Non-binary LDPC codes for Data Storage

TL;DR

This paper analyzes non-binary LDPC codes over q-ary fields for data storage, introducing the concept of ultimate distance, and provides bounds, algorithms, and code constructions demonstrating their effectiveness against erasures.

Contribution

It introduces the notion of ultimate distance for non-binary LDPC codes, derives bounds, and presents algorithms and constructions to improve erasure correction in data storage.

Findings

Derived a random-coding bound on non-correctable erasure patterns.

Presented algorithms for calculating ultimate and minimum distances.

Constructed codes achieving the ultimate distance.

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.