Rates of linear codes with low decoding error probability

TL;DR

This paper investigates the limits of binary linear codes constructed from bipartite graphs, showing that if the decoding error probability diminishes and the average parity node degree stays bounded, then the code rate approaches zero as message length grows.

Contribution

It establishes a fundamental limitation on the rate of such codes under low error probability and bounded degree conditions.

Findings

Decoding error probability tends to zero

Code rate approaches zero under given conditions

Bounded parity node degree constrains code efficiency

Abstract

Consider binary linear codes obtained from bipartite graphs as follows. There are~\(k \geq 1\) left nodes each representing a message bit and there are~\(m = m(k)\) right nodes each representing a parity bit, generated from the corresponding set of message node neighbours. Both the message and the parity bits are sent through a memoryless binary input channel that either retains, flips or erases each transmitted bit, independently. Based on the received set of symbols, the decoder at the receiver obtains an estimate of the original message sent. If the decoding error probability~\(P_k \longrightarrow 0\) and the average degree per parity node remains bounded as~\(k \rightarrow \infty,\) then the rate of the code~\(\frac{k}{k+m} \longrightarrow 0\) as~\(k \rightarrow \infty.\)