Linear Complementary dual codes and Linear Complementary pairs of AG codes in function fields

TL;DR

This paper constructs explicit linear complementary pairs and LCD codes from algebraic function fields, enhancing coding theory and cryptography applications by leveraging divisors in various algebraic curves.

Contribution

It introduces new methods for constructing LCPs and LCD codes from function fields of genus g ≥ 1, including Kummer extensions and elliptic curves.

Findings

Constructed LCPs of algebraic geometry codes from function fields.

Developed explicit divisors for non-special divisors of degree g-1.

Applied results to hyperelliptic and elliptic curve codes.

Abstract

In recent years, linear complementary pairs (LCP) of codes and linear complementary dual (LCD) codes have gained significant attention due to their applications in coding theory and cryptography. In this work, we construct explicit LCPs of codes and LCD codes from function fields of genus $g \geq 1$. To accomplish this, we present pairs of suitable divisors giving rise to non-special divisors of degree $g-1$ in the function field. The results are applied in constructing LCPs of algebraic geometry codes and LCD algebraic geometry (AG) codes in Kummer extensions, hyperelliptic function fields, and elliptic curves.