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A new approach to the Berlekamp-Massey-Sakata Algorithm. Improving Locator Decoding

José Joaquín Bernal, Juan Jacobo Simón

TL;DR

The paper tackles the problem of computing a Groebner basis for the ideal ${\mathbf {\Lambda}}(U)$ of linear recurrences of a doubly periodic array by identifying a special index set ${\mathcal{S}}(t)$ that guarantees the final BMSa output is a Groebner basis. It then reframes locator decoding in a unified framework via the locator ideal $L(e)$ and shows $L(e)=\overline{ {\mathbf {\Lambda}}(U) }$ for a suitable syndrome table $U$, enabling recovery of the error polynomial from a compact index set. Sufficient conditions and a complexity-bounded strategy are provided for terminating the BMSa with a valid Groebner basis, and the results are applied to abelian codes to achieve reliable decoding up to $t$ errors when the defining set contains $\tau+{\mathcal{S}}(t)$, linking to BCH-type bounds. Overall, the work delivers a practical, algebraic refinement of locator decoding and a concrete termination criterion for BMSa in the bivariate setting.

Abstract

We study the problem of the computation of Groebner basis for the ideal of linear recurring relations of a doubly periodic array. We find a set of indexes such that, along with some conditions, guarantees that the set of polynomials obtained at the last iteration in the Berlekamp-Massey-Sakata algorithm is exactly a Groebner basis for the mentioned ideal. Then, we apply these results to improve locator decoding in abelian codes.

A new approach to the Berlekamp-Massey-Sakata Algorithm. Improving Locator Decoding

TL;DR

The paper tackles the problem of computing a Groebner basis for the ideal of linear recurrences of a doubly periodic array by identifying a special index set that guarantees the final BMSa output is a Groebner basis. It then reframes locator decoding in a unified framework via the locator ideal and shows for a suitable syndrome table , enabling recovery of the error polynomial from a compact index set. Sufficient conditions and a complexity-bounded strategy are provided for terminating the BMSa with a valid Groebner basis, and the results are applied to abelian codes to achieve reliable decoding up to errors when the defining set contains , linking to BCH-type bounds. Overall, the work delivers a practical, algebraic refinement of locator decoding and a concrete termination criterion for BMSa in the bivariate setting.

Abstract

We study the problem of the computation of Groebner basis for the ideal of linear recurring relations of a doubly periodic array. We find a set of indexes such that, along with some conditions, guarantees that the set of polynomials obtained at the last iteration in the Berlekamp-Massey-Sakata algorithm is exactly a Groebner basis for the mentioned ideal. Then, we apply these results to improve locator decoding in abelian codes.
Paper Structure (9 sections, 24 theorems, 41 equations)

This paper contains 9 sections, 24 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. The Berlekamp-Massey-Sakata algorithm
  4. Delta sets and defining points.
  5. The Berlekamp-Massey-Sakata algorithm
  6. A new framework for locator ideals
  7. Sufficient conditions to find Groebner basis for $\mathbf {\Lambda}(U)$
  8. Applications to locator decoding in abelian codes.
  9. Conclusions

Key Result

Lemma 8

Let $U$ be a doubly periodic array, $l\in \Sigma_0$, $u=u^l\subset U$ and $f,g\in \mathbb L[\bf X]$, with $LP(f)=s_f$ and $LP(g)=s_g$. If $s_f<_T s_g$ and $n\in \Sigma_{s_g}^l\neq \emptyset$ then $n-s_g+s_f\in \Sigma_{s_f}^l$ and $(f+g)[u]_n=f[u]_{(n-s_g+s_f)}+g[u]_n$.

Theorems & Definitions (54)

  • Definition 1
  • Definition 2
  • Definition 3: see Sakata 2
  • Remark 5
  • Definition 6
  • Remark 7
  • Lemma 8
  • proof
  • Lemma 9
  • proof
  • ...and 44 more