Two-dimensional constacyclic codes over finite chain rings
2026-07-10 • Information Theory
Information Theory
AI summaryⓘ
The authors study a special type of error-correcting codes called two-dimensional $(λ,μ)$-constacyclic codes over a specific kind of mathematical structure known as finite chain rings. They explore how these codes can be described and generated by breaking down the ring into simpler pieces called primitive idempotents. Additionally, the authors identify the conditions under which these codes achieve the best possible error detection, measured by something called maximum Hamming distance with respect to rank. Their work helps better understand the algebraic foundation of these codes in coding theory.
two-dimensional constacyclic codesfinite chain ringsprimitive idempotentsmaximum Hamming distance rank (MHDR)residue fieldgeneratorserror-correcting codesalgebraic structure
Authors
Vaishali Singh, Sucheta Dutt, Ridhima Thakral
Abstract
The main focus of this paper is on the algebraic structure of two-dimensional $(λ,μ)$-constacyclic codes of length $\ell\mathrm{m}$ over finite chain rings with residue field $\mathbb{F}_q$, where $q \equiv 1 \pmod{r\mathrm{m}}$ and $r$ denotes the multiplicative order of $\barμ$. In this paper, the structure of two-dimensional $(λ,μ)$-constacyclic codes is obtained. Our approach relies on analysing primitive idempotents within the finite chain ring to determine the generators of these codes. We also find the condition under which two-dimensional constacyclic codes are maximum Hamming distance with respect to rank (MHDR) over finite chain rings.