A Two-Branch Finite-Field Construction for Regular CSS LDPC Bases

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. The resulting instance is a ([[10240,4108,,10\le d\le32]]) 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\times10^{-7}).

Source: arXiv:2605.23894v1 - http://arxiv.org/abs/2605.23894v1 PDF: https://arxiv.org/pdf/2605.23894v1 Original Link: http://arxiv.org/abs/2605.23894v1