Generating linear spans over finite fields

Abstract

(1) sk = { vk if k ≤ n , ∑n−1 i=0 aisk−n+i if k > n , consists of all nonzero elements of V for k = 1, . . . , p − 1. Such generating patterns are of interest because they provide simple algorithms for generating the linear span of independent subsets of vector spaces over Fp (see [1] for details). In this paper we generalize a number of the results from [1] by working over Fq where Fq is the finite field of order q and by showing that if a0 6= 0, (a0, . . . , an−1) is an n-dimensional generating pattern over Fq if and only if f(x) = x − ∑n−1 i=0 aix i is a primitive polynomial over Fq. More generally, we show that the number of distinct elements generated by a linear recurring sequence is related to the order of its characteristic polynomial. For q = p < 10 with p ≤ 97, we indicate when one can find an optimal n-dimensional generating pattern over Fp with weight two, i.e. with two nonzero ai’s (in [1] the length is defined to be the number of nonzero ai’s but a more natural term is Hamming weight). If V is an n-dimensional vector space over Fq then V is isomorphic to Fqn as a vector space over Fq. Consequently, instead of considering vectors in V as in [1], we may assume that the elements of the sequence are in Fqn . We will make this identification throughout the remainder of the paper.