An open problem of \(k\)-Galois hulls and its application. (English) Zbl 1521.94086
Summary: The \(k\)-Galois hull of a linear code \(C\) over the finite field \(\mathbb{F}_q\) is defined as \(\mathrm{Hull}_k(C) = C \cap C^{\bot_k}\), where \(q = p^e\), \(p\) is prime and \(C^{\bot_k}\) is the \(k\)-Galois dual code of \(C\) with \(0 \leq k < e\). In this paper, by considering an equation of the matrix rank over finite fields, one of the two problems proposed by M. Cao about \(k\)-Galois hull of a linear code in [“Galois hulls of MDS codes and their quantum error correction”, Preprint, arXiv:2002.12892 (2020)] is settled. Then we construct nine new classes of MDS EAQECCs with flexible parameters. In addition, our obtained MDS EAQECCs have parameters better than the ones available in the literature.
MSC:
| 94B05 | Linear codes (general theory) |
| 11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |
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