Linear Complementary Pairs of Algebraic Geometry Codes via Kummer Extensions

TL;DR

This paper develops explicit methods for constructing linear complementary pairs of algebraic geometry codes using Kummer extensions, including new families of MDS and LCD codes from elliptic and hyperelliptic function fields.

Contribution

It introduces explicit constructions of non-special divisors on Kummer extensions and applies these to generate new LCPs of AG Codes, including MDS and LCD codes from various function fields.

Findings

Constructed explicit non-special divisors on Kummer extensions.

Developed methods for constructing LCPs of AG Codes from various function fields.

Generated families of MDS and LCD AG Codes from elliptic and hyperelliptic fields.

Abstract

Due to their widespread applications, linear complementary pairs (LCPs) have attracted much attention in recent years. In this paper, we determine explicit construction of non-special divisors of degree $g$ and $g-1$ on Kummer extensions with specific properties. In addition, we present several methods for constructing LCPs of algebraic geometry codes (AG Codes) via Kummer extensions. These results are applied in constructing explicit LCPs of AG Codes from subcovers of the BM curve, elliptic function fields, hyperelliptic function fields and other function fields. It is important to mention that we construct several families LCPs of MDS AG Codes from elliptic function fields and we obtain some linear complementary dual (LCD) codes from certain maximal elliptic function fields and hyperelliptic function fields.