Algebraic geometry codes with complementary duals exceed the asymptotic Gilbert-Varshamov bound

TL;DR

This paper demonstrates that certain algebraic geometry codes over fields of even characteristic can be equivalent to LCD codes and can surpass the asymptotic Gilbert-Varshamov bound, indicating improved code performance.

Contribution

It establishes a new class of LCD codes derived from algebraic geometry codes that exceed the longstanding asymptotic GV bound.

Findings

Existence of LCD codes surpassing the GV bound.

Equivalence between algebraic geometry codes and LCD codes in even characteristic fields.

Enhanced asymptotic performance of LCD codes.

Abstract

It was shown by Massey that linear complementary dual (LCD for short) codes are asymptotically good. In 2004, Sendrier proved that LCD codes meet the asymptotic Gilbert-Varshamov (GV for short) bound. Until now, the GV bound still remains to be the best asymptotical lower bound for LCD codes. In this paper, we show that an algebraic geometry code over a finite field of even characteristic is equivalent to an LCD code and consequently there exists a family of LCD codes that are equivalent to algebraic geometry codes and exceed the asymptotical GV bound.