WEIGHTS OF p - ARY ABELIAN CODES

Generalizing a theorem of McEliece, one obtains the highest power of p dividing all weights of a p-ary Abelian code, as a function of the set of nonzeros of that code. 1. Introduetion The important theorem of McEliece 7.8) on the weights of cyclic codes can be stated as follows. Let C be a cyclic code, of length n:;i: 0 (mod p), over the prime field GF(y) and let 'I' be the least positive integer for which it is possible to find 'I' nonzeros of C whose product is equal to I. Then, the "weights" of every code vector of C are all divisible by pm, with m = [('1'1)I(p I)] *). In the present paper, we extend these results to the more general family of p-ary Abelian codes, i.e., the codes that are ideals in the group algebra of a finite Abelian group over GF(p), as defined by MacWiIIiams 6). We prove that [('1'1)1(yI)] is in fact the greatest integer m for which all weights of C are divisible by pm. A similar result has recently been obtained by McEliece (private comm.) for binary cyclic codes. As an application of the theory, we show that either m = k I, or m :::;; [k12 I/Cp I)], where k is the dimension of C, and we describe two classes of codes meeting exactly that bound. The notations used here for group characters and group algebras are the same as in our paper on Abelian codes 4). 2. Weights of linear codes Let F = GF(y) be the field of p elements, where p is a prime, and let F" denote the set of all n-tuples over F. Then, F" becomes a linear algebra of dimension n over F, when the sum and product of two n-tuples, or vectors, of FR are defined componentwise in F. A linear code of length n and dimension k over F is a k-dimensional subspace of F". For a vector a = (a(1), a(2), ... , a(n»)of F" and a nonzero element w of F, we denote by N(w,a) the number ofpositions i (1 :::;;i ~n) for which a(1) = w. The p I integers N(w,a) are called the weights of a. In the binary case, N(1,a) is simply the Hamming weight of a. ' • ,I, Let us now consider the t-tuples w = (W1> W2, ••• , Wt) of elements WI of *) One denotes by [xl, as usual, the greatest integer not exceeding x. Mt(ë,ä) = ~ Nlö>,ii), (1) 146 P.DELSARTE F as the points of a Euclidean geometry EG(t,p). For a t-tuple ä = (al> a2, ... , at) of vectors aj E P, we denote by Nt(W,ä) the number of positions i (1 ~ i ~ n) for which al(1) = W1, a2(1) = W2' ... , at(l) = Wt. Next, for any nonzero point ë of EG(t,p) and for a fixed nonzero element W of F, we set Mt(ë,ä) = N(w, BI al + ... + Bt at). It is then obvious that the numbers Mt(ë,ä) can be expressed as sums of the NrCw,ä), for a given ä. In fact, one has where the sum is calculated for all points w such that wl BI + ... + wt e, = os. Let A be the incidence matrix of points (distinct from the origin) and Euclidean hyperplanes (not through the origin) in EG(t,p). Then, eq. (1) can be written as m(ä) = A n(ä), where m(ä) and n(ä) denote the column vectors, of order pt -1, of the numbers Mt(ë,ä) and Nt(w,ä), respectively, with w, ë =1= o. Theorem 1. Let m,t be integers, with 1 ~ t ~ m, and let al> ... , at be vectors of pn. Assume that Mlë,ä) == 0 (mod p") for every nonzero point ë of EG(t,p). Then, one has Nt(öj,ä) == 0 (modpm-t+l), for every nonzero point w of EG(t,p). Proof. Set v =pt 1 and u = v/(P 1). From known properties of Euclidean geometries, cf. for instance Carmichael 2), it can be shown that A is permutation-equivalent to a matrix satisfying both (2) and (3):