Expander-like Codes based on Finite Projective Geometry

TL;DR

This paper introduces a new error correcting code based on finite projective geometry with a novel decoding algorithm, demonstrating superior error correction performance and efficient FPGA implementation for storage media applications.

Contribution

The paper proposes a new expander-like code based on finite projective geometry and Reed-Solomon codes, with a modified decoding algorithm and proven improved error correction bounds.

Findings

Error correction capability surpasses previous graph codes.

Simulation shows 10x better average performance than geometric bounds.

FPGA implementation achieves optimal throughput.

Abstract

We present a novel error correcting code and decoding algorithm which have construction similar to expander codes. The code is based on a bipartite graph derived from the subsumption relations of finite projective geometry, and Reed-Solomon codes as component codes. We use a modified version of well-known Zemor's decoding algorithm for expander codes, for decoding our codes. By derivation of geometric bounds rather than eigenvalue bounds, it has been proved that for practical values of the code rate, the random error correction capability of our codes is much better than those derived for previously studied graph codes, including Zemor's bound. MATLAB simulations further reveal that the average case performance of this code is 10 times better than these geometric bounds obtained, in almost 99% of the test cases. By exploiting the symmetry of projective space lattices, we have designed a corresponding decoder that has optimal throughput. The decoder design has been prototyped on Xilinx Virtex 5 FPGA. The codes are designed for potential applications in secondary storage media. As an application, we also discuss usage of these codes to improve the burst error correction capability of CD-ROM decoder.