New quantum codes from constacyclic codes over finite chain rings

TL;DR

This paper introduces new quantum codes derived from constacyclic codes over finite chain rings, utilizing novel Gray maps and constructions to achieve self-orthogonality properties for improved quantum error correction.

Contribution

It develops new Gray maps and construction methods to generate quantum codes from constacyclic codes over finite chain rings, expanding the types of quantum codes available.

Findings

New classes of $2^{m}$-ary quantum codes constructed

New classes of $p^{m}$-ary quantum codes constructed

Enhanced quantum error correction capabilities demonstrated

Abstract

Let $R$ be the finite chain ring $\mathbb{F}_{p^{2m}}+{u}\mathbb{F}_{p^{2m}}$, where $\mathbb{F}_{p^{2m}}$ is the finite field with $p^{2m}$ elements, $p$ is a prime, $m$ is a non-negative integer and ${u}^{2}=0.$ In this paper, we firstly define a class of Gray maps, which changes the Hermitian self-orthogonal property of linear codes over $\mathbb{F}_{2^{2m}}+{u}\mathbb{F}_{2^{2m}}$ into the Hermitian self-orthogonal property of linear codes over $\mathbb{F}_{2^{2m}}$. Applying the Hermitian construction, a new class of $2^{m}$-ary quantum codes are obtained from Hermitian constacyclic self-orthogonal codes over $\mathbb{F}_{2^{2m}}+{u}\mathbb{F}_{2^{2m}}.$ We secondly define another class of maps, which changes the Hermitian self-orthogonal property of linear codes over $R$ into the trace self-orthogonal property of linear codes over $\mathbb{F}_{p^{2m}}$. Using the Symplectic construction, a new class of $p^{m}$-ary quantum codes are obtained from Hermitian constacyclic self-orthogonal codes over $R.$