Repeated-Root Constacyclic Codes of Length $3p^s$ over the Finite Non-Chain Ring $\frac{\mathbb{F}_{p^m}[u, v]}{\langle u^2, v^2, uv-vu\rangle}$ and their Duals
Divya Acharya, Prasanna Poojary, Vadiraja Bhatta G R
TL;DR
This work analyzes $\alpha$-constacyclic codes of length $3p^s$ over the finite non-chain ring $\mathcal{R}=\mathbb{F}_{p^m}[u,v]/\langle u^2,v^2,uv-vu\rangle$ for $p\neq3$, distinguishing when $\alpha$ is a cube in $\mathcal{R}$ versus a non-cube. For cube $\alpha$, the authors factor $x^{3p^s}-\alpha$ and apply the Chinese Remainder Theorem to decompose the ambient ring into simpler components, reducing the problem to known length-$p^s$ and length-$2p^s$ cases and yielding direct-sum descriptions of all codes. For non-cubes, they study the resulting quotient rings, show they are local but not chain rings, and classify ideals into four types (A–D) to completely describe the algebras of constacyclic codes, including counts of codewords and dual codes. The duals are obtained via annihilator and reciprocal-polynomial techniques, with explicit generators and size relations ensuring $|\mathcal{C}|\cdot|\mathcal{C}^{\perp}|=|\mathcal{R}|^{3p^s}$. Overall, the paper extends constacyclic-code theory to a broad family of repeated-root codes over a non-chain ring, providing a comprehensive classification and duals across all cube and most non-cube subcases.
Abstract
This study aims to determine the algebraic structures of $α$-constacyclic codes of length $3p^s$ over the finite commutative non-chain ring $\mathcal{R}=\frac{\mathbb{F}_{p^m}[u, v]}{\langle u^2, v^2, uv-vu\rangle}$, for a prime $p \neq 3.$ For the unit $α$, we consider two different instances: when $α$ is a cube in $\mathcal{R}$ and when it is not. Analyzing the first scenario is relatively easy. When $α$ is not a unit in $\mathcal{R}$, we consider several subcases and determine the algebraic structures of constacyclic codes in those cases. Further, we also provide the number of codewords and the duals of $α$-constacyclic codes.