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arXiv:2610.19451

High-Girth QLDPC Codes from Circulant and Affine Permutation Matrices

Vahid Nourozi, Roxana Smarandache, David G. M. Mitchell

Published Oct 14, 2026• 0 citations
arXivHTMLPDF

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 $N\ge37$. 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.

DOI
10.48550/arxiv.2610.19451

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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 \code6N2N+26\code{6N}{2N+2}{6}\code6N2N+26 with Tanner graph girth twelve for N=31N=31N=31 and every N≥37N\ge37N≥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]][[192,16,8]][[192,16,8]], [[288,16,12]][[288,16,12]][[288,16,12]], [[384,16,14]][[384,16,14]][[384,16,14]], and [[576,32,20]][[576,32,20]][[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.
We construct a quantum low-density parity-check code family from a length-512512512 Calderbank--Shor--Steane base matrix pair. The base pair is permutation-equivalent to the known SPC(3) product CSS code, and the present affine-coset description gives a direct proof that both Tanner graphs are (3,8)(3,8)(3,8)-regular with girth 888. The base code has parameters [[512,174,8]][[512,174,8]][[512,174,8]]. We then apply circulant permutation matrix (CPM) lifts. The main decoding experiment uses the CPM-lifted code with lift factor P=32P=32P=32, which has parameters [[16384,4142,18≤d≤32]][[16384,4142,18\le d\le32]][[16384,4142,18≤d≤32]], under the code-capacity depolarizing model. Complete enumeration excludes nontrivial logical representatives of weight at most 161616 on both sides, giving dX,dZ≥18d_X,d_Z\ge18dX​,dZ​≥18. Explicit weight-323232 non-stabilizer logical representatives give the upper bounds dX,dZ≤32d_X,d_Z\le32dX​,dZ​≤32. A belief-propagation decoder with post-processing achieved frame error rate about 10−810^{-8}10−8 at p=0.085p=0.085p=0.085; an independently observed logical residual of weight 404040 is consistent with the sharper structural bound.
We study square-base Calderbank--Shor--Steane (CSS) hypergraph-product codes as a finite-length class for regular high-girth quantum low-density parity-check (LDPC) design. For base matrices of small column weight, we give checkable conditions for regularity, rank deficiency, and short-cycle exclusion, and we present explicit column-weight-three and column-weight-four examples with Tanner girth 6 and 8. We also analyze circulant permutation matrix (CPM) lifts of this class. Using the standard voltage-sum criterion, we identify orthogonality-forced Tanner 8-cycles and show that CPM lifting cannot raise the Tanner girth beyond 8 when these cycles are present. As a representative finite-length instance, a randomized CPM lift of the girth-8 base construction gives a [[28800,62,18≤d≤192]][[28800,62,18\le d\le192]][[28800,62,18≤d≤192]] girth-8 (3,6)(3,6)(3,6)-regular CSS-LDPC code. Complete enumeration excludes nontrivial logical supports of weight at most 161616 on both sides, giving dX,dZ≥18d_X,d_Z\ge18dX​,dZ​≥18; explicit weight-192192192 representatives give dX,dZ≤192d_X,d_Z\le192dX​,dZ​≤192. Under degeneracy-aware belief-propagation decoding with optional ordered-statistics-decoding-lite post-processing, this code produced zero decoding failures in 2.993×1082.993\times 10^82.993×108 independent trials at depolarizing probability p=0.1402p=0.1402p=0.1402; the Wilson 95\% upper confidence bound is 1.28×10−81.28\times 10^{-8}1.28×10−8.
We construct a rate-2/32/32/3 quantum low-density parity-check (LDPC) code from a (3,18)(3,18)(3,18)-regular two-branch finite-field base and a circulant-permutation-matrix (CPM) lift of degree P=101P=101P=101. The resulting code is a Calderbank--Shor--Steane (CSS) code with parameters [[34542,23032,18]][[34542,23032,18]][[34542,23032,18]]. Its distance is established by a computer-assisted proof: an explicit nontrivial logical operator gives the upper bound, and a complete symmetry-reduced enumeration excludes every nonzero vector of weight below 18 in both parity-check kernels. We also prove that every affine-in-ttt CPM lift of the same base satisfying CSS orthogonality has distance at most 18 for every prime lift degree P>19P>19P>19. The construction has row weight 18 and column weight 3, and the Tanner graphs of HXH_XHX​ and HZH_ZHZ​ separately have girth 8. Decoder experiments with log-likelihood-ratio (LLR) joint belief propagation (BP) and deterministic post-processing show no failures in 10810^8108 trials at p=0.01p=0.01p=0.01, and a finite-length frame error rate (FER) sweep estimates the transition near p=0.029p=0.029p=0.029.
We introduce the pair-partition (PP) construction of binary Calderbank--Shor--Steane quantum low-density parity-check codes from circulant permutation matrices. A square array of pair partitions imposes linear paired-difference equations on the CPM exponents and thereby guarantees CSS orthogonality. Pairing graphs derived from this array allow the combinatorial design to be screened before exponent search. We further give and prove a complete algorithm for verifying lower bounds on the quantum minimum distance of a fixed CSS lift. The algorithm searches for zero-syndrome vectors outside the opposing stabilizer row space, uses only rigorously valid pruning rules, and finds no vector through a prescribed weight if and only if the corresponding distance exceeds that weight. For fixed column weight, row weight, and search limit, cyclic symmetry makes the combinatorial support-enumeration bound independent of the lift size, although matrix preprocessing and row-space tests can still depend on the lift size. Thus the construction stage and the distance-verification stage are both specified by directly checkable finite procedures.
We propose a method for constructing quantum error-correcting codes using non-binary low-density parity-check codes whose Tanner graphs have girth 16. While conventional constructions based on circulant permutation matrices are limited to a maximum girth of 12, our approach leverages affine permutation matrices combined with a randomized sequential selection strategy to eliminate short cycles and achieve girth 16. Numerical experiments show that the proposed codes significantly reduce the number of low-weight codewords. Joint belief propagation decoding over depolarizing channels reveals that although a slight degradation appears in the waterfall region, a substantial improvement is achieved in the error floor performance. We also evaluated the minimum distance of the proposed codes and found that they achieve a larger upper bound than conventional constructions.
Classical low-density parity-check (LDPC) codes are a widely deployed and well-established technology, forming the backbone of modern communication and storage systems. It is well known that, in this classical setting, increasing the girth of the Tanner graph while maintaining regular degree distributions leads simultaneously to good belief-propagation (BP) decoding performance and large minimum distance. In the quantum setting, however, this principle does not directly apply because quantum LDPC codes must satisfy additional orthogonality constraints between their parity-check matrices. When one enforces both orthogonality and regularity in a straightforward manner, the girth is typically reduced and the minimum distance becomes structurally upper bounded. In this work, we overcome this limitation by using permutation matrices with controlled commutativity and by restricting the orthogonality constraints to only the active part of the construction, while preserving regular check-matrix structures. This design circumvents conventional structural distance limitations induced by parent-matrix orthogonality, and enables the construction of quantum LDPC codes with large girth while avoiding latent low-weight logical operators. As a concrete demonstration, we construct a girth-8, (3,12)-regular [[9216,4612,≤48]][[9216,4612, \leq 48]][[9216,4612,≤48]] quantum LDPC code and show that, under BP decoding combined with a low-complexity post-processing algorithm, it achieves a frame error rate as low as 10−810^{-8}10−8 on the depolarizing channel with error probability 4%4 \%4%.