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Apparent Distance and a Notion of BCH Multivariate Codes

José Joaquín Bernal, Diana H. Bueno-Carreño, Juan Jacobo Simón

TL;DR

This work generalizes the BCH-distance bound to multivariate Abelian codes by introducing a strong notion of apparent distance and a hypermatrix-based algorithm that computes the minimum apparent distance efficiently via generating-idempotent data. It proves that the apparent distance of an Abelian code equals the maximum over defining roots of the minimum apparent distance of the associated Fourier-coefficient hypermatrices, enabling a direct link between algebraic structure and combinatorial bounds. The authors extend classical BCH results to the multivariate setting, define multivariate BCH codes, provide dimension and distance bounds, and demonstrate practical code-design strategies, including dimension-multiplication and HD-code construction for prescribed bounds. Collectively, these contributions yield scalable methods for designing high-dimensional Abelian codes with guaranteed lower-distance performance, with concrete algorithmic gains in the bivariate case and a broad multivariate BCH theory for applications in coding theory and communications.

Abstract

This paper is devoted to studying two main problems: 1) computing the apparent distance of an Abelian code and 2) giving a notion of Bose, Ray-Chaudhuri, Hocquenghem (BCH) multivariate code. To do this, we first strengthen the notion of an apparent distance by introducing the notion of a strong apparent distance; then, we present an algorithm to compute the strong apparent distance of an Abelian code, based on some manipulations of hypermatrices associated with its generating idempotent. Our method uses less computations than those given by Camion and Sabin; furthermore, in the bivariate case, the order of computation complexity is reduced from exponential to linear. Then, we use our techniques to develop a notion of a BCH code in the multivariate case, and we extend most of the classical results on cyclic BCH codes. Finally, we apply our method to the design of Abelian codes with maximum dimension with respect to a fixed apparent distance and a fixed length.

Apparent Distance and a Notion of BCH Multivariate Codes

TL;DR

This work generalizes the BCH-distance bound to multivariate Abelian codes by introducing a strong notion of apparent distance and a hypermatrix-based algorithm that computes the minimum apparent distance efficiently via generating-idempotent data. It proves that the apparent distance of an Abelian code equals the maximum over defining roots of the minimum apparent distance of the associated Fourier-coefficient hypermatrices, enabling a direct link between algebraic structure and combinatorial bounds. The authors extend classical BCH results to the multivariate setting, define multivariate BCH codes, provide dimension and distance bounds, and demonstrate practical code-design strategies, including dimension-multiplication and HD-code construction for prescribed bounds. Collectively, these contributions yield scalable methods for designing high-dimensional Abelian codes with guaranteed lower-distance performance, with concrete algorithmic gains in the bivariate case and a broad multivariate BCH theory for applications in coding theory and communications.

Abstract

This paper is devoted to studying two main problems: 1) computing the apparent distance of an Abelian code and 2) giving a notion of Bose, Ray-Chaudhuri, Hocquenghem (BCH) multivariate code. To do this, we first strengthen the notion of an apparent distance by introducing the notion of a strong apparent distance; then, we present an algorithm to compute the strong apparent distance of an Abelian code, based on some manipulations of hypermatrices associated with its generating idempotent. Our method uses less computations than those given by Camion and Sabin; furthermore, in the bivariate case, the order of computation complexity is reduced from exponential to linear. Then, we use our techniques to develop a notion of a BCH code in the multivariate case, and we extend most of the classical results on cyclic BCH codes. Finally, we apply our method to the design of Abelian codes with maximum dimension with respect to a fixed apparent distance and a fixed length.
Paper Structure (11 sections, 19 theorems, 35 equations)

This paper contains 11 sections, 19 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Apparent distance and BCH bounds
  4. The apparent distance of polynomials and matrices
  5. The apparent distance of an abelian code
  6. Computing the minimum apparent distance of a hypermatrix
  7. Multivariate BCH bound and BCH code
  8. Applications
  9. Multiplying dimension in abelian codes
  10. Designing maximum dimensional abelian codes (HD codes) for prescribed bounds
  11. Conclusion

Key Result

Theorem 8

Let $s,q$, $r_1,\dots,r_s$, $I$, and $A_q(r_1,\dots,r_s)$ be as above. For any polynomial $f\in A_q(r_1,\dots,r_s)$ with coefficient hypermatrix $M(f)$, the equality $d^\ast(f)=d^\ast(M(f))$ holds.

Theorems & Definitions (50)

  • Definition 1
  • Example 2
  • Definition 3
  • Definition 5
  • Definition 6
  • Theorem 8
  • proof
  • Definition 9
  • Theorem 10: CamionCamion
  • Definition 11
  • ...and 40 more