Beating the Gilbert-Varshamov Bound for Online Channels

TL;DR

This paper demonstrates that for online channels with certain corruption levels, it is possible to construct codes that outperform the classical Gilbert-Varshamov bound, thus achieving higher communication rates.

Contribution

The authors establish a new lower bound on online channel capacity that exceeds the Gilbert-Varshamov bound using random codes and nearest neighbor decoding.

Findings

Achieves capacity above Gilbert-Varshamov bound for p where H(2p)<0.5

Uses random coding and nearest neighbor decoding

Proves high probability of exceeding classical bounds

Abstract

In the online channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword x =(x_1,...,x_n) in {0,1}^n bit by bit via a channel limited to at most pn corruptions. The channel is online in the sense that at the ith step the channel decides whether to flip the ith bit or not and its decision is based only on the bits transmitted so far, i.e., (x_1,...,x_i). This is in contrast to the classical adversarial channel in which the corruption is chosen by a channel that has full knowledge on the sent codeword x. The best known lower bound on the capacity of both the online channel and the classical adversarial channel is the well-known Gilbert-Varshamov bound. In this paper we prove a lower bound on the capacity of the online channel which beats the Gilbert-Varshamov bound for any positive p such that H(2p) < 0.5 (where H is the binary entropy function). To do so, we prove that for any such p, a code chosen at random combined with the nearest neighbor decoder achieves with high probability a rate strictly higher than the Gilbert-Varshamov bound (for the online channel).