A Two-Branch Finite-Field Construction for Regular CSS LDPC Bases
Abstract
This paper develops a two-branch multiplicative-coset construction for regular Calderbank--Shor--Steane (CSS) quantum low-density parity-check base matrices. For a target column weight J and an even row weight L, the method reduces regularity, CSS orthogonality, and same-type 4-cycle exclusion to explicit quotient-coset conditions over a finite field. A normalized exhaustive search for these conditions produces base matrices for several (J,L) pairs, so the construction is not tied to a single degree distribution. The construction separates the finite-length design into two stages: the base matrix fixes the degree distribution and the first girth constraints, and a cyclic lift randomizes edge connections subject to exact algebraic checks. As a detailed example, we carry one (3,10)-regular base through the lift and decoding stages. For this example, the selected 64-fold lift gives a code whose same-type Tanner graphs have girth at least eight, and it also excludes a specified weight-16 nondegenerate logical-support orbit. Complete support enumeration excludes all nontrivial logical representatives of weight at most 16 on both sides. The resulting instance is a [[10240,4108, 18≤ d≤32]] CSS code. For decoding, we use joint log-domain belief propagation together with low-complexity deterministic post-processing rules for small residual syndromes, including repairs for residual patterns with two unsatisfied checks. The frame error rate (FER) measurements provide finite-length decoding data for this detailed example; at depolarizing probability p=0.058, the post-processing FER is 1.0×10−7.
Related
- Rate-2/3 Girth-8 (3,18)-Regular Quantum LDPC Codes from Two-Branch Finite-Field Bases and CPM Lifts
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- Efficient Construction of T-Mesh Bases via a Constraint Transformation Matrix
- Two-element factors and finite bases for additively idempotent semirings
BibTeX
@misc{okada2026twobranch,
title = {{A Two-Branch Finite-Field Construction for Regular CSS LDPC Bases}},
author = {Koki Okada and Kenta Kasai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.23894},
eprint = {2605.23894},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.23894},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Two-Branch Finite-Field Construction for Regular CSS LDPC Bases
AU - Koki Okada
AU - Kenta Kasai
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2605.23894
UR - https://arxiv.org/abs/2605.23894
AB - This paper develops a two-branch multiplicative-coset construction for regular Calderbank--Shor--Steane (CSS) quantum low-density parity-check base matrices. For a target column weight J and an even row weight L, the method reduces regularity, CSS orthogonality, and same-type 4-cycle exclusion to explicit quotient-coset conditions over a finite field. A normalized exhaustive search for these conditions produces base matrices for several (J,L) pairs, so the construction is not tied to a single degree distribution. The construction separates the finite-length design into two stages: the base matrix fixes the degree distribution and the first girth constraints, and a cyclic lift randomizes edge connections subject to exact algebraic checks. As a detailed example, we carry one (3,10)-regular base through the lift and decoding stages. For this example, the selected 64-fold lift gives a code whose same-type Tanner graphs have girth at least eight, and it also excludes a specified weight-16 nondegenerate logical-support orbit. Complete support enumeration excludes all nontrivial logical representatives of weight at most 16 on both sides. The resulting instance is a [[10240,4108, 18≤ d≤32]] CSS code. For decoding, we use joint log-domain belief propagation together with low-complexity deterministic post-processing rules for small residual syndromes, including repairs for residual patterns with two unsatisfied checks. The frame error rate (FER) measurements provide finite-length decoding data for this detailed example; at depolarizing probability p=0.058, the post-processing FER is 1.0×10^−7.
ER -