Hamming and Symbol-Pair Distances of Constacyclic Codes of Length $2p^s$ over $\frac{\mathbb{F}_{p^m}[u, v]}{\langle u^2, v^2, uv-vu\rangle}$

TL;DR

This work determines the Hamming distance $d_H$ and symbol-pair distance $d_{sp}$ for $(\alpha_1 + \alpha_2 u + \alpha_3 v + \alpha_4 uv)$-constacyclic codes of length $2p^s$ over the non-chain ring $\mathcal{R}=\frac{\mathbb{F}_{p^m}[u, v]}{\langle u^2, v^2, uv-vu\rangle}$ with $p$ an odd prime. The authors classify the ideals when $\alpha$ is a unit into Type A–D, analyze the torsion and residue components via a reduction $\psi_u$, and obtain the codeword counts $\eta_{\mathcal{C}}=|\mathcal{C}|=|\text{Tor}(\mathcal{C})|\cdot|\text{Res}(\mathcal{C})|$. They then derive explicit, piecewise expressions for the Hamming distance across these types (including reductions to corresponding $\mathbb{F}_{p^m}$-codes) and analogously determine the symbol-pair distances, enriching the distance profiles for repeated-root constacyclic codes over a non-chain ring. The results have implications for code design in symbol-pair read channels and extend the landscape of exact distance determinations for constacyclic codes over finite rings.

Abstract

Let $p$ be an odd prime. In this paper, we have determined the Hamming distances for constacyclic codes of length $2p^s$ over the finite commutative non-chain ring $\mathcal{R}=\frac{\mathbb{F}_{p^m}[u, v]}{\langle u^2, v^2, uv-vu\rangle}$. Also their symbol-pair distances are completely obtained.

Theorems & Definitions (30)

• Proposition 2.1
• Proposition 2.2
• Theorem 3.1
• Theorem 4.1
• Theorem 4.2
• proof
• Theorem 4.3
• proof
• Theorem 4.4
• proof