Quasi-cyclic Hermitian construction of binary quantum codes

TL;DR

This work develops a sufficient condition for 2-generator self-orthogonal quasi-cyclic codes of index 2 over $F_4$ under the Hermitian inner product and leverages the Hermitian construction to produce binary quantum codes. By characterizing dual-containing QC codes via generator polynomials $g_1(x), g_2(x)$ and a linking polynomial $ u(x)$, the authors derive a family of pure binary quantum codes with parameters $[[2n, 2n-2\\deg(g_1(x)) - 2\\deg(g_2(x)), d]]_{2}$, where $d$ is the minimum weight of the QC code. They provide explicit constructions with $n=42,74,78$, achieving $d=9$, along with 24 derived codes that beat existing lower bounds in Grassl's table. The results demonstrate the viability of 2-generator QC codes and the Hermitian construction for advancing quantum error correction, offering new high-distance binary quantum codes and motivating further algebraic investigations into structure–distance trade-offs.

Abstract

In this paper, we propose a sufficient condition for a family of 2-generator self-orthogonal quasi-cyclic codes with respect to Hermitian inner product. Supported in the Hermitian construction, we show algebraic constructions of good quantum codes. 30 new binary quantum codes with good parameters improving the best-known lower bounds on minimum distance in Grassl's code tables \cite{Grassl:codetables} are constructed.