Multivariate Goppa codes

TL;DR

This paper introduces multivariate Goppa codes, explores their structure and relationships with other codes, and demonstrates their applications in quantum error correction and code construction.

Contribution

It defines multivariate Goppa codes, provides their parity check matrices, and links them to tensor products of Reed-Solomon and augmented codes, revealing new properties and applications.

Findings

Multivariate Goppa codes are subfield subcodes of augmented Cartesian codes.

Hulls of multivariate Goppa codes can also be multivariate Goppa codes.

Applications include entanglement-assisted quantum codes and families of LCD, self-dual, or self-orthogonal codes.

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.