Abstract A perfect matroid design (PMD) is a matroid whose flats of the same rank all have the same size. In this paper we introduce the q -analogue of a PMD and its properties. In order to do so, we first establish a new cryptomorphic definition for q -matroids. We show that q -Steiner systems are examples of q -PMD’s and we use this q -matroid structure to construct subspace designs from q -Steiner systems. We apply this construction to the only known q -Steiner system, which has parameters S (2, 3, 13; 2), and hence establish the existence of a new subspace design with parameters 2-(13, 4, 5115; 2).

In this paper, we describe a new variation of the interpolation algorithm by Möller, proposed in a way that completely avoids Gröbner bases and does not need a term order, but only a well order on terms. This algorithm takes a set of functionals describing a Macaulay chain, namely, roughly speaking, the functionals are chosen and ordered in such a way that the first functional defines a zero-dimensional ideal and all the sets one gets by adding the functionals one after the other define zero-dimensional ideals as well. Starting from this set, the algorithm describes the zero-dimensional ideals of the Macaulay chain via a basis of the quotient algebra and Auzinger-Stetter matrices.

Abstract Linear error-correcting codes can be used for constructing secret sharing schemes; however, finding in general the access structures of these secret sharing schemes and, in particular, determining efficient access structures is difficult. Here we investigate the properties of certain algebraic hypersurfaces over finite fields, whose intersection numbers with any hyperplane only takes a few values; these varieties give rise to q-divisible linear codes with at most 5 weights. Furthermore, for q odd, these codes turn out to be minimal and we characterize the access structures of the secret sharing schemes based on their dual codes. Indeed, the secret sharing schemes thus obtained are democratic, that is each participant belongs to the same number of minimal access sets and can easily be described.