Vaishali Singh | Searcharxiv

arXiv2026

Two-dimensional constacyclic codes over finite chain rings

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.

cs.IT↗

arXiv2026

On the structure of constacyclic codes over finite chain rings

In the present paper, we provide an explicit construction for generators of a $λ$-constacyclic code $\mathcal{C}$ of arbitrary length $\ell$ over a finite chain ring(FCR) $\mathcal{R}$ in terms of certain minimum degree polynomials of the ring $\mathcal{R}[x]/ \langle x^{\ell}-λ\rangle$. Moreover, the proposed construction achieves the minimum possible number of generators. We prove certain properties of this set of generators, using which we obtain a minimal spanning set of $\mathcal{C}$. We also obtain that the rank of $\mathcal{C}$ is $\ell-n_0$, where $n_0$ is the degree of the minimal degree polynomial in $\mathcal{C}$. Finally, we derive necessary and sufficient conditions under which an arbitrary length $λ$-constacyclic code $\mathcal{C}$ over $\mathcal{R}$ is Maximum Hamming Distance with respect to Rank(MHDR) as well as Maximum Distance Separable(MDS) in terms of a torsion code of $\mathcal{C}$ over the residue field $\mathbb{F}_q$ of $\mathcal{R}$. We further determine the exact values for $n_0$ for which $\mathcal{C}$ over $\mathcal{R}$ is MHDR.

cs.IT↗