Multivariate Goppa Codes

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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.
Funding Information
  • American Mathematical Society (AMS)–Simons Travel Grant
  • NSF (DMS-1855136, DMS-2037833)
  • Commonwealth Cyber Initiative

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