Generalized twisted Gabidulin codes. (English) Zbl 1427.94105
Summary: Let \(\mathcal{C}\) be a set of \(m\) by \(n\) matrices over \(\mathbb{F}_q\) such that the rank of \(A - B\) is at least \(d\) for all distinct \(A, B \in \mathcal{C}\). Suppose that \(m \le n\). If \(\# \mathcal{C} = q^{n(m - d + 1)}\), then \(\mathcal{C}\) is a maximum rank distance (MRD for short) code. Until 2016, there were only two known constructions of MRD codes for arbitrary \(1 < d < m - 1\). One was found by P. Delsarte in 1978 [J. Comb. Theory, Ser. A 25, 226–241 (1978; Zbl 0397.94012)] and È. M. Gabidulin Probl. Inf. Transm. 21, 1–12 (1985; Zbl 0585.94013); independently, and it was later generalized by A. Kshevetskiy and È. M. Gabidulin in 2005 [The new construction of rank codes, Proceedings of the International Symposium on Information Theory, ISIT 2005, 2105–2108 (2005; doi:10.1109/ISIT.2005.1523717)]. We often call them (generalized) Gabidulin codes. Another family was recently obtained by J. Sheekey [Adv. Math. Commun. 10, No. 3, 475–488 (2016; Zbl 1348.94087)] and its elements are called twisted Gabidulin codes. In the same paper, Sheekey also proposed a generalization of the twisted Gabidulin codes. However the equivalence problem for it is not considered, whence it is not clear whether there exist new MRD codes in this generalization. We call the members of this putative larger family generalized twisted Gabidulin codes.
In this paper, we first compute the Delsarte duals and adjoint codes of them, then we completely determine the equivalence between different generalized twisted Gabidulin codes. In particular, it can be proven that, up to equivalence, generalized Gabidulin codes and twisted Gabidulin codes are both proper subsets of this family.
In this paper, we first compute the Delsarte duals and adjoint codes of them, then we completely determine the equivalence between different generalized twisted Gabidulin codes. In particular, it can be proven that, up to equivalence, generalized Gabidulin codes and twisted Gabidulin codes are both proper subsets of this family.
MSC:
| 94B25 | Combinatorial codes |
References:
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