Large Constant Dimension Codes and Lexicodes

TL;DR

This paper explores large constant dimension codes used in network coding, introduces efficient search methods, and provides a formula for calculating distances between subspaces of varying dimensions.

Contribution

It presents a more efficient computer search approach for large constant dimension codes and introduces a formula for subspace distance computation.

Findings

Some lexicodes are larger than previously known codes

The new distance formula improves search efficiency

Enhanced methods facilitate the discovery of larger codes

Abstract

Constant dimension codes, with a prescribed minimum distance, have found recently an application in network coding. All the codewords in such a code are subspaces of $\F_q^n$ with a given dimension. A computer search for large constant dimension codes is usually inefficient since the search space domain is extremely large. Even so, we found that some constant dimension lexicodes are larger than other known codes. We show how to make the computer search more efficient. In this context we present a formula for the computation of the distance between two subspaces, not necessarily of the same dimension.