Abstract-In this paper we introduce the class of Spread Codes for the use in random network coding. Spread Codes are based on the construction of spreads in finite projective geometry. The major contribution of the paper is an efficient decoding algorithm of spread codes up to half the minimum distance.

This paper contributes to the study of rank-metric codes from an algebraic and combinatorial point of view. We introduce q-polymatroids, the q-analogue of polymatroids, and develop their basic properties. We associate a pair of q-polymatroids to a rank-metric codes and show that several invariants and structural properties of the code, such as generalized weights, the property of being MRD or an optimal anticode, and duality, are captured by the associated combinatorial object.We start by establishing the notation and the definitions used throughout the paper.Notation 1.1. In the sequel, we fix integers n, m ≥ 2 and a prime power q. For an integer t, we let [t] := {1, ..., t}. We denote by F q the finite field with q elements. The space of n × m matrices with entries in F q is denoted by Mat. Up to transposition, we assume without loss of generality that n ≤ m. We let Mat(J, c) = {M ∈ Mat | colsp(M ) ⊆ J} and Mat(J, r) = {M ∈ Mat | rowsp(M ) ⊆ J}.Throughout the paper, we only consider linear codes. All dimensions are computed over F q , unless otherwise stated.

Optimal rank-metric codes in Ferrers diagrams are considered. Such codes consist of matrices having zeros at certain fixed positions and can be used to construct good codes in the projective space. Four techniques and constructions of Ferrers diagram rank-metric codes are presented, each providing optimal codes for different diagrams and parameters.

We study the family of ideals defined by mixed size minors of two-sided ladders of indeterminates. We compute their Gröbner bases with respect to a skew-diagonal monomial order, then we use them to compute the height of the ideals. We show that these ideals correspond to a family of irreducible projective varieties, that we call mixed ladder determinantal varieties. We show that these varieties are arithmetically Cohen-Macaulay, and we characterize the arithmetically Gorenstein ones. Our main result consists in proving that mixed ladder determinantal varieties belong to the same G-biliaison class of a linear variety.

Abstract. We give an optimal-size representation for the elements of the trace zero subgroup of the Picard group of an elliptic or hyperelliptic curve of any genus, with respect to a field extension of any prime degree. The representation is via the coefficients of a rational function, and it is compatible with scalar multiplication of points. We provide efficient compression and decompression algorithms, and complement them with implementation results. We discuss in detail the practically relevant cases of small genus and extension degree, and compare with the other known compression methods.