Elementary 2-Group Character Codes Cunsheng Ding 1, David Kohel 2, and San Ling Abstract In this correspondence we describe a class of codes over GF (q), where q is a power of an odd prime. These codes are analogues of the binary Reed-Muller codes and share several features in common with them. We determine the minimum weight and determine properties of these codes. For a subclass of codes we nd the weight distribution and prove that the minimum nonzero weight codewords give 1-designs. Index Terms Linear codes, Reed-Muller codes, group character codes. 1 Introduction In this correspondence we describe a class of group character codes C q (r; n), dened over GF (q), with parameters 2 n ; s n (r); 2 r+1, where q is a power of an odd prime and s n (k) = P k i=0? n i. The codes Cq (r; n) are dened in analogy with the binary Reed-Muller codes and have the same parameters []. Moreover, as for Reed-Muller codes, C q (r; n) is generated by minimum weight codewords, and the dual of C q (r; n) is equivalent to C q (n? r? 1; n), which is the analogue of the equality R(r; n)? = R(n? r? 1; n). The purpose of this correspondence is to describe this class of codes, to determine the dimensions and minimum distances, to characterize the dual codes, and to nd the weight distribution and associated 1-designs for the subclass C q (1; n). 2 Abelian Group Character Codes Let A be an additive abelian group of exponent m and order N, with 0 as the identity element. Let K be a nite eld whose characteristic does not divide N and which contains the mth roots of unity. Let K denote the multiplicative group of nonzero elements of K and let M denote the 1 Department of Computer Science, School of Computing, National University of Singapore, Lower Kent Ridge Road, Singapore 119260. Email dingcs@comp.nus.edu.sg 2 Department of Mathematics, National University of Singapore, Lower Kent Ridge Road, Singapore 119260. Email kohel@math.nus.edu.sg. Department of Mathematics, National University of Singapore, Lower Kent Ridge Road, Singapore 119260. Email matlings@math.nus.edu.sg. C. Ding and S. Ling are partially supported by grant number RP 960668/M. 1
multiplicative group of characters from A to K. The group M is isomorphic noncanonically to A [4, Chapter VI]. In particular we have jmj = jaj = N. The following lemma is a well-known result, known as the orthogonality relations in character theory [4, Chapter VI, Proposition 4]. Lemma 1 (Orthogonality Relations) Let A be a nite additive abelian group of order N and let M be the group of characters of A. For characters f; g in M and elements x; y in A, we have 1. P x2a f(x)g(x) = N if f = g?1 0 if f 6= g?1 2. P f 2M f(x)f(y) = N if x =?y 0 if x 6=?y Let M = ff 0 ; f 1 ; ; f N?1 g, where f 0 is the trivial character. For any subset X of A, we dene a linear code C X over K as C X = n (c 0 ; c 1 ; ; c N?1 ) 2 K N N X?1 i=0 c i f i (x) = 0 for all x 2 X Let X = fx 0 ; x 1 ; ; x t?1 g be a subset of A and let X c be the complement of X in A, indexed such that A = fx 0 ; x 1 ; ; x N?1 g. Proposition 2 Let A and X be as above. For 0 i N? 1, let v i denote the vector (f 0 (x i ); f 1 (x i ); ; f N?1 (x i )). Then the set fv 0 ; v 1 ; ; v N?1 g is linearly independent. In particular, h i H = f j?1 (x i?1 ) 1it; 1jN has rank t and is a parity check matrix of C X, h i G = f j?1 (?x t?1+i ) 1iN?t; 1jN has rank N? t and is a generator matrix for C X, so C X is an [N; N? t] linear code over K. Moreover, h i f j?1 (?x i?1 ) 1it; 1jN is a generator matrix for C X c and C X C X c = K N Proof The linear independence of the set fv 0 ; v 1 ; ; v N?1 g follows from Lemma 1. The other conclusions of the proposition then follow from this linear independence. o 2
Elementary 2-Group Character Codes Hereafter we let A be the elementary 2-abelian group (Z=2Z) n, for which we prescribe a basis fe 1 ; ; e n g of generators. Moreover we denote the neutral element of A by e 0. For the eld K we take a nite eld GF (q) for q odd. Since f1g is contained in GF (q), the character group M of A is dened. Relative to P the basis for A we can dene characters by f j (e i ) = (?1) ji?1, n?1 where j = k=0 j k2 k for 0 j < 2 n. One easily veries that this gives an indexing M = ff 0 ; ; f 2?1g on the group of characters from A to GF (q). n Theorem For any subset X of A, let X c = A n X. Then the dual code C? X equals C X c. Proof This follows from the orthogonality relations of Lemma 1, the description of a generator matrix for C X c of Proposition 2, and the fact that every element of A is its own inverse. The classes of codes over GF (q) described in this section have many subclasses of codes. Dierent choices of the set X give dierent subclasses of codes over GF (q). We now describe a subclass of codes over GF (q) which have the same parameters as the binary Reed-Muller codes. Denitions. The Hamming weight jjajj of an element a = a 1 e 1 + + a n e n of A is dened to be the number of nonzero a k. For?1 r n, let X(r; n) = fa 2 A jjajj > rg; and let C q (r; n) denote the code C X(r;n) over GF (q). For a word c = (c 0 ; ; c 2 n?1) in K 2n, let the support of c be dened as Supp(c) = fi 0 i < 2 n ; and c i 6= 0g; and let its weight wt(c) be dened as jsupp(c)j. By convention we dene the minimum distance of the zero code to be 1, which we represent by any integer larger than the block length of the code. We use j to denote concatenation of codewords, and for two sets U and V of codewords dene U jj V to be the set fu j (u + v) u 2 U; v 2 V g Lemma 4 The dimension of the code C q (r; n) is s n (r) = P r j=0? n j. Proof This follows from Proposition 2 and the denition of X(r; n).
Lemma 5 The code C q (r + 1; n + 1) decomposes as the direct sum C q (r + 1; n) jj C q (r; n) = C q (r + 1; n) jj f0g f0g jj C q (r; n) where f0g is the zero code in K 2n. Proof For vector subspaces U and V of K 2n it is clear from the denitions that U jj V = U jj f0g f0g jj V. Moreover by the combinatorial equality s n (r) + s n (r + 1) = s n+1 (r + 1); Lemma 4 and the linear independence of C q (r+1; n) jj f0g and f0g jj C q (r; n) imply the result provided that both are contained in C q (r + 1; n + 1). Let A n be the elementary 2-abelian group generated by fe 1 ; ; e n g, identied as a subgroup of A n+1 = A n Z=2Z e n+1. The character f j on A n identies with the character f j on A n+1 by setting f j (e n+1 ) = 1 for all 0 j 2 n? 1. Then by denition f j+2 n = f j f 2 n, where f 2 n(e i ) =?1 if i = n + 1 1 otherwise. The words in C q (r + 1; n) jj f0g are of the form cjc such that c = (c 0 ; ; c 2?1) and n 2X n?1 j=0 c j f j (a) = 0 for all a in X(r + 1; n). One readily veries that C q (r + 1; n) jj f0g = C X 1, where X 1 = X(r+1; n)[(a n +e n+1 ). Similarly f0g jj C q (r; n) = C X 2, where X 2 = A n [ (X(r; n)+ e n+1 ). Since X 1 and X 2 contain X(r + 1; n + 1), both C X 1 and C X2 are contained in C q(r + 1; n + 1) and the result holds. Theorem 6 C q (r; n) is a [2 n ; s n (r); 2 n?r ] code over GF (q). Proof It only remains to prove the correctness of the minimum distance. Since X(0; n) = Anfe 0 g, a generator matrix for C q (0; n) is h f 0 (e 0 ) f 1 (e 0 ) f 2 n?1(e 0 ) Therefore, C q (0; n) has minimum weight 2 n. i = h 1 1 i 4
At the other extreme, X(n; n) is empty, so C q (n; n) = K 2n and C q (n; n) has minimum weight 1. In particular it follows that the minimum weight is correct for n = 1 and 0 r 1. Suppose now that Theorem 6 gives the correct minimum distance for some n 1 and all 0 r n. By Lemma 5, a word c in C q (r + 1; n + 1) is of the form c = u j (u + v); where u and v are codewords in C q (r +1; n) and C q (r; n) respectively. Then wt(c) 2 wt(u) + wt(v)? 2 jsupp(u) \ Supp(v)j wt(v) 2 n?r Conversely f0g jj C q (r; n) is a subcode of C q (r + 1; n + 1) with minimum distance 2 n?r, so the minimum distance of C q (r + 1; n + 1) is 2 n?r. Theorem 7 The minimum nonzero weight codewords generate C q (r; n). Proof Let M q (r; n) denote the set of minimum nonzero weight codewords of C q (r; n). The result is clear for n = 1 and all r, and likewise for r =?1 and all n. Moreover C q (r + 1; n + 1) is generated by M q (r + 1; n) jj f0g [ f0g jj M q (r; n); by the direct sum of Lemma 5. This set is contained in M q (r + 1; n + 1), from which the result holds by induction. Let G be dened as the subgroup in the group of linear automorphisms of K 2n generated by permutations of coordinates and by multiplications of coordinates by elements of K. Two codes C and C 0 are called equivalent if and only if C 0 = (C) for some 2 G. Theorem 8 The dual code C q (r; n)? is equivalent to C q (n? r? 1; n). Proof The theorem clearly holds for r = n, so we assume henceforth that r < n. Let = e 1 + + e n. Then the equality X(r; n) c = + X(n? r? 1; n) follows immediately from the denitions. By Theorem and the denitions of the codes, one easily veries that the map K 2n?! K 2n (c 0 ; ; c 2?1) n 7?! (f 0 ()c 0 ; ; f 2?1()c n 2?1) n 5
is an automorphism of K 2n of order two, which induces an equivalence of C q (r; n)? and C q (n? r? 1; n) and vice versa. Corollary 9 C q (r; n)? is a [2 n ; s n (n? r? 1); 2 r+1 ] code which is generated by its minimum nonzero weight codewords. Proof The conclusions follow from Theorems 6, 7, and 8. Remark. The Reed Muller codes R(r; n) are well-known to have the same parameters [2 n ; s n (r); 2 n?r ] as the codes C q (r; n), so the latter can be viewed as analogues over GF (q) of the corresponding binary Reed-Muller codes. Moreover, we have R(r; n)? = R(n? r? 1; n) of which Theorem 8 is its analogue. While strict equality of C q (r; n)? and C q (n? r? 1; n) does not hold, the proof shows that the equivalence is a simple coordinate twist. Stated formally, this says that C q (n? r? 1; n) is the dual of C q (r; n) with respect to the twisted inner product hu; vi = X 2 n?1 j=0 f j () u j v j ; on vectors u = (u 0 ; ; u 2 n?1) and v = (v 0 ; ; v 2 n?1) in K 2n. Example 1 Consider n =. Then C (1; ) is a [8; 4; 4] ternary code with generator matrix 2 6 4 1 1?1?1?1?1 1 1 1?1 1?1?1 1?1 1 1?1?1 1 1?1?1 1 1?1?1?1 1 1 1?1 and weight enumerator polynomial 1 + 24x 4 + 16x 5 + 2x 6 + 8x 8. This is one of the best possible ternary codes of length 8 and dimension 4, cf. [2]. In Theorem 17, we show that the support of the codewords of minimum nonzero weight form a 1-(8; 4; 6) design. Example 2 Consider n = 4. Then C (2; 4) is a [16; 5; 8] ternary code with generator matrix 2 6 4 1?1?1?1 1 1 1?1 1?1?1?1 1 1 1?1 1?1?1 1?1 1?1 1?1 1?1 1?1 1 1?1 1?1 1?1?1?1 1 1?1?1 1 1 1?1 1?1 1 1?1?1?1?1?1?1 1 1 1 1 1 1?1?1 1?1?1?1?1 1 1 1 1 1 1?1?1?1?1 1 6 7 5 7 5
and weight enumerator polynomial 1 + 40x 8 + 80x 10 + 2x 11 + 80x 12 + 10x 16. The best ternary codes of length 16 and dimension 5 have minimum distance 9, so this is almost the best ternary code of these parameters [2]. Its dual code C (2; 4)? is a [16; 11; 4] code with weight enumerator polynomial 1 + 200x 4 + 52x 5 + 2544x 6 + 5600x 7 + 1740x 8 + 2840x 9 + 4272x 10 + 6480x 11 + 0840x 12 + 18400x 1 + 8720x 14 + 1824x 15 + 4x 16 The code C (2; 4)? is one of the best possible ternary codes of these parameters. 4 The Weight Distribution in C q (1; n) Let the characters f j be as dened previously for 0 j < 2 n and dene vectors v i = (f 0 (e i ); ; f 2?1(e n i )) for 0 i n. In particular v 0 is the vector (1; ; 1) and fv 0 ; v 1 ; ; v n g is a basis for the code C q (1; n). Let V = GF (q) n and for a in V let jjajj denote its Hamming weight. For a = (a 1 ; ; a n ) in V, set a (v 1 ; ; v P n ) equal P to the dot product n a n iv i. The collection of vectors of the form a iv i constitutes a subcode Cq 0 (1; n) of codimension 1. We rst describe the weight distribution in Cq 0(1; n) then treat the nontrivial cosets a 0v 0 + Cq 0 (1; n). We begin with a series of lemmas which reduce the analysis of the weight distribution to the weights of codewords in subsets of Cq 0 (1; n). Lemma 10 Let a have Hamming weight m in V. Then the vector v = a (v 1 ; ; v n ) has Hamming weight 2 n? 2 n?m jw j, where W = n b = (b 1 ; ; b n ) 2 V b i = a i for all i and nx o b i = 0 Proof Clearly a (v 1 ; ; v n ) has weight 2 n less the number of j for which nx a i f j (e i ) = 0 (1) For any such j we associate the vector b = (a 1 f j (e 1 ); ; a n f j (e n )) in W. But for any b in W there are 2 n?m indices j for which (1) holds and has associated P P n?1 vector b. Namely, if we let j = i=0 j i2 i be one such value, then n?1 k = i=0 k i2 i is another if and only if k i?1 = j i?1 whenever a i 6= 0. 7
Let s = (q? 1)=2 and let f 1 ; ; q?1 g be an indexing of the elements of GF (q) such that i+s =? i. For an element a = (a 1 ; a 2 ; ; a n ) of V and each i in 1 i s, we dene the multiplicities of i in a. n i (a) = fk 1 k n and a k = i g; Lemma 11 For a in V, let n 1 = n 1 (a); ; n s = n s (a) be the associated multiplicities. Then the weight of the vector v = a (v 1 ; ; v n ) is X sy 2 n? 2 n?m ni?n i m i n i Pm i n i mod2 s m i i =0 n i +m i 2 In particular, the weight of v depends only on the multiplicities n 1 ; ; n s, and not on the ordering of the coordinates of a. Proof By Lemma 10, it suces to determine jw j. Let b be in W, and for each i let u i be the number of occurences of i in the coordinates of b. Then the number of occurences of? i in this vector is n i? u i. It follows that nx b i = sx (u i i? (n i? u i ) i ) = sx Q s and we set m i = 2u i? n i. Since there are associated multiplicities n 1 ; ; n s, we obtain from which the lemma follows. X jw j = P0u i n i s m i i =0 sy ni ; u i (2u i? n i ) i ;? ni u i vectors b in W with The following lemma gives the weight of vectors in the nontrivial cosets of C 0 q (1; n) in C q (1; n), and is proved similarly. Lemma 12 For a in V, let n 1 = n 1 (a); ; n s = n s (a) be the associated multiplicities. Then the weight of a vector v = a 0 v 0 + a (v 1 ; ; v n ), with a 0 6= 0, is 2 n? 2 n?m X?n i m i n i Pm i n i mod2 s m i i =?1 sy ni n i +m i 2 8
In particular, the weight of v depends only on the multiplicities n 1 ; ; n s, and not on the ordering of the coordinates of a. When q = the product in Lemmas 11 and 12 consists of one term. In this case we obtain a precise formula for the weight distribution in C 0 (1; n) and C (1; n). First we prove a result concerning sums of binomial coecients in progressions. Lemma 1 Let m be a positive integer. Dene X m a = ; X m b = ; X m c = j j j 0jm j0mod 1. If m = k, then 2. If m = k + 1, then. If m = k + 2, then a = 2m + (?1) k 2 0jm j1mod a = b = 2m + (?1) k b = 2m + (?1) k 2 0jm j2mod ; b = c = 2m? (?1) k ; c = 2m? (?1) k 2 ; a = c = 2m? (?1) k Proof Let m be of the form k + i, with 0 i 2, and let be a primitive third root of unity. We note that has trace?1 and that + 1 is a cube root of?1 whose square is. Expanding the binomial expression we obtain mx m ( + 1) m = i?j j j=0 In the three cases i = 0, 1, and 2 respectively, we obtain (?1) k = a + c + b 2 ; (?1) k ( + 1) = b + a + c 2 ; (?1) k = c + b + a 2 Taking the trace of 1 and times the corresponding equation, together with the equation 2 m = a + b + c, gives the asserted result. 9
? n m 2 m 0 m n Weight Frequency m = k 2 n? 2 n?m (2 m + (?1) k 2)=? m = k + 1 2 n? 2 n?m (2 m? (?1) k 2)= n m 2 m? m = k + 2 2 n? 2 n?m (2 m + (?1) k 2)= n m 2 m Table 1 The weight distribution in C 0 (1; n)? n m 2 m+1 0 m n Weight Frequency m = k 2 n? 2 n?m (2 m? (?1) k )=? m = k + 1 2 n? 2 n?m (2 m + (?1) k )= n m 2 m+1? m = k + 2 2 n? 2 n?m (2 m? (?1) k )= n m 2 m+1 Table 2 The weight distribution in C (1; n)nc 0 (1; n). Theorem 14 The weight distribution in the code C 0 (1; n) is given by Table 1 and in C (1; n)nc 0 (1; n) by Table 2. Proof The weights follow by applying Lemma 1 to the weight formulas of Lemmas 11 and 12. The frequency of a given weight is determined by counting the number of a in V with Hamming weight m. Corollary 15 The weight distribution of the dual code C (n?2; n)? is given in Tables 1 and 2. Proof This follows from the equivalence with C (1; n) of Theorem 8. Having determined the weight distribution in C (1; n), we now determine the minimum nonzero weight codewords. Theorem 16 The minimum nonzero weight codewords in C (1; n) are precisely the distinct words av i + bv j, for a; b in GF () and 0 i < j n. In particular, they are 2n(n + 1) in number, and two words have the same support if and only if one is a multiple of the other. Proof By Theorem 6 the minimum nonzero weight is 2 n?1. By Theorem 14 the weight 2 n?1 codewords are those of the form av i + bv j, of which there are 2n(n + 1). The nal statement is an immediate consequence. 10
Corollary 17 The set of supports of the minimum nonzero weight codewords of C (1; n) is a 1-(2 n ; 2 n?1 ; n(n + 1)=2) design. Proof By Theorem 6 the minimum nonzero weight supports are subsets of the 2 n coordinate positions of size 2 n?1, and by Theorem 16 are n(n + 1) in number. Since the k-th coordinate of v i is f k?1 (e i ), it is clear that exactly one of the pair fav i bv j g has nonzero k-th coordinate. Thus exactly half of the supports of minimum nonzero weight codewords contains a given k, and the result follows. 5 Concluding Remarks We have described a class C q (r; n) of group character codes over GF (q) and determined their dimensions and minimum weights. For each n and r, the length, dimension, and minimum weight of C q (r; n) agrees with that of the the binary Reed-Muller code R(r; n). For the codes C q (1; n) we have explicitly determined the weight distribution and proved that the minimum nozero weight codewords give 1-designs. The class of codes over GF () contains some ternary codes with best possible error correcting capacity and others near the best possible. Some of these are listed in Table, of which ve are best possible for ternary codes of their block length and dimension. (n; r) Parameters of C (r; n) Comment Parameters of best ternary codes (; 1) [8; 4; 4] best possible [8; 4; 4] (4; 1) [16; 5; 8] almost best [16; 5; 9] (4; 2) [16; 11; 4] best possible [16; 11; 4] (5; 1) [2; 6; 16] very good [2; 6; 18] (5; ) [2; 26; 4] best possible [2; 26; 4] (6; 4) [64; 57; 4] best possible [64; 57; 4] (7; 5) [128; 120; 4] best possible [128; 120; 4] Table Examples of good codes of the form C (r; n). It remains an open problem for the class of codes C q (r; n) described here to determine the weight distribution for r greater than 1. 11
References [1] I. F. Blake and R. C. Mullin, The Mathematical Theory of Coding. New York Academic Press, 1975. [2] A. E. Brouwer, \Bounds on the size of linear codes," in Handbook of Coding Theory, Pless and Human, eds. Amsterdam Elsevier, 1998. [] F. J. MacWilliams, N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam North-Holland, 1977. [4] J.-P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics 7. New York Springer-Verlag, 197. 12