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.
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]] 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 10−8 on the depolarizing channel with error probability 4%.
We construct a quantum low-density parity-check code family from a length-512 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)-regular with girth 8. The base code has parameters [[512,174,8]]. We then apply circulant permutation matrix (CPM) lifts. The main decoding experiment uses the CPM-lifted code with lift factor P=32, which has parameters [[16384,4142,18≤d≤32]], under the code-capacity depolarizing model. Complete enumeration excludes nontrivial logical representatives of weight at most 16 on both sides, giving dX,dZ≥18. Explicit weight-32 non-stabilizer logical representatives give the upper bounds dX,dZ≤32. A belief-propagation decoder with post-processing achieved frame error rate about 10−8 at p=0.085; an independently observed logical residual of weight 40 is consistent with the sharper structural bound.
Quasi-cyclic low-density parity-check (QC-LDPC) codes are widely used in modern communication standards due to their simple structure and efficient encoder and decoder implementations. However, the commutativity of the circulant permutation matrices used in their construction significantly limits their girth and minimum distance, affecting iterative decoding performance. In this paper, we investigate protograph-based codes constructed using permutation polynomials of higher degree, focusing on a closed family of non-commuting binomial permutation polynomials that enables these constructions to exceed both of these girth and minimum distance upper bounds while maintaining a simple representation. Simulation results demonstrate improved decoding performance compared to constructions based on affine permutation polynomials, which include QC-LDPC codes.
This study proposes an explicit construction method for quantum quasi-cyclic low-density parity-check (QC-LDPC) codes with a girth of 12. The proposed method designs parity-check matrices that maximize the girth while maintaining an orthogonal structure suitable for quantum error correction. By utilizing algebraic techniques, short cycles are eliminated, which improves error correction performance. Additionally, this method is extended to non-binary LDPC codes and spatially-coupled LDPC codes, demonstrating that both the girth and orthogonality can be preserved. The results of this study enable the design of high-performance quantum error-correcting codes without the need for random search.
Low-density parity-check (LDPC) codes based on affine permutation matrices, APM-LDPC codes, have been attracted recently, because of some advantages rather than QC-LDPC codes in minimum-distance, cycle distribution and error-rate performance. In this letter, circulant and anti-circulant permutation matrices are used to define a class of LDPC codes, called AQC-LDPC codes, which can be considered as an special case of APM-LDPC codes. In fact, each AQC-LDPC code can be verified by a sign matrix and a slope matrix which are helpful to show each cycle in the Tanner graph by a modular linear equation. For the normal sign matrix A , if -1 ϵ, A, it is shown that the corresponding AQC-LDPC code has maximum-girth 8. Finally, two explicit constructions for AQC-LDPC codes with girths 6, 8 are presented which have some benefits rather than the explicitly constructed QC and APM LDPC codes in minimum-distance, cycle distributions and bit-error-rate performances.
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]] girth-8 (3,6)-regular CSS-LDPC code. Complete enumeration excludes nontrivial logical supports of weight at most 16 on both sides, giving dX,dZ≥18; explicit weight-192 representatives give dX,dZ≤192. Under degeneracy-aware belief-propagation decoding with optional ordered-statistics-decoding-lite post-processing, this code produced zero decoding failures in 2.993×108 independent trials at depolarizing probability p=0.1402; the Wilson 95\% upper confidence bound is 1.28×10−8.