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October 16, 2025Open Access

Duality of Codes over Non-unital Rings of Order Six

Key Points

The study identifies duality between codes over non-unital rings, enhancing understanding of coding structures.
Self-orthogonal, self-dual, and quasi self-dual codes are classified, yielding insight into their properties.
The construction of cyclic and linear complementary dual codes adds new dimensions to code theory over these rings.
Building-up methods for self-orthogonal codes highlight potential applications in coding theory.

Abstract

We present some basic theory on the duality of codes over two non-unital rings of order 6, namely H₂₃ and H₃₂. For a code C over these rings, we associate a binary code Cₐ and a ternary code Cb. We characterize self-orthogonal, self-dual and quasi self-dual (QSD) codes over these rings using the codes Cₐ and Cb. In addition, we present a building-up construction for self-orthogonal codes, introduce cyclic codes and linear complementary dual (LCD) codes. We also gave a classification of self-orthogonal codes for short lengths.

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Cite This Study

Alshuhail et al. (Tue,) studied this question.

synapsesocial.com/papers/68f0d5eb105731330a2b212a https://doi.org/https://doi.org/10.48550/arxiv.2502.18069

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