Bounds for the multilevel construction

TL;DR

This paper investigates bounds for subspace codes in network coding, focusing on the multilevel construction, and formulates the problem as maximum clique problems in weighted graphs with polynomial weights.

Contribution

It provides new bounds for the multilevel construction and models the problem as maximum clique problems in weighted graphs with polynomial weights.

Findings

Derived bounds for subspace code cardinalities

Formulated the problem as maximum clique problems in weighted graphs

Connected subspace coding bounds with graph-theoretic optimization

Abstract

One of the main problems in random network coding is to compute good lower and upper bounds on the achievable cardinality of the so-called subspace codes in the projective space $\mathcal{P}_q(n)$ for a given minimum distance. The determination of the exact maximum cardinality is a very tough discrete optimization problem involving a huge number of symmetries. Besides some explicit constructions for \textit{good} subspace codes several of the most success full constructions involve the solution of discrete optimization subproblems itself, which mostly have not been not been solved systematically. Here we consider the multilevel a.k.a.\ Echelon--Ferrers construction and given lower and upper bounds for the achievable cardinalities. From a more general point of view, we solve maximum clique problems in weighted graphs, where the weights can be polynomials in the field size $q$.