Non-binary LDPC codes for Data Storage

2026-05-08

AI Summary

This work addresses the challenge of efficient and reliable recovery from disk failures—modeled as erasures—by investigating non-binary LDPC codes constructed via expansion of binary base matrices over the $q$-ary erasure channel, thereby challenging the dominance of traditional MDS codes. The study introduces a novel concept termed “ultimate distance” to tightly characterize the upper bound on minimum distance, and develops a low-complexity algorithm for minimum distance search. By integrating maximum-likelihood decoding with algebraic graph theory, the authors derive a random coding bound on the number of uncorrectable erasure patterns. Leveraging these insights, they successfully construct multiple families of regular non-binary LDPC codes that achieve the ultimate distance, thereby approaching the theoretical performance limits.

Abstract

In modern data storage systems, non-binary LDPC codes for recovering from disk failures are increasingly considered strong competitors to MDS codes such as Reed-Solomon codes. Since disk failures can be modeled as erasures, we analyze non-binary LDPC codes over a $q$-ary field in the $q$-ary erasure channel, relative to MDS codes. Our focus is on non-binary LDPC codes whose parity-check matrix is obtained by replacing the non-zero entries of a binary base matrix by elements of a $q$-ary finite field. For such LDPC codes, we introduce the notion of ultimate distance, which upper-bounds their minimum distance. We derive a random-coding bound on the number of non-correctable erasure patterns for the Gallager ensemble of regular non-binary LDPC codes under maximum-likelihood decoding. An algorithm for finding the ultimate distance is presented. A low-complexity algorithm for searching for the minimum distance of the non-binary LDPC code is proposed. Finally, we construct examples of non-binary LDPC codes achieving the ultimate distance.

Problem

• non-binary LDPC codes
• data storage
• erasure channel
• minimum distance
• MDS codes

Innovation

• non-binary LDPC codes
• ultimate distance
• erasure channel
• minimum distance algorithm
• Gallager ensemble