Distribution coefficients

Abstract

The maximum iiumber of ones arid zeros and their distribution among the coefficients of some polynomials is given. It is shown that the coefficients of the poly- nomials gq(x) = (Xn + l)/p(x) and g2(x) = (Xn + 1)/((X + l)p(x)) and other related polynomials have special distribution when p(x) is a primitive polynomial, and these coefficients contain the maximum possible number of ones and zeros. Introduction. Let F be the field of two elements, 0 and 1, and let g(x) be a polynomial of degree r over F. Let n be the least integer for which g(x) divides xn + 1. We suppose g(x) to have no multiple zeros, so that n is odd. Denote by Rn the polynomial ring F(x)/(xn + 1). Let k = n - r. The set of polynomials I = (f(x)q(x) mod (x' + 1)) = Rng(x) is an ideal (cyclic code) of Rn. The following facts about I are used repeatedly (1), (2). (a) I is a subspace of the vector space of Rn ; vector addition is ordinary polynomial addition (coefficients in F). (b) a(x) E I implies xta(x) (mod (xn + 1)) E I for i = 1, 2, * * *, n -1. In words, I contains with a(x) every cyclic shift of a(x). (c) g(x) is the unique polynomial of least degree (r) in I. (d) It follows from (b) and (c) that iio polynomial of I can contain a run of k ( = n - r) consecutive zero coefficieiits; otherwise a suitable cyclic shift would make it into a polynomial of degree < r. The first part of this paper is concerned with the distribution of ones and zeros among the coefficients of g(x). We then take up a special case. Let p (x) be an irreducible factor of degree k of xn + 1, and a a zero of p(x). F(x)/p(x) = F(a) is isomorphic to the finite field GF(2k). If a is a generator of the multiplicative group of GF(2k), we say that p(x) is primi- tive. In this case n = 2k _ 1. We consider in particular the cases in which p(x) is primitive, and (i) gl(X) = (xn + l)/p(x), (ii) g2(x) = (xn + 1)/(p(x)(x + 1)) = gi(x)/(x + 1). In general,