Self-orthogonal quasi-abelian codes are asymptotically good

Let F be the finite field with q = p(s) elements, where p is an odd prime and s is a positive integer. Suppose that g(q)(-1) (x) is the inverse function of g(q)(x) = 1 - h(q)(x), where h(q)(x) is the q-ary entropy. In this paper we construct a class of random self-orthogonal quasi-abelian codes of index 2p over the finite field F, characterize the cumulative weight enumerator of such random codes by means of a blend of representation theory and probabilistic arguments, and then prove that for any given delta is an element of(0, g(q)(-1)(1/p)), the probability that the cumulative weight enumerator is at most delta converges to 0. As a consequence, the class of self-orthogonal quasi-abelian codes of index 2p is asymptotically good. (C) 2021 Elsevier Inc. All rights reserved.