Galois Hulls of a Kind of Goppa Codes With Applications to EAQECCs

This is the authors' abstract. We don't add key points for this paper.

Galois hulls of MDS codes can be applied to construst MDS entanglement-assisted quantum error-correcting codes (EAQECCs). Goppa codes over <tex-math notation="LaTeX">F_qᵐ </tex-math> are generalized Reed-Solomon codes (GRS codes) when <tex-math notation="LaTeX">$m=1$ </tex-math>. In this paper, we give a necessary and sufficient condition for the Galois dual codes of Goppa codes when <tex-math notation="LaTeX">$m=1$ </tex-math> 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.