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 \[[10240,4108, 10le dle32\]] 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 1.0times10^-7.