New quantum codes from dual-containing cyclic codes over finite rings

TL;DR

This paper introduces new quantum error-correcting codes derived from dual-containing cyclic codes over a finite ring, expanding the types of codes available for quantum information protection.

Contribution

The paper constructs two new families of quantum codes from dual-containing cyclic codes over a specific finite ring, including new Gray maps and conditions for code existence.

Findings

Developed a new Gray map over the ring R.

Established necessary and sufficient conditions for dual-containing cyclic codes over R.

Generated new families of quantum codes over 2^m-ary and binary systems.

Abstract

Let $R=\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}+\cdots+u^{k}\mathbb{F}_{2^{m}}$ , where $\mathbb{F}_{2^{m}}$ is a finite field with $2^{m}$ elements, $m$ is a positive integer, $u$ is an indeterminate with $u^{k+1}=0.$ In this paper, we propose the constructions of two new families of quantum codes obtained from dual-containing cyclic codes of odd length over $R$. A new Gray map over $R$ is defined and a sufficient and necessary condition for the existence of dual-containing cyclic codes over $R$ is given. A new family of $2^{m}$-ary quantum codes is obtained via the Gray map and the Calderbank-Shor-Steane construction from dual-containing cyclic codes over $R.$ Furthermore, a new family of binary quantum codes is obtained via the Gray map, the trace map and the Calderbank-Shor-Steane construction from dual-containing cyclic codes over $R.$