Low density majority codes and the problem of graceful degradation

TL;DR

This paper introduces Low Density Majority Codes (LDMCs), a new class of non-linear sparse-graph codes that improve error correction performance over traditional linear codes on binary erasure channels by providing smoother degradation curves.

Contribution

The paper proposes LDMCs, analyzes their performance with belief propagation, and establishes new bounds on achievable error rates for different code subclasses.

Findings

LDMCs outperform linear systematic codes with few BP iterations.

LDMCs exhibit smooth input-output BER curves, enabling graceful degradation.

New two-point converse bounds are tighter than existing area theorem bounds.

Abstract

We study a problem of constructing codes that transform a channel with high bit error rate (BER) into one with low BER (at the expense of rate). Our focus is on obtaining codes with smooth ("graceful'') input-output BER curves (as opposed to threshold-like curves typical for long error-correcting codes). This paper restricts attention to binary erasure channels (BEC) and contains three contributions. First, we introduce the notion of Low Density Majority Codes (LDMCs). These codes are non-linear sparse-graph codes, which output majority function evaluated on randomly chosen small subsets of the data bits. This is similar to Low Density Generator Matrix codes (LDGMs), except that the XOR function is replaced with the majority. We show that even with a few iterations of belief propagation (BP) the attained input-output curves provably improve upon performance of any linear systematic code. The effect of non-linearity bootstraping the initial iterations of BP, suggests that LDMCs should improve performance in various applications, where LDGMs have been used traditionally. Second, we establish several \textit{two-point converse bounds} that lower bound the BER achievable at one erasure probability as a function of BER achieved at another one. The novel nature of our bounds is that they are specific to subclasses of codes (linear systematic and non-linear systematic) and outperform similar bounds implied by the area theorem for the EXIT function.