Cyclic codes over a commutative non-unitary ring of order 4

Marvin Olavides; Jon-Lark Kim

coding theory

Cyclic codes over a commutative non-unitary ring of order 4

arXiv:2607.11965

summary

The paper investigates cyclic codes over the commutative non‑unitary ring I₂ of order 4, introducing twisted and untwisted variants, linking them to binary quasi‑cyclic codes via Gray maps, studying duality, and classifying permutation‑inequivalent codes for lengths up to 7.

Abstract

Let $I_{2}$ be the commutative non-unitary ring of order $4$ arising in the classification of Fine. In this paper, we investigate cyclic codes over $I_{2}$ through their associated residue and torsion codes over $F_{2}$ . We introduce the notions of twisted and untwisted cyclic codes and characterize cyclicity in terms of a compatibility condition involving the twist map and the cyclic shift. Connections between cyclic codes over $I_{2}$ and binary quasi-cyclic codes are established via Gray maps. In particular, we show that the Gray image of a cyclic code over $I_{2}$ is a binary quasi-cyclic code of index $2$ . We also study duality properties of cyclic codes over $I_{2}$ and prove that the dual of a cyclic code is again cyclic. Finally, we classify permutation inequivalent cyclic codes over $I_{2}$ for lengths $n \leq 7$ and determine various structural properties of these codes.

29 pages

Topics & keywords

#cyclic codes #non-unitary ring #gray map #binary quasi-cyclic codes #code duality #code classificationI₂ ringtwist mapresidue codetorsion codeGray imagepermutation inequivalence