dd and n ≤ t + 1 + log t log( q − 1) A q ( n, d ) ≤ q n − 1 P d − 2 2 i =0  n − 1 i  ( q − 1) i (4) Pr o of: Under the con ditions o f th is lemma, N q ( n, t ) = q P d − 2 2 i =0  n − 1 i  ( q − 1) i [3, Eq. 1.7 ]. Th e resu lt th en f ollows from Corollary 1. I V . B O U N D S F O R C O N S TA N T W E I G H T C O D E S Let A ( n, 2 δ, w ) be th e maxim um possible nu mber of code- words in a b inary code of length n , constant weight w an d minimum distance 2 δ [2], [10]. Define the graph K ( n, 2 δ, w ) as th e graph whose vertices are all the binary sequen ces of length n an d weig ht w and where two vertices u, v are adjacent iff d ( u, v ) ≥ 2 δ . It can be easily seen that A ( n, 2 δ, w ) = ω ( K ( n, 2 δ, w )) . Let  [ n ] w  denote the s et of all subsets of [ n ] = { 1 , 2 , . . . , n } of order w . There is a natural bijection ν between V( K ( n, 2 δ, w )) and  [ n ] w  . Namely , ∀ u ∈ V( K ( n, 2 δ, w )) , ν ( u ) = U = { i ; u ( i ) = 1 } . Lemma 8: ∀ p, q ∈ V( K ( n, 2 δ, w )) , p ∼ q if f | P ∩ Q | ≤ w − δ whe re P = ν ( q ) an d Q = ν ( q ) . Pr o of: 2 δ ≤ d ( p, q ) = | ( P ∩ ¯ Q ) ∪ ( ¯ P ∩ Q ) | = 2 w − 2 | P ∩ Q | . Lemma 9: K ( n, 2 δ, w ) is vertex transitive. Pr o of: For any two vertices p, q o f K , any b ijection on [ n ] su ch that the imag e of P = ν ( p ) is Q = ν ( q ) , takes p to q and belo ngs to Aut ( K ) . The first r esult that follows directly fr om Lemma 9 is the Bassalygo-Elias inequality [10]. W e first recall some additional results in gr aph theor y . Definition 5 ( Graph H omomorph ism): Let X and Y be tw o graphs. A mappin g f from V( X ) to V( Y ) is a h omomo rphism if ∀ x, y ∈ V( X ) x ∼ y ⇒ f ( x ) ∼ f ( y ) . Theor em 2: If Y is vertex tra nsiti ve and there is a homo- morph ism fr om X to Y , then | V ( X ) | α ( X ) ≤ | V ( Y ) | α ( Y ) Pr o of: An app lication of Lemma 7 .14.2 in [7]. Lemma 10 (Bassalygo-Elias in equality): A ( n, d ) ≤ 2 n  n w  A ( n, d, w ) Pr o of: Consider the two graph s Y = ¯ H ( n, d ) and X = ¯ K ( n, d, w ) . Y is vertex transitive. Since X is an indu ced subgrap h of Y , the inclusion map is a h omomo rphism that takes X to Y . The result then f ollows from ap plying Th m. 2. By the same token, we can show the below eq ualities Lemma 11: A ( n, d, w ) ≤ n − w + 1 w A ( n, d + 2 , w − 1) (5) A ( n, d, w ) ≤ n + 1 w + 1 A ( n + 1 , d + 2 , w + 1) (6) A ( n, d, w ) ≤ n w A ( n − 1 , d, w − 1) (7) A ( n, d, w ) ≤ n n − w A ( n − 1 , d, w ) (8) Pr o of: W e start by provin g inequality 5. Let φ b e a mapping fr om  [ n ] w − 1  to  [ n ] w  , such that ∀ P ∈  [ n ] w − 1  , P ⊂ φ ( P ) . φ is a ho momor phism f rom K ( n, d + 2 , w − 1) to K ( n, d, w ) . I n fact, ∀ P, Q ∈ K ( n, d + 2 , w − 1 ) such that P ∼ Q, | φ ( P ) ∩ φ ( Q ) | ≤ | P ∩ Q | + 2 ≤ w − 1 − ( d + 2) / 2 + 2 = w − d/ 2 (by Lemm a 8). Therefore, φ ( P ) ∼ φ ( Q ) . T he inequality then fo llows by ap plying Thm. 2. T o prove inequ ality 6, take the homo morph ism φ from K ( n + 1 , d + 2 , w + 1) to K ( n, d, w ) to be φ ( X ) = X \ { max x ∈ X x } , ∀ X ∈  [ n +1] w +1  . The r est of the ineq ualities can be pr oved similarly by considerin g the correspon ding graphs an d taking th e ho mo- morph ism to b e the inclu sion map. The fir st two inequalities are new , whereas inequalities 7 and 8 were fir st proven by Johnson in [11]. Similarly , we can show the following ine qualities regarding A q ( n, d ) . Lemma 12: A q ( n, d ) ≤ 1 q A q ( n + 1 , d + 1 ) A q ( n, d ) ≤ q A q ( n − 1 , d ) A q ( n, d ) ≤ q n ( q − 1) n A q − 1 ( n, d, w ) Lemma 13: Let t = w − δ + 1 . A ( n, 2 δ, w ) ≤  n w   n − t w − t  (9) Pr o of: Let G = K ( n, d, w ) . Since G is vertex transitive, we have A ( n, 2 δ, w ) α ( G ) ≤ | V ( G ) | =  n w  . Define M ( n, w , s ) as in [4] to be the maximum numbe r of sub sets of [ n ] of o rder w that intersect pairwise in at least s elemen ts. By Lemma 8, α ( G ) = M ( n, w , t ) . But, M ( n, w , t ) ≥  n − t w − t  (for instance, consider the system of all subsets of [ n ] of o rder w that contain the set { 1 , 2 , . . . , t } ) . The b ound of L emma 13 is actually the same as the one in Thm. 12 in [10] which was given with a different proo f. One can impr ove on the b ound of Le mma 13 by using the exact value of M ( n, w , t ) [4]. It is k nown that for n ≥ ( w − t + 1)( t + 1 ) , M ( n, w, t ) =  n − t w − t  [13], [1 4]. Howe ver , this is not the case f or lower values of n . Lemma 14: Let t = w − δ + 1 and r = max { 0 , ⌈ δ ( w − δ ) n − d − 1 ⌉} , then A ( n, 2 δ, w ) ≤  n w  P w i = t + r  t +2 r i  n − t − 2 r w − i  ; (10) with  n k  = 0 when k > n . Pr o of: (sketch) A ( n, d, w ) ≤ ( n w ) M ( n,w, t ) , then use the exact value o f M ( n, w, t ) given by the main theo rem of [4]. V . B O U N D S F O R D O U B L Y B O U N D E D W E I G H T C O D E S Let T ( w 1 , n 1 , w 2 , n 2 , d ) be the maximum nu mber of code - words in a dou bly constant weight b inary code of minimum distance d , length n = n 1 + n 2 and constan t weight w = w 1 + w 2 , wher e the first n 1 entries of ea ch codewords ha ve exactly w 1 ones [12]. T ′ ( w 1 , n 1 , w 2 , n 2 , d ) is defin ed similarly but where the first n 1 entries of ea ch cod ew ords h av e at mo st w 1 ones [10]. Lemma 15: A ( n, d, w ) ≤  n 1 + n 2 w 1 + w 2   n 1 w 1  n 2 w 2  T ( w 1 , n 1 , w 2 , n 2 , d ) (11) A ( n, d ) ≤ 2 n P w 1 i =0  n 1 i  n 2 w 1 + w 2 − i  T ′ ( w 1 , n 1 , w 2 , n 2 , d ) (12) Pr o of: Same as L emma 10. Note that inequality (11) was first proven in [12], whereas inequality (12) is new . Se veral oth er bounds on T ( w 1 , n 1 , w 2 , n 2 , d ) kn own in literature, such as T heorem 36 in [1 0], can b e also easily ob tained in th e same way . The next lemma establishes so me addition al new bound s. Lemma 16: T ( w 1 , n 1 , w 2 , n 2 , d ) ≤ n 2 w 2 ! A ( n 1 , w 1 , d − 2 w 2 ) if d − 2 w 2 ≥ 0 T ( w 1 , n 1 , w 2 , n 2 , d ) ≤ n 1 w 1 ! A ( n 2 , w 2 , d − 2 w 1 ) if d − 2 w 1 ≥ 0 T ( w 1 , n 1 , w 2 , n 2 , d ) ≤ n 1 − w 1 + 1 w 1 T ( w 1 − 1 , n 1 , w 2 , n 2 , d + 2) T ( w 1 , n 1 , w 2 , n 2 , d ) ≤ n 1 + 1 w 1 + 1 T ( w 1 + 1 , n 1 + 1 , w 2 , n 2 , d + 2) T ( w 1 , n 1 , w 2 , n 2 , d ) ≤ n 2 − w 2 + 1 w 2 T ( w 1 , n 1 , w 2 − 1 , n 2 , d + 2) T ( w 1 , n 1 , w 2 , n 2 , d ) ≤ n 2 + 1 w 2 + 1 T ( w 1 , n 1 , w 2 + 1 , n 2 + 1 , d + 2) V I . O T H E R A P P L I C A T I O N S In this section we demonstrate how the above tech niques can be help ful in solving othe r problem s. F or instance, we show how to c ompute N q ( n, 1) , the maximu m nu mber of q - ary sequen ces of len gth n intersecting pairwise in a t lea st one position [3]. Lemma 17: N q ( n, 1) = q n − 1 Pr o of: Let G = H q ( n, n ) ; N q ( n, 1) = α ( G ) . Now , consider the set of q sequenc es where the entries in the i -th sequence a re all the same and equal to i , hen ce ω ( G ) ≥ q . But ω ( G ) ≤ q since the first entries of all sequen ces in a clique in G sh ould con tain different letters. Therefor e, ω ( G ) = q . By Lemma 2, w e get N q ( n, 1) ≤ q n − 1 . But N q ( n, 1) ≥ q n − 1 (see the pr oof o f Lem ma 3). The n ext lem ma gives the ch romatic nu mber of ce rtain Hamming grap hs. Lemma 18: χ ( H q ( n, d )) = q n − d +1 , for q ≥ n − d + 2 , 1 ≤ d ≤ n . Pr o of: From the definitions, it f ollows that fo r any g raph G , χ ( G ) ≥ | V( G ) | α ( G ) . But, α ( H q ( n, d )) = q d − 1 [5, Coro llary 1]. Therefo re, χ ( H q ( n, d )) ≥ q n q d − 1 = q n − d +1 . Let φ b e a mappin g from Σ n to Σ n − d +1 consisting of deleting the last d − 1 entries of a sequence. φ is a homo- morph ism fro m H q ( n, d ) to H q ( n − d + 1 , 1) = K n − d +1 , where K ℓ is the complete graph on ℓ vertices. Ther efore, χ ( H q ( n − d + 1 , 1 )) ≤ χ ( K n − d +1 ) = q n − d +1 [7, Lemma 1.4.1] . Let v ( G ) b e the Lov ´ asz up per bound [15] on the zero e rror capacity Θ ( G ) [16] of a grap h G . W e recall the following two results of [ 15]. Lemma 19: α ( G ) ≤ Θ( G ) ≤ v ( G ) Theor em 3: If G ( V , E ) is vertex transiti ve then v ( G ) v ( ¯ G ) = | V | . In th e following, we give a p artial answer to a question raised in the conclusion o f [15], nam ely “Fin d fu rther graphs with v ( G ) = Θ ( G ) ”. Lemma 20: T he following graphs satisfy v ( G ) = Θ( G ) 1) H q ( n, d ) when there exists a q-ary perfect code of length n and m inimum distance d . 2) H q ( n, d ) when q ≥ n − d + 2 and there exists a q- ary MDS code o f length n an d minimum distance d . 3) H q ( n, n ) . Pr o of: Let G be a vertex transitive grap h such that α ( G ) α ( ¯ G ) = | V( G ) | . Then, ap plying L emma 19 to G and ¯ G a nd m ultiplying the two resulting equation s we get Θ( G ) v ( G ) = v ( ¯ G ) Θ( ¯ G ) ≥ 1 . Theref ore, Θ( G ) = v ( G ) . One can check that the g raphs G belong ing to the three families men tioned ab ove satisfy α ( G ) α ( ¯ G ) = | V( G ) | . V I I . C O N C L U S I O N W e constructed vertex tran siti ve grap hs where a code cor- respond s to a clique and an an ti-code to an indep endent set. Thus, we established a conne ction between th e max imal order of cod es and that o f anti-cod es. Using intersection theor ems for systems of fi nite sets a nd th at of finite sequ ences, we provided a framework wh ere several k nown bounds on code size follow easily and new inequalities can b e derived. Sev eral questions naturally arise h ere. 1) What ar e the zero error c apacities of the gra phs H and K and their compleme nts ¯ H and ¯ K ? W hat are the values of the v fu nction o f the se grap hs. No te, tha t these quantities can be useful to d eriv e boun ds for A q ( n, d ) and A ( n, d, w ) usin g Lemma 19 and Thm. 3. 2) From a graph theor etical standpoin t, trying to extend the result o f Lem ma 1 8 b y find ing the ch romatic nu mber o f the above g raphs is also an in teresting qu estion, and can have app lications to coding theo ry and cryp tograph y . 3) Perfect cod es ar e co des who achieve the Hamming bound . 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