🤖 AI Summary
This work addresses the challenge of efficiently determining the dimension and distance of quantum Bicycle LDPC codes, which traditionally rely on group-algebraic constructions that hinder the discovery of high-performance short codes. The authors reformulate the construction of cyclic Bicycle codes as a purely algebraic problem in the polynomial ring $\mathbb{F}_2[x]/(x^\ell - 1)$. Exploiting the fact that self-orthogonality is automatically satisfied, they show that the code dimension is directly given by a polynomial greatest common divisor, while the distance can be precisely verified via the Calderbank correspondence mapping to additive codes over $\mathbb{F}_4$. This enables efficient pre-screening and systematic enumeration, bypassing group-theoretic limitations. For the first time, an exhaustive algebraic search of Bicycle codes becomes feasible, revealing the onset boundary of the co-set phenomenon and yielding new high-performance codes such as $[[66,20,7]]_2$ ($kd^2/n=14.85$) and $[[46,2,8]]_2$. A complete census at $n=48$ disproves the existence of a $[[48,10,5]]_2$ code, thereby clarifying fundamental construction limits.
📝 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 $kd^2/n$, including $[[66,20,7]]_2$ with $kd^2/n=14.85$, above the bivariate bicycle code $[[144,12,12]]_2$ ($kd^2/n=12$) 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.