Galois Hulls of a Kind of Goppa Codes With Applications to EAQECCs
Jingge Liu, Hongwei Liu · IEEE Transactions on Information Theory · 2025
Galois hulls of MDS codes can be applied to construst MDS entanglement-assisted quantum error-correcting codes (EAQECCs). Goppa codes over$\mathbb {F}_{q^{m}}$are generalized Reed-Solomon codes (GRS codes) when$m=1$. In this paper, we give a necessary and sufficient condition for the Galois dual codes of Goppa codes when$m=1$to be Goppa codes with the same locator sets. Furthermore, we show that the Galois hulls of the above codes are still the Goppa codes and determine their Goppa polynomials and dimensions. In particular, we characterize Galois linear complementary dual (LCD), Galois self-orthogonal, Galois dual-containing and Galois self-dual codes among such codes. Moreover, we apply these results to EAQECCs.