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Fractional decoding of algebraic geometry codes over extension fields

Eduardo Camps-Moreno, Gretchen L. Matthews, Welington Santos

TL;DR

The work presents a unified framework for fractional decoding of algebraic geometry codes over extension fields by exploiting virtual projections to base-field codes. By formulating $t$-virtual projections $\mathcal{VP}_t(G,D)$ and organizing evaluation points via annihilator polynomials, the authors show how to recover $\mathrm{ev}(f)$ from a reduced amount of base-field data, with concrete error bounds tied to pole-divisor degrees. They further leverage interleaved-code perspectives to potentially decode more errors with high probability, using established interleaved-AG decoding methods and power-decoding strategies. The approach is applicable to broad families of AG codes, including those from Kummer extensions and Castle curves, and offers practical benefits for distributed storage and network-constrained settings where decoding bandwidth is critical.

Abstract

In this paper, we study algebraic geometry codes from curves over $\mathbb{F}_{q^\ell}$ through their virtual projections which are algebraic geometric codes over $\mathbb{F}_q$. We use the virtual projections to provide fractional decoding algorithms for the codes over $\mathbb{F}_{q^\ell}$. Fractional decoding seeks to perform error correction using a smaller fraction of $\mathbb{F}_q$-symbols than a typical decoding algorithm. In one instance, the bound on the number of correctable errors differs from the usual lower bound by the degree of a pole divisor of an annihilator function. In another, we view the virtual projections as interleaved codes to, with high probability, correct more errors than anticipated.

Fractional decoding of algebraic geometry codes over extension fields

TL;DR

The work presents a unified framework for fractional decoding of algebraic geometry codes over extension fields by exploiting virtual projections to base-field codes. By formulating -virtual projections and organizing evaluation points via annihilator polynomials, the authors show how to recover from a reduced amount of base-field data, with concrete error bounds tied to pole-divisor degrees. They further leverage interleaved-code perspectives to potentially decode more errors with high probability, using established interleaved-AG decoding methods and power-decoding strategies. The approach is applicable to broad families of AG codes, including those from Kummer extensions and Castle curves, and offers practical benefits for distributed storage and network-constrained settings where decoding bandwidth is critical.

Abstract

In this paper, we study algebraic geometry codes from curves over through their virtual projections which are algebraic geometric codes over . We use the virtual projections to provide fractional decoding algorithms for the codes over . Fractional decoding seeks to perform error correction using a smaller fraction of -symbols than a typical decoding algorithm. In one instance, the bound on the number of correctable errors differs from the usual lower bound by the degree of a pole divisor of an annihilator function. In another, we view the virtual projections as interleaved codes to, with high probability, correct more errors than anticipated.
Paper Structure (9 sections, 8 theorems, 50 equations)

This paper contains 9 sections, 8 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Virtual projections of algebraic geometry codes
  3. Notation
  4. Codes from constant field extensions
  5. Virtual projections from particular partitions
  6. Decoding via virtual projections
  7. Fractional decoding directly from virtual projections
  8. Interleaved codes from virtual projections
  9. Fractional decoding via interleaved algebraic geometry codes

Key Result

Proposition 4

Given divisors $G$ and $D$ whose supports consist only of $\mathbb{F}_q$-rational points, the $t$-virtual projection $\mathcal{VP}_{t}(G,D)$ of $\mathcal{C}_{l}(G,D)$ is a subcode of $\mathbb{F}_q^n$. In particular,

Theorems & Definitions (18)

  • Definition 1
  • Definition 3
  • Proposition 4
  • proof
  • Theorem 5
  • proof
  • Corollary 6
  • proof
  • Example 7
  • Theorem 8
  • ...and 8 more