Additive Conjucyclic Codes over $\F_{q^2}$: Trace Correspondence and Applications to Quantum Codes

TL;DR

This paper develops a trace-based algebraic framework for additive conjucyclic codes over finite fields, enabling their structural analysis, enumeration, and application to quantum error-correcting codes.

Contribution

It introduces a trace isomorphism linking additive conjucyclic codes to cyclic codes, and characterizes dual-containing codes for quantum code construction.

Findings

Establishes a bijection between additive conjucyclic codes and cyclic codes.

Provides explicit generator and parity-check matrices for these codes.

Constructs quantum error-correcting codes from dual-containing additive conjucyclic codes.

Abstract

Additive conjucyclic codes over $\F_{q^2}$ are closed under the conjugated cyclic shift and play an important role in constructing quantum error-correcting codes (QECCs). However, a systematic algebraic theory for such codes over general finite fields has been lacking. In this paper, we develop a unified framework by establishing a trace-based $\F_q$-linear isomorphism between $\F_{q^2}^n$ and $\F_q^{2n}$. This correspondence shows that additive conjucyclic codes of length $n$ correspond bijectively to $q$-ary linear cyclic codes of length $2n$, translating their structural analysis to the well-understood setting of cyclic codes. Using this isomorphism, we determine the enumeration of such codes and give explicit forms of their generator matrices. We then introduce an alternating inner product on $\F_{q^2}^n$, which is shown to be compatible with the symplectic inner product on $\F_q^{2n}$ under the trace isomorphism. Based on this inner product, we characterize the dual-containing condition for additive conjucyclic codes and derive explicit parity-check matrices. Finally, we construct $q$-ary QECCs from dual-containing additive conjucyclic codes. Our results unify and generalize previous studies on quaternary additive conjucyclic codes and present a construction method for $q$-ary QECCs from additive conjucyclic codes, together with an illustrative example.