Polycyclic codes over serial rings and their annihilator CSS construction

TL;DR

This work extends polycyclic codes to the serial ring $\mathscr R=R[x_1,\ldots,x_s]/\langle t_1(x_1),\ldots,t_s(x_s)\rangle$ with $R$ a chain ring, establishing a comprehensive framework that decomposes codes into chain-ring components, defines $f(x)$-polycyclic-Q2DP/QsDP structures, and links Euclidean and annihilator dualities. It develops an annihilator dual theory that preserves polycyclic structure, provides necessary and sufficient conditions for annihilator self-duality/orthogonality/LCD/dual-containing, and proves a non-degenerate annihilator product with $|\mathscr C|\,|\mathscr C^{\circ}|=|\mathscr R|^n$. Building on this, the paper introduces an annihilator CSS construction that yields quantum stabilizer codes from annihilator dual-preserving $f(x)$-polycyclic codes over Frobenius rings, including a concrete $[[5,3,2]]_{3^4}$ example via a Gray map. Together, these results broaden the class of polycyclic codes available for quantum code construction and provide a principled duality framework suited to serial-ring settings.

Abstract

In this paper, we investigate the algebraic structure for polycyclic codes over a specific class of serial rings, defined as $\mathscr R=R[x_1,\ldots, x_s]/\langle t_1(x_1),\ldots, t_s(x_s) \rangle$, where $R$ is a chain ring and each $t_i(x_i)$ in $R[x_i]$ for $i\in\{1,\ldots, s\}$ is a monic square-free polynomial. We define quasi-$s$-dimensional polycyclic codes and establish an $R$-isomorphism between these codes and polycyclic codes over $\mathscr R$. We provide necessary and sufficient conditions for the existence of annihilator self-dual, annihilator self-orthogonal, annihilator linear complementary dual, and annihilator dual-containing polycyclic codes over this class of rings. We also establish the CSS construction for annihilator dual-preserving polycyclic codes over the chain ring $R$ and use this construction to derive quantum codes from polycyclic codes over $\mathscr{R}$.

Theorems & Definitions (50)

• Proposition 1: Moro_2006
• Proposition 2
• Theorem 1
• Lemma 1
• proof
• Proposition 3
• proof
• Remark 1
• Theorem 2
• proof