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Duality and decoding of linearized Algebraic Geometry codes

Elena Berardini, Xavier Caruso, Fabrice Drain

Abstract

We design a polynomial time decoding algorithm for linearized Algebraic Geometry codes with unramified evaluation places, a family of sum-rank metric evaluation codes on division algebras over function fields. By establishing a Serre duality and a Riemann-Roch theorem for these algebras, we prove that the dual codes of such linearized Algebraic Geometry codes, that we term linearized Differential codes, coincide with the linearized Algebraic Geometry codes themselves over the adjoint algebra, and that our decoding algorithm is correct.

Duality and decoding of linearized Algebraic Geometry codes

Abstract

We design a polynomial time decoding algorithm for linearized Algebraic Geometry codes with unramified evaluation places, a family of sum-rank metric evaluation codes on division algebras over function fields. By establishing a Serre duality and a Riemann-Roch theorem for these algebras, we prove that the dual codes of such linearized Algebraic Geometry codes, that we term linearized Differential codes, coincide with the linearized Algebraic Geometry codes themselves over the adjoint algebra, and that our decoding algorithm is correct.
Paper Structure (16 sections, 31 theorems, 96 equations, 3 figures, 1 algorithm)

This paper contains 16 sections, 31 theorems, 96 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Organisation of the paper.
  2. Riemann--Roch theorem on division algebras
  3. The case of numerical divisors
  4. Riemann theorem for extended divisors
  5. Differentials and residues
  6. Definitions
  7. The residue formula
  8. Riemann--Roch spaces
  9. Serre duality and Riemann--Roch Theorem
  10. Application to linearized Algebraic Geometry codes
  11. Duality
  12. Decoding
  13. An auxiliary divisor
  14. Error localization
  15. Syndrome equation
  16. ...and 1 more sections

Key Result

Theorem 1

Let $A$ be a divisor as above, and satisfying $2g -2< \operatorname{deg}A < sr$. We assume the existence of a non-split non-evaluation rational place of $\mathbb{K}$ (see Hypothesis HYPO2). Set Then, our algorithm decodes any error relative to the code $\operatorname{LAG}_{\mathbb{D}}(A,D)$ of sum-rank weight at most $\rho_{\mathrm{algo}}$, with polynomial complexity in $s$ and $r$.

Figures (3)

  • Figure 1: Description of the tested codes, where $z_n$ denotes a generator of $\mathbb{F}^\times_{p^n}$
  • Figure 2: CPU Timings in seconds for constructing a code.
  • Figure 3: CPU Timings in seconds for decoding a single random message in a code.

Theorems & Definitions (85)

  • Theorem 1: Theorem \ref{['LAGDECODING']}
  • Theorem 2: Theorem \ref{['RROCHTHEO']}
  • Theorem 3: Theorem \ref{['NONCAN']}
  • Remark 1.0.1
  • Lemma 1.1.1
  • proof
  • Definition 1.1.2
  • Definition 1.1.3
  • Definition 1.1.4: 1
  • Definition 1.1.5
  • ...and 75 more