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

Alonso S. Castellanos, Adler V. Marques, Luciane Quoos

TL;DR

The work presents explicit constructions of Linear Complementary Pairs (LCP) and Linear Complementary Dual (LCD) codes arising from algebraic-geometry codes on function fields with genus $g\ge1$. It leverages non-special divisors of degree $g-1$ and, via gcd/lmd divisor data, establishes general LCP criteria and Kummer-based explicit families, then extends to Hermitian-generalized, maximal, hyperelliptic, and elliptic curves. Key contributions include Theorems that yield concrete code parameters for LCPs and LCDs on Kummer extensions, hyperelliptic curves, and elliptic curves, plus new LCD instances on maximal subcovers of Hermitian curves. The results broaden the catalog of AG-code-based LCP/LCD constructions with practical length–dimension–distance ranges, underpinned by divisor-theoretic conditions that also clarify security implications for cryptographic use.

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.

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

TL;DR

The work presents explicit constructions of Linear Complementary Pairs (LCP) and Linear Complementary Dual (LCD) codes arising from algebraic-geometry codes on function fields with genus . It leverages non-special divisors of degree and, via gcd/lmd divisor data, establishes general LCP criteria and Kummer-based explicit families, then extends to Hermitian-generalized, maximal, hyperelliptic, and elliptic curves. Key contributions include Theorems that yield concrete code parameters for LCPs and LCDs on Kummer extensions, hyperelliptic curves, and elliptic curves, plus new LCD instances on maximal subcovers of Hermitian curves. The results broaden the catalog of AG-code-based LCP/LCD constructions with practical length–dimension–distance ranges, underpinned by divisor-theoretic conditions that also clarify security implications for cryptographic use.

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 . To accomplish this, we present pairs of suitable divisors giving rise to non-special divisors of degree 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.
Paper Structure (10 sections, 25 theorems, 106 equations)

This paper contains 10 sections, 25 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function fields
  4. Linear and Algebraic Geometry Codes
  5. LCPs of algebraic geometry codes
  6. LCD AG codes and LCPs of AG codes over Kummer extensions
  7. LCD AG codes and LCPs of AG codes over a generalization of the Hermitian function field
  8. LCD AG codes and LCPs of AG codes over a maximal curve covered by the Hermitian function field
  9. LCPs of AG codes and LCD codes over the curve yq+1=x2+x, q odd
  10. LCPs of AG codes over elliptic function fields

Key Result

Proposition 2.4

STICH2008 The code $\mathcal{C}_\mathcal{L}(D, G)$ has parameters $[n, k, d]$ which satisfy In particular, for $2g - 2 < \deg G < n$ it holds that $k=\ell(G)=\deg(G) + 1 - g$.

Theorems & Definitions (53)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 3.1
  • proof
  • ...and 43 more