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Faster List Decoding of AG Codes

Peter Beelen, Vincent Neiger

TL;DR

This work substantially speeds up Guruswami-Sudan list decoding for algebraic-geometry codes by reworking the interpolation step via efficient univariate polynomial-matrix computations. The authors generalize the interpolant module $\mathcal{M}_{s,\ell,\boldsymbol{r}}$ with flexible generators, and construct polynomial-matrix bases $\boldsymbol{B}_{s,\ell,\boldsymbol{r}}$ and a small-degree basis $\boldsymbol{M}_{s,\ell,\boldsymbol{r}}$ that enable fast $\boldsymbol{d}$-Popov reductions. The resulting decoding algorithm achieves a running time of $\tilde{O}(s^2\ell^{\omega-1}\mu^{\omega-1}(n+g) + \ell^{\omega}\mu^{\omega})$ with a refined root-finding cost of $\tilde{O}(s\ell\mu^{\omega-1}(n+g))$, improving prior bounds and extending efficient precomputation reuse. These advances, particularly the small-average-degree bases and the generalized multiplication maps, offer practical gains for decoding AG codes and also tighten the Reed-Solomon special case. Overall, the paper advances the state of the art in algebraic-geometry code decoding by marrying advanced polynomial-matrix techniques with the GS framework.

Abstract

In this article, we present a fast algorithm performing an instance of the Guruswami-Sudan list decoder for algebraic geometry codes. We show that any such code can be decoded in $\tilde{O}(s^2\ell^{ω-1}μ^{ω-1}(n+g) + \ell^ωμ^ω)$ operations in the underlying finite field, where $n$ is the code length, $g$ is the genus of the function field used to construct the code, $s$ is the multiplicity parameter, $\ell$ is the designed list size and $μ$ is the smallest positive element in the Weierstrass semigroup of some chosen place.

Faster List Decoding of AG Codes

TL;DR

This work substantially speeds up Guruswami-Sudan list decoding for algebraic-geometry codes by reworking the interpolation step via efficient univariate polynomial-matrix computations. The authors generalize the interpolant module with flexible generators, and construct polynomial-matrix bases and a small-degree basis that enable fast -Popov reductions. The resulting decoding algorithm achieves a running time of with a refined root-finding cost of , improving prior bounds and extending efficient precomputation reuse. These advances, particularly the small-average-degree bases and the generalized multiplication maps, offer practical gains for decoding AG codes and also tighten the Reed-Solomon special case. Overall, the paper advances the state of the art in algebraic-geometry code decoding by marrying advanced polynomial-matrix techniques with the GS framework.

Abstract

In this article, we present a fast algorithm performing an instance of the Guruswami-Sudan list decoder for algebraic geometry codes. We show that any such code can be decoded in operations in the underlying finite field, where is the code length, is the genus of the function field used to construct the code, is the multiplicity parameter, is the designed list size and is the smallest positive element in the Weierstrass semigroup of some chosen place.
Paper Structure (16 sections, 21 theorems, 51 equations, 1 algorithm)

This paper contains 16 sections, 21 theorems, 51 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function fields and AG codes
  4. The Guruswami-Sudan list decoder
  5. Reminders on univariate polynomial matrices
  6. The interpolant module M_{s,l,r} and polynomial matrix representations of it
  7. A more flexible description of M_{s,l,r}
  8. Inclusion and multiplication maps, and their matrices
  9. A first polynomial matrix basis of M_{s,l,r}
  10. A small-degree polynomial matrix basis of M_{s,l,r}
  11. Efficient construction of a polynomial matrix basis of M_{s,l,r}
  12. Computing multiplication maps
  13. Computing a small-degree Fq[x]-basis of M[s,l](D,G)
  14. Decoder with better complexity
  15. The decoding algorithm
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $A$ be a divisor and $E = E_1 + \cdots + E_N$ for distinct rational places $E_1,\dots,E_N$ of $F$ different from $P_{\infty}$ such that $\mathop{\mathrm{supp}}\nolimits(A) \cap \mathop{\mathrm{supp}}\nolimits(E)= \emptyset$. For any $(w_1,\dots,w_N) \in \mathbb{F}_q^N$ there exists an $a \in \ma such that $a(E_j) = w_j$ for $j = 1,\dots,N$.

Theorems & Definitions (51)

  • Lemma 2.1: BRS2022
  • Definition 2.2
  • Lemma 2.3: BRS2022
  • Lemma 2.4: BRS2022
  • Theorem 2.5: Instance of Guruswami-Sudan
  • Theorem 2.6: BRS2022
  • Definition 2.7
  • Definition 2.8
  • Lemma 2.9
  • proof
  • ...and 41 more