Asymptotic Improvement of the Binary Gilbert-Varshamov Bound on the Code Rate

TL;DR

This paper demonstrates that certain binary linear codes constructed via a greedy method surpass the classical Gilbert-Varshamov bound, challenging the conjecture about its asymptotic optimality.

Contribution

It introduces a greedy construction method for binary linear codes that asymptotically outperforms the GV bound, providing new insights into code optimality.

Findings

Codes improve the GV bound on size and rate

Counterexample to the conjecture on GV bound's asymptotic optimality

Method for constructing better binary codes

Abstract

We compute the code parameters for binary linear codes obtained by greedy constructing the parity check matrix. Then we show that these codes improve the Gilbert-Varshamov (GV) bound on the code size and rate. This result counter proves the conjecture on the asymptotical exactness of the binary GV bound.