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 intersects 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 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 obtained osculating spaces of Veronese varieties are equidistant in the above metric. The parameters of the resulting linear network codes are determined.