Generalized Twisted Gabidulin Codes

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 $m\leqslant n$. 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 Delsarte (1978) and Gabidulin (1985) independently, and it was later generalized by Kshevetskiy and Gabidulin (2005). We often call them (generalized) Gabidulin codes. Another family was recently obtained by Sheekey (2016), 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.Comment: One missing case (n=4) has been included in the appendix. Typos are corrected, Journal of Combinatorial Theory, Series A, 201

Slices of the unitary spread

We prove that slices of the unitary spread of Q(+)(7, q), q equivalent to 2 (mod 3), can be partitioned into five disjoint classes. Slices belonging to different classes are non-equivalent under the action of the subgroup of P Gamma O+(8, q) fixing the unitary spread. When q is even, there is a connection between spreads of Q(+)(7, q) and symplectic 2-spreads of PG(5, q) (see Dillon, Ph.D. thesis, 1974 and Dye, Ann. Mat. Pura Appl. (4) 114, 173-194, 1977). As a consequence of the above result we determine all the possible non-equivalent symplectic 2-spreads arising from the unitary spread of Q(+)(7, q), q = 2(2h+1). Some of these already appeared in Kantor, SIAM J. Algebr. Discrete Methods 3(2), 151-165, 1982. When q = 3(h), we classify, up to the action of the stabilizer in P Gamma O(7, q) of the unitary spread of Q(6, q), those among its slices producing spreads of the elliptic quadric Q(-)(5, q)

On sets of subspaces with two intersection dimensions and a geometrical junta bound

In this article, constant dimension subspace codes whose codewords have subspace distance in a prescribed set of integers, are considered. The easiest example of such an object is a {\it junta}; i.e. a subspace code in which all codewords go through a common subspace. We focus on the case when only two intersection values for the codewords, are assigned. In such a case we determine an upper bound for the dimension of the vector space spanned by the elements of a non-junta code. In addition, if the two intersection values are consecutive, we prove that such a bound is tight, and classify the examples attaining the largest possible dimension as one of four infinite families

Spreads of PG(3,q)PG(3,q) and ovoids of polar spaces

To any spread S of PG(3,q) corresponds a family of locally hermitian ovoids of the Hermitian surface H(3, q^2), and conversely; if in addition S is a semifield spread, then each associated ovoid is a translation ovoid, and conversely. In this paper we calculate the translation group of the locally hermitian ovoids of H(3,q^2) arising from a given semifield spread, and we characterize the p-semiclassical ovoid constructed by Cossidente, Ebert, Marino and Siciliano as the only translation ovoid of H(3,q^2) whose translation group is abelian. If S is a spread of PG(3,q) and O(S) is one of the associated ovoids of H(3,q^2), then using the duality between H(3,q^2) and Q^-(5, q) , another spread of PG(3,q) , say S_1, can be constructed. On the other hand, using the Barlotti-Cofman representation of H(3,q^2), one more spread of a 3-dimensional projective space, say S_2, arises from the ovoid O(S). Lunardon has posed some questions on the relations among S, S_1 and S_2; here we prove that the three spreads are always isomorphic