Multivariate Goppa Codes
Document Type
Article
Publication Date
1-2023
Publication Title
IEEE Transactions on Information Theory
Abstract
In this paper, we introduce multivariate Goppa codes, which contain, as a particular 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 a tensor product 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 some instances, 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.
Repository Citation
Lopez, Hiram H. and Matthews, Gretchen L., "Multivariate Goppa Codes" (2023). Mathematics and Statistics Faculty Publications. 355.
https://engagedscholarship.csuohio.edu/scimath_facpub/355
DOI
10.1109/TIT.2022.3201692
Volume
69
Issue
1
