Quantum Error Correction with Girth-16 Non-Binary LDPC Codes via Affine Permutation Construction

Kenta Kasai

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.

Kasai Kenta, Manabu Hagiwara, Hideki Imai +1 more

Koki Okada, Kouki OKADA, Kenta Kasai

Mohammad Gholami, Masoumeh Alinia

Low‐density parity‐check (LDPC) codes from affine permutation matrices , called APM‐LDPC codes, are a class of LDPC codes whose parity‐check matrices consist of zero matrices or APMs of the same orders. APM‐LDPC codes are not quasi‐cyclic (QC), in general. In this study, necessary and sufficient conditions are provided for an APM‐LDPC code to have cycles of length 2 l , l ≥ 2, and a deterministic algorithm is given to generate APM‐LDPC codes with a given girth. Unlike Type‐I conventional QC‐LDPC codes, the constructed ( J , L ) APM‐LDPC codes with the J × L all‐one base matrix can achieve minimum distance greater than ( J + 1)! and girth larger than 12. Moreover, the lengths of the constructed APM‐LDPC codes, in some cases, are smaller than the best known lengths reported for QC‐LDPC codes with the same base matrices. Another significant advantage of the constructed APM‐LDPC codes is that they have remarkably fewer cycle multiplicities compared with QC‐LDPC codes with the same base matrices and the same lengths. Simulation results show that the binary and non‐binary constructed APM‐LDPC codes with lower girth outperform QC‐LDPC codes with larger girth.

学 萩原, Manabu Hagiwara, Kenta Kasai +3 more

Jaeil Lim, Jaewon Chung, Donghun Jeong +2 more

!Non-binary low density parity check (NB-LDPC) codes are known to have higher correction capability than their binary counterparts, low density parity check (LDPC) codes. The latest technique for analyzing NB-LDPC codes is to measure generalized girth. The generalized girth cannot be measured in a binary LDPC code, and can be considered an indicator for measuring the performance of NB-LDPC code. In this paper, we propose a new analysis method to represent NB-LDPC code as a graph. This graph is called a permutation graph. In this paper, we present and prove a reasoning for analyzing generalized girth using the permutation graph. This is a faster and simpler method than the existing generalized girth analysis method. The proposed code construction algorithm constructs NB-LDPC codes to have an arbitrary generalized girth length. Because the permutation graph-based analysis is applied, which is more effective than existing methods, the complexity of code construction is significantly reduced. This enables more efficient code construction. The decoding simulations show that the proposed method improves the correction performance of NB-LDPC codes compared to the existing state-of-the-art.