On Eisenstein additive codes over chain rings and linear codes over mixed alphabets
Leijo Jose, Anuradha Sharma
TL;DR
The paper establishes a duality-preserving correspondence between Eisenstein additive codes over the chain ring $\mathcal{R}_e$ and mixed-alphabet linear codes over $\mathbb{Z}_{p^e}\mathbb{Z}_{p^{e-1}}$, linking character-theoretic duals to Euclidean duals. Using this bridge, it develops construction methods and exact enumeration formulas for self-orthogonal, self-dual, and complementary-dual (LCD) codes over $\mathcal{R}_e$, with explicit results for $e=2,3$ and a recursive approach for $e\ge 4$, plus an application to ACD codes and plots that demonstrate optimality under homogeneous weight. The work translates monomial equivalence of Eisenstein additive codes into $\ast$-equivalence on the mixed-alphabet level, enabling complete classifications of these codes for small lengths via computational tools. Overall, it advances both the algebraic understanding and practical construction of optimized codes over chain rings with respect to the homogeneous metric, offering a robust framework for exploring optimal additive and mixed-alphabet codes.
Abstract
Let $\mathcal{R}_e=GR(p^e,r)[y]/\langle g(y),p^{e-1}y^t\rangle$ be a finite commutative chain ring, where $p$ is a prime number, $GR(p^e,r)$ is the Galois ring of characteristic $p^e$ and rank $r,$ $t$ and $k$ are positive integers satisfying $1\leq t\leq k$ when $e \geq 2,$ while $t=k$ when $e=1,$ and $g(y)=y^k+p(g_{k-1}y^{k-1}+\cdots+g_1y+g_0)\in GR(p^e,r)[y]$ is an Eisenstein polynomial with $g_0$ as a unit in $GR(p^e,r).$ In this paper, we first establish a duality-preserving 1-1 correspondence between additive codes over $\mathcal{R}_e$ and $\mathbb{Z}_{p^e}\mathbb{Z}_{p^{e-1}}$-linear codes, where the character-theoretic dual codes of additive codes over $\mathcal{R}_e$ correspond to the Euclidean dual codes of $\mathbb{Z}_{p^e}\mathbb{Z}_{p^{e-1}}$-linear codes, and vice versa. This correspondence gives rise to a method for constructing additive codes over $\mathcal{R}_e$ and their character-theoretic dual codes, as unlike additive codes over $\mathcal{R}_e,$ $\mathbb{Z}_{p^e}\mathbb{Z}_{p^{e-1}}$-linear codes can be completely described in terms of generator matrices. We also list additive codes over the chain ring $\mathbb{Z}_4[y]/\langle y^2-2,2y \rangle$ achieving the Plotkin's bound for homogeneous weights, which suggests that additive codes over $\mathcal{R}_e$ is a promising class of error-correcting codes to find optimal codes with respect to the homogeneous metric.