Asymptotically Good Additive Cyclic Codes Exist

TL;DR

This paper demonstrates the existence of asymptotically good additive cyclic codes over finite fields, building on recent results that connect primitive roots and code properties, thus addressing a long-standing open problem.

Contribution

It extends recent findings on quasi-cyclic and self-dual codes to establish the asymptotic goodness of long additive cyclic codes over various finite fields.

Findings

Long additive cyclic codes are asymptotically good over many finite fields.

The results depend on recent advances related to Artin primitive root conjecture.

Addresses a fifty-year-old open problem in coding theory.

Abstract

Long quasi-cyclic codes of any fixed index $>1$ have been shown to be asymptotically good, depending on Artin primitive root conjecture in (A. Alahmadi, C. G\"uneri, H. Shoaib, P. Sol\'e, 2017). We use this recent result to construct good long additive cyclic codes on any extension of fixed degree of the base field. Similarly self-dual double circulant codes, and self-dual four circulant codes, have been shown to be good, also depending on Artin primitive root conjecture in (A. Alahmadi, F. \"Ozdemir, P. Sol\'e, 2017) and ( M. Shi, H. Zhu, P. Sol\'e, 2017) respectively. Building on these recent results, we can show that long cyclic codes are good over $\F_q,$ for many classes of $q$'s. This is a partial solution to a fifty year old open problem.