Quantum Bicycle LDPC Codes with High $k{d}^2/n$ from Divisor-Driven Search

Abstract

Bicycle (two-block circulant) quantum low-density parity-check (LDPC) codes include some of the best known small quantum codes, yet their design has relied on group-algebra formulations in which the dimension and distance are accessible only through matrix computation. We show that in the cyclic case the construction collapses into the polynomial ring $\F_2[x]/(x^{l}-1)$: self-orthogonality is automatic, the quantum dimension is read off from a polynomial gcd, and the minimum distance is certified exactly through the Calderbank correspondence to additive codes over $\F_4$, turning code search into an algebraically pre-filtered enumeration that reaches parameter regimes poorly covered by existing tables. A computer search based on this framework recovers the short codes $[[42,12,4]{]}_2$ and $[[62,12,4]{]}_2$ and produces a family of codes with competitive figure of merit $k{d}^2/n$, including $[[66,20,7]{]}_2$ with $k{d}^2/n=14.85$, above the bivariate bicycle code $[[144,12,12]{]}_2$ ($k{d}^2/n=12$) at less than half the block length, together with $[[46,2,8]{]}_2$, $[[66,2,9]{]}_2$, $[[66,4,8]{]}_2$, $[[66,6,8]{]}_2$ and, at $n=90$, $[[90,16,6]{]}_2$, $[[90,18,6]{]}_2$, $[[90,20,6]{]}_2$. An exhaustive census at $n=48$ delineates the boundary of this picture: we exhibit a $[[48,10,6]{]}_2$ code from a minimal $48$-element group (the Aydin--Tamo--Barg realization uses $72$ elements), and prove that distance $5$ forces a stabilizer-rank loss, which excludes $[[48,10,5]{]}_2$ from the weight-$8$ symmetric coset family. The framework thus opens a systematic route to bicycle-type quantum LDPC codes beyond the reach of group-theoretic searches, and identifies exactly where genuinely coset-theoretic phenomena begin.