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Good iso-dual AG-codes from towers of function fields

María Chara, Ricardo Podestá, Luciane Quoos, Ricardo Toledano

TL;DR

The paper develops a simple, flexible framework to construct asymptotically good iso-dual AG-codes by lifting codes across towers of function fields. The key idea is a lifting operation that preserves iso-duality under explicit conditions on splitting, ramification, and evenness of different exponents, enabling asymptotically good sequences to be built from recursive towers. The authors prove existence results including a TVZ-bound-attaining sequence over non-prime fields and a characteristic-2 construction over $\mathbb{F}_8$, with explicit, scalable towers providing more direct constructions than prior Galois-closure approaches. These results extend the reach of iso-dual AG-codes in the asymptotic regime and offer practical, explicit code families for applications requiring iso-duality properties.

Abstract

We present a simple method to establish the existence of asymptotically good sequences of iso-dual AG-codes. A key advantage of our approach, beyond its simplicity, is its flexibility, allowing it to be applied to a wide range of towers of function fields. As a result, we present a novel example of an asymptotically good sequence of iso-dual AG-codes over a finite field with 8 elements.

Good iso-dual AG-codes from towers of function fields

TL;DR

The paper develops a simple, flexible framework to construct asymptotically good iso-dual AG-codes by lifting codes across towers of function fields. The key idea is a lifting operation that preserves iso-duality under explicit conditions on splitting, ramification, and evenness of different exponents, enabling asymptotically good sequences to be built from recursive towers. The authors prove existence results including a TVZ-bound-attaining sequence over non-prime fields and a characteristic-2 construction over , with explicit, scalable towers providing more direct constructions than prior Galois-closure approaches. These results extend the reach of iso-dual AG-codes in the asymptotic regime and offer practical, explicit code families for applications requiring iso-duality properties.

Abstract

We present a simple method to establish the existence of asymptotically good sequences of iso-dual AG-codes. A key advantage of our approach, beyond its simplicity, is its flexibility, allowing it to be applied to a wide range of towers of function fields. As a result, we present a novel example of an asymptotically good sequence of iso-dual AG-codes over a finite field with 8 elements.

Paper Structure

This paper contains 7 sections, 6 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lifting iso-dual AG-codes on function field extensions
  4. Good sequences of iso-dual AG-codes from towers
  5. A sequence of iso-dual AG-codes attaining the TVZ-bound
  6. An asymptotically good sequence of iso-dual codes in characteristic 2
  7. Final remarks: successive liftings

Key Result

Theorem 3.1

Let $F'/F$ be a finite separable extension of degree $m\geq 2$ of function fields over $\mathbb{F}_{q}$ with genera $g_{_{F'}}$ and $g_{_F}$, respectively. Let $n, r \in \mathbb{N}$ with $n$ even and suppose that $\{P_1, \dots, P_n\}$ and $\{Q_1, \dots, Q_r\}$ are two disjoint set of places of $F$ s Let $(\beta_1, \ldots, \beta_r) \in \mathbb{Z}^r$ be a non-zero $r$-tuple, and consider the disjoin

Theorems & Definitions (13)

  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1: CPQT2024
  • Corollary 3.2: CPQT2024
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • proof
  • Proposition 4.3
  • proof
  • ...and 3 more