A Construction of Infinite Families of Self-Orthogonal Quasi-Cyclic Codes Using Constituent Codes

TL;DR

This paper introduces a method to construct infinite families of self-orthogonal quasi-cyclic codes over field extensions, which are useful for quantum error correction, with proven bounds on their minimum distance.

Contribution

The paper presents a novel construction of self-orthogonal quasi-cyclic codes over field extensions, including their parameters and applications to quantum error correction.

Findings

Constructed infinite families of self-orthogonal quasi-cyclic codes.

Derived lower bounds for minimum distance that follow a square-root-like pattern.

Demonstrated the application of these codes in quantum error-correcting code construction.

Abstract

Quasi-cyclic codes have been recently employed in the constructions of quantum error-correcting codes. In this paper, we propose a construction of infinite families of quasi-cyclic codes which are self-orthogonal with respect to the Euclidean and Hermitian inner products. In particular, their dimension and a lower bound for their minimum distance are computed using their constituent codes defined over field extensions of $\mathbb{F}_q$. We also show that the lower bound for the minimum distance satisfies the square-root-like lower bound and also show how self-dual quasi-cyclic codes can arise from our construction. Using the CSS construction, we show the existence of quantum error-correcting codes with good parameters.