Construction of Self-Orthogonal Quasi-Cyclic Codes and Their Application to Quantum Error-Correcting Codes

TL;DR

This paper establishes conditions for self-orthogonality of quasi-cyclic codes under various inner products and constructs quantum error-correcting codes with optimal parameters based on these conditions.

Contribution

It provides necessary and sufficient conditions for self-orthogonality of 2-generator QC codes and applies these to construct high-performance quantum codes.

Findings

Derived conditions for self-orthogonality under Euclidean, Hermitian, and symplectic inner products.

Constructed quantum stabilizer and synchronizable codes with parameters matching best-known codes.

Generalized many known codes through the study of dual codes of 2-generator QC codes.

Abstract

In this paper, necessary and sufficient conditions for the self-orthogonality of t-generator quasi-cyclic (QC) codes are presented under the Euclidean, Hermitian, and symplectic inner products, respectively. Particularly, by studying the structure of the dual codes of a class of 2-generator QC codes, we derive necessary and sufficient conditions for the QC codes to be dual-containing under the above three inner products. This class of 2-generator QC codes generalizes many known codes in the literature. Based on the above conditions, we construct several quantum stabilizer codes and quantum synchronizable codes with good parameters, some of which share parameters with certain best-known codes listed in Grassl's code table.