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.
翻译：在本文中,我们引入了多种变式的戈帕代码,其中作为特例包含了众所周知的古典戈帕代码。我们以通用Reed-Solomon代码的发分产品为多变式戈帕代码提供了一个对等检查矩阵。我们证明,多变式戈帕代码是强化的笛卡尔法典的子字段子代码。我们通过展示这个新的代码组合如何与通用Reed-Solomon代码和扩展代码的发分产品相关联。我们获得了关于多变式戈帕代码的参数、子代码、双倍代码和船体的信息。我们发现,在某些情况下,多变式戈帕代码的船体(resp.,通用Reed-Solomon代码的发数)也是多变式的戈帕代码(除了通用Reed-Solomon代码的发数项产品 )。我们使用多变式戈帕代码来获取内嵌辅助的量子错误校正代码,并构建长LCD、自上或自上式代码的家庭。
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