Equidistant Codes in the Grassmannian

Equidistant codes over vector spaces are considered. For $k$-dimensional subspaces over a large vector space the largest code is always a sunflower. We present several simple constructions for such codes which might produce the largest non-sunflower codes. A novel construction, based on the Pl\"{u}cker embedding, for 1-intersecting codes of $k$-dimensional subspaces over \F_q^n, $n \geq \binom{k+1}{2}$, where the code size is $\frac{q^{k+1}-1}{q-1}$ is presented. Finally, we present a related construction which generates equidistant constant rank codes with matrices of size $n \times \binom{n}{2}$ over \F_q, rank $n-1$, and rank distance $n-1$.Comment: 16 page

Problems on q-Analogs in Coding Theory

The interest in $q$-analogs of codes and designs has been increased in the last few years as a consequence of their new application in error-correction for random network coding. There are many interesting theoretical, algebraic, and combinatorial coding problems concerning these q-analogs which remained unsolved. The first goal of this paper is to make a short summary of the large amount of research which was done in the area mainly in the last few years and to provide most of the relevant references. The second goal of this paper is to present one hundred open questions and problems for future research, whose solution will advance the knowledge in this area. The third goal of this paper is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author

Osculating Spaces of Varieties and Linear Network Codes

Abstract. We present a general theory to obtain good linear network codes utilizing the osculating nature of algebraic varieties. In particular, we obtain from the osculating spaces of Veronese varieties explicit families of equidimensional vector spaces, in which any pair of distinct vector spaces intersect in the same dimension. Linear network coding transmits information in terms of a basis of a vector space and the information is received as a basis of a possible altered vector space. Ralf Koetter and Frank R. Kschischang [KK08] introduced a metric on the set af vector spaces and showed that a minimal distance decoder for this metric achieves correct decoding if the dimension of the intersection of the transmitted and received vector space is sufficiently large. The proposed osculating spaces of Veronese varieties are equidistant in the above metric. The parameters of the resulting linear network codes are determined. Notation – Fq is the finite field with q elements of characteristic p. – F = Fq is an algebraic closure of Fq. – Rd = F[X0,..., Xn]d and Rd(Fq) = Fq[X0,..., Xn]d the homogenous polynomials of degree d with coefficients in F and Fq. – R = F[X0,..., Xn] = ⊕dRd and R(Fq) = Fq[X0,..., Xn] = ⊕dRd(Fq) – AffCone(Y) ⊆ FM+1 denotes the affine cone of the subvariety Y ⊆ PM and AffCone(Y)(Fq) its Fq-rational points. – Ok,X,P ⊆ PM is the embedded osculating space of a variety X ⊆ PM at the point P ∈ X and Ok,X,P (Fq) its Fq-rational points, see 1.2. – V = σd(Pn) ⊆ PM with M = () d+n − 1 is the Veronese variety, n see 1.1. For generalities on algebraic geometry we refer to [Har77]. Part of this work was done while visiting Institut de Mathématiques de Luminy