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

2026-05-22

AI Summary

This work addresses the construction of quantum LDPC base matrices satisfying regularity, CSS orthogonality, and the absence of 4-cycles of the same type by proposing a finite-field-based dual-branch multiplicative coset method. The design is decomposed into two stages: base matrix construction and cyclic lifting. The former explicitly encodes degree distribution and girth constraints via quotient coset conditions, while the latter employs algebraic randomization to establish edge connections. The proposed framework accommodates various $(J,L)$ degree distributions, offering both flexibility and structural rigor. Experimentally, a CSS code with parameters $[[10240,4108,10\leq d\leq32]]$ is successfully constructed, achieving a frame error rate of $1.0\times10^{-7}$ at a physical error rate of $p=0.058$, exhibiting a Tanner graph girth of at least 8, and excluding weight-16 non-degenerate logical error support orbits.

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. 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 \(1.0\times10^{-7}\).

Problem

• CSS codes
• LDPC
• finite-field construction
• regularity
• girth constraints

Innovation

• two-branch construction
• finite-field coset
• regular CSS LDPC
• girth optimization
• syndrome post-processing