Quantum Bicycle LDPC Codes with High $kd^2/n$ from Divisor-Driven Search
2026-08-10 • Information Theory
Information Theory
AI summaryⓘ
The authors study a special family of quantum error-correcting codes called bicycle quantum LDPC codes, which were previously understood mostly through complicated group-based methods. They show that when the codes are cyclic, the construction simplifies to working with polynomials, making it easier to determine key code properties like dimension and minimum distance. Using this approach, they perform computer searches that find new quantum codes with good performance metrics and clarify limitations on certain code parameters. Their work provides a clearer and more algebraic way to design these quantum codes and highlights where more complex group-based behaviors appear.
quantum error correctionLDPC codesbicycle codestwo-block circulant matricescyclic codespolynomial ringminimum distanceCalderbank correspondenceadditive codes over F4stabilizer codes
Authors
Liangdong Lu, Guanmin Guo, Yang Liu, Ruipan Yang
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.