Multivariate Goppa Codes
Multivariate Goppa Codes
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DOI：
10.1109/tit.2022.3201692
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发表时间：
2021-12
影响因子：
2.5
通讯作者：
H. L'opez;Gretchen L. Matthews
中科院分区：
文献类型：
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作者：
H. L'opez;Gretchen L. Matthews
在本文中，我们引入多元Goppa码，其中包含，作为一个特殊的情况下，著名的经典Goppa码。我们提供了一个奇偶校验矩阵的多元Goppa码的广义Reed-Solomon码的张量积。证明了多元Goppa码是增广笛卡尔码的子域子码。通过展示这个新的家庭的代码涉及到广义里德-所罗门码和增广码的张量积，我们获得的信息的参数，子码，双，和外壳的多变量Goppa码。我们看到，在某些情况下，多变量Goppa码的外壳（分别为，广义Reed-Solomon码张量积）也是多变量Goppa码（分别广义Reed-Solomon码的张量积）。我们利用多元Goppa码获得纠缠辅助量子纠错码，并建立家庭的长LCD，自对偶，或自正交码。
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.