Osculating Spaces of Varieties and Linear Network Codes

TL;DR

This paper develops a theoretical framework using osculating spaces of algebraic varieties, especially Veronese varieties, to construct linear network codes with desirable intersection properties for improved decoding.

Contribution

It introduces a novel method to generate linear network codes from osculating spaces of algebraic varieties, providing explicit families with equidistant properties.

Findings

Constructed explicit families of equidimensional vector spaces with uniform intersection dimensions.

Demonstrated that osculating spaces of Veronese varieties are equidistant under the network coding metric.

Determined parameters of the resulting 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.