From Skew-Cyclic Codes to Asymmetric Quantum Codes

TL;DR

This paper introduces a new additive map transforming classical codes over  into structures useful for constructing asymmetric quantum codes, revealing new relationships between cyclic codes and quantum error correction.

Contribution

It presents a novel additive map with structural properties linking classical -codes to quantum code construction, especially for cyclic and module -cyclic codes.

Findings

The map $S$ transforms linear -codes into additive codes with specific parameters.

$S(C)$ is self-orthogonal under the trace Hermitian inner product, aiding quantum code design.

The map preserves nestedness, useful for constructing asymmetric quantum codes.

Abstract

We introduce an additive but not $\mathbb{F}_4$-linear map $S$ from $\mathbb{F}_4^{n}$ to $\mathbb{F}_4^{2n}$ and exhibit some of its interesting structural properties. If $C$ is a linear $[n,k,d]_4$-code, then $S(C)$ is an additive $(2n,2^{2k},2d)_4$-code. If $C$ is an additive cyclic code then $S(C)$ is an additive quasi-cyclic code of index $2$. Moreover, if $C$ is a module $\theta$-cyclic code, a recently introduced type of code which will be explained below, then $S(C)$ is equivalent to an additive cyclic code if $n$ is odd and to an additive quasi-cyclic code of index $2$ if $n$ is even. Given any $(n,M,d)_4$-code $C$, the code $S(C)$ is self-orthogonal under the trace Hermitian inner product. Since the mapping $S$ preserves nestedness, it can be used as a tool in constructing additive asymmetric quantum codes.