Systematic Non-Binary Extension of LDPC-CSS Codes Preserving Orthogonality

2025-10-29

AI Summary

This work addresses the systematic construction of non-binary LDPC-CSS codes. We propose a general framework based on finite field extension: starting from an arbitrary binary CSS code, we lift its parity-check matrix to an arbitrary finite field while preserving the original support structure (i.e., zero/non-zero pattern) and strictly maintaining inter-row orthogonality (even overlap). Unlike conventional LDPC constructions, our method transcends sparsity constraints and applies universally to any binary CSS code without altering its underlying topology. We provide rigorous theoretical guarantees that the extended codes retain sparsity, orthogonality, and CSS compatibility. Experimental results demonstrate improved encoding flexibility and potential gains in error-correction performance. The key innovation lies in establishing, for the first time, an algebraic extension paradigm that simultaneously preserves support structure and enforces orthogonality conservation—yielding a universal tool for designing high-dimensional quantum LDPC codes.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Quantum Machine Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Extends binary LDPC-CSS codes to non-binary fields while preserving support
• Maintains orthogonality condition between parity-check matrices during extension
• Generalizes construction method to arbitrary CSS codes beyond sparse settings

Innovation

• Extends LDPC-CSS codes to non-binary fields
• Preserves binary support structure of parity-check matrices
• Maintains orthogonality condition in finite-field generalization