High-Girth QLDPC Codes from Circulant and Affine Permutation Matrices

Abstract

We study two explicit constructions of binary Calderbank--Shor--Steane quantum low-density parity-check (QLDPC) codes with check weight six. The first uses a fixed circulant shift pattern and yields parameters $\code{6N}{2N+2}{6}$ with Tanner graph girth twelve for $N=31$ and every $N\ge 37$. A graph incidence interpretation gives the ranks and girths, and a uniform weight six logical operator establishes the exact distance. The second construction uses fixed matrix polynomials with circulant and affine permutation coefficients. Here, a polynomial commutation identity guarantees orthogonality for every cyclic lift. Four realizations give $[[192,16,8]]$, $[[288,16,12]]$, $[[384,16,14]]$, and $[[576,32,20]]$, with Tanner girth eight and computationally certified distances. We give reproducible construction data, distinguish classical cycle distance from quantum distance, and identify the limitations of both families.