Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multivariate Goppa codes

Hiram H. López, Gretchen L. Matthews

TL;DR

This work generalizes classical Goppa codes to a multivariate setting by defining $\Gamma(\mathcal{S},g)$ over a Cartesian product $\mathcal{S}$ and a factorable polynomial $g=g_1\cdots g_m$. It provides a parity-check description of the codes using a tensor product of generalized Reed-Solomon codes via $\mathrm{T}(\mathcal{S},\mathbf{k},g)$ and shows $\Gamma(\mathcal{S},g)$ is the $\mathbb{F}_q$-subfield subcode of the dual of $\mathrm{T}(\mathcal{S},g)$, and also that $\Gamma(\mathcal{S},g)$ is a subfield subcode of the augmented Cartesian code $\mathrm{ACar}(\mathcal{S},g)$. By relating these three families, the paper derives parameter bounds, characterizes hulls and subcodes, and constructs entanglement-assisted quantum error-correcting codes and LCD/self-orthogonal/self-dual codes, including long-length families not bounded by the field size. The results unify and extend prior work on Goppa codes, GRS tensors, and augmented codes, with practical implications for quantum coding and code design.

Abstract

In this paper, we introduce multivariate Goppa codes, which contain as a special case the well-known, classical Goppa codes. We provide a parity check matrix for a multivariate Goppa code in terms of a tensor product of generalized Reed-Solomon codes. We prove that multivariate Goppa codes are subfield subcodes of augmented Cartesian codes. By showing how this new family of codes relates to tensor products of generalized Reed-Solomon codes and augmented codes, we obtain information about the parameters, subcodes, duals, and hulls of multivariate Goppa codes. We see that in certain cases, the hulls of multivariate Goppa codes (resp., tensor product of generalized Reed-Solomon codes), are also multivariate Goppa codes (resp. tensor product of generalized Reed-Solomon codes). We utilize the multivariate Goppa codes to obtain entanglement-assisted quantum error-correcting codes and to build families of long LCD, self-dual, or self-orthogonal codes.

Multivariate Goppa codes

TL;DR

This work generalizes classical Goppa codes to a multivariate setting by defining over a Cartesian product and a factorable polynomial . It provides a parity-check description of the codes using a tensor product of generalized Reed-Solomon codes via and shows is the -subfield subcode of the dual of , and also that is a subfield subcode of the augmented Cartesian code . By relating these three families, the paper derives parameter bounds, characterizes hulls and subcodes, and constructs entanglement-assisted quantum error-correcting codes and LCD/self-orthogonal/self-dual codes, including long-length families not bounded by the field size. The results unify and extend prior work on Goppa codes, GRS tensors, and augmented codes, with practical implications for quantum coding and code design.

Abstract

In this paper, we introduce multivariate Goppa codes, which contain as a special case the well-known, classical Goppa codes. We provide a parity check matrix for a multivariate Goppa code in terms of a tensor product of generalized Reed-Solomon codes. We prove that multivariate Goppa codes are subfield subcodes of augmented Cartesian codes. By showing how this new family of codes relates to tensor products of generalized Reed-Solomon codes and augmented codes, we obtain information about the parameters, subcodes, duals, and hulls of multivariate Goppa codes. We see that in certain cases, the hulls of multivariate Goppa codes (resp., tensor product of generalized Reed-Solomon codes), are also multivariate Goppa codes (resp. tensor product of generalized Reed-Solomon codes). We utilize the multivariate Goppa codes to obtain entanglement-assisted quantum error-correcting codes and to build families of long LCD, self-dual, or self-orthogonal codes.
Paper Structure (6 sections, 15 theorems, 68 equations, 1 figure)

This paper contains 6 sections, 15 theorems, 68 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A parity check matrix given by the tensor product of GRS codes
  3. As subfield subcodes of augmented codes
  4. Subcodes, intersections, and hulls
  5. Quantum, LCD, self-orthogonal and self-dual codes
  6. Conclusion

Key Result

Theorem 4

Given a multivariate Goppa code $\Gamma(\mathcal{S}, g)$, where $\operatorname{T}$ is a generator matrix of $\operatorname{T}(\mathcal{S},g)$; that is, $\Gamma(\mathcal{S}, g)$ is a subfield subcode of the dual of a tensor product of GRS codes via Goppa codes.

Figures (1)

  • Figure 1: The code $\textrm{ACar}(S_1 \times S_2, (2,2),h),$ with $h=1$ and $K=\mathbb{F}_{17}$ in Example \ref{['21.01.01']} is generated by the vectors the vectors $\operatorname{ev}(S_1 \times S_2,h)(\text{$M$}),$ where $M$ is a point in (a). The dual code $\textrm{ACar}(S_1 \times S_2, (2,2),h)^{\perp}$ is generated by the vectors $\operatorname{ev}(S_1 \times S_2,L)(\text{$M$}),$ where $M$ is a point in (b).

Theorems & Definitions (38)

  • Definition 1
  • Remark 2
  • Remark 3
  • Theorem 4
  • proof
  • Corollary 5
  • Example 6
  • Definition 7
  • Remark 8
  • Lemma 9
  • ...and 28 more