Classification of LCD and self-dual codes over a finite non-unital local ring

Anup Kushwaha; Indibar Debnath; Om Prakash; Patrick Solé

paper

Classification of LCD and self-dual codes over a finite non-unital local ring

arXiv:2501.03016

Abstract

This work explores LCD and self-dual codes over a noncommutative non-unital ring $E_{p} = ⟨ r, s ∣ p r = p s = 0, r^{2} = r, s^{2} = s, r s = r, sr = s ⟩$ of order $p^{2}$ where $p$ is a prime. Initially, we study the monomial equivalence of two free $E_{p}$ -linear codes. In addition, a necessary and sufficient condition is derived for a free $E_{p}$ -linear code to be MDS and almost MDS (shortly AMDS). Then, we use these results to classify MDS and AMDS LCD codes over $E_{2}$ and $E_{3}$ under monomial equivalence for lengths up to $6$ . Subsequently, we study left self-dual codes over the ring $E_{p}$ and classify MDS and AMDS left self-dual codes over $E_{2}$ and $E_{3}$ for lengths up to $12$ . Finally, we study self-dual codes over the ring $E_{p}$ and classify MDS and AMDS self-dual codes over $E_{2}$ and $E_{3}$ for short lengths.

20 pages