Quaternary Conjucyclic Codes with an Application to EAQEC Codes

TL;DR

This paper studies additive conjucyclic codes over GF(4), deriving their duals, hulls, and conditions for duality, and applies these findings to construct entanglement-assisted quantum error-correcting codes.

Contribution

It introduces new properties and conditions for additive conjucyclic codes over GF(4), including duals, hulls, and applications to quantum error correction.

Findings

Derived duals of ACC codes using trace inner product

Established conditions for additive code duality and trace code properties

Constructed effective entanglement-assisted quantum error-correcting codes

Abstract

Conjucyclic codes are part of a family of codes that includes cyclic, constacyclic, and quasi-cyclic codes, among others. Despite their importance in quantum error correction, they have not received much attention in the literature. This paper focuses on additive conjucyclic (ACC) codes over $\mathbb{F}_4$ and investigates their properties. Specifically, we derive the duals of ACC codes using a trace inner product and obtain the trace hull and its dimension. Also, establish a necessary and sufficient condition for an additive code to have a complementary dual (ACD). Additionally, we identify a necessary condition for an additive conjucyclic complementary pair of codes over $\mathbb{F}_4$. Furthermore, we show that the trace code of an ACC code is cyclic and provide a condition for the trace code of an ACC code to be LCD. To demonstrate the practical application of our findings, we construct some good entanglement-assisted quantum error-correcting (EAQEC) codes using the trace code of ACC codes.