Self-orthogonal quasi-abelian codes are asymptotically good

Authors：Zhang, Guanghui; Chen, Bocong^*

Source：Finite Fields and their Applications, 2022, 78: 101958.

DOI：10.1016/j.ffa.2021.101958

• Random quasi-abelian codes
• Asymptotically good codes
• Self-orthogonal codes

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.