Systematic Non-Binary Extension of LDPC-CSS Codes Preserving Orthogonality
Kenta Kasai
TL;DR
The paper addresses extending binary CSS codes to non-binary representations over a finite field while preserving the original parity-check support and the orthogonality condition. It recasts the problem as a sparse system of linear congruences on exponent variables and develops efficient solution strategies, including Smith normal form-based analysis and a fraction-free row-elimination approach, to obtain feasible non-binary assignments. A canonical separable assignment (CSA) is presented as a baseline that guarantees orthogonality for any even-overlap pattern but preserves all binary codewords, highlighting the need for solving exponent-congruence equations to improve distance properties. The proposed framework enables systematic construction of non-binary LDPC-CSS and related codes, with potential impact on decoding performance via joint belief-propagation and applicability to hypergraph-product and QC-LDPC code families.
Abstract
We study finite-field extensions that preserve the same support as the parity-check matrices defining a given binary CSS code. Here, an LDPC-CSS code refers to a CSS code whose parity-check matrices are orthogonal in the sense that each pair of corresponding rows overlaps in an even (possibly zero) number of positions, typically at most twice in sparse constructions. Beyond the low-density setting, we further propose a systematic construction method that extends to arbitrary CSS codes, providing feasible finite-field generalizations that maintain both the binary support and the orthogonality condition.