Coding Theory and Projective Spaces

TL;DR

This paper explores error-correcting codes in projective spaces, introducing new formulas, bounds, and constructions for constant dimension codes, and analyzing their properties and applications in coding theory.

Contribution

It introduces a multilevel construction using Ferrers diagram rank-metric codes and provides new bounds and efficient encoding/decoding methods for constant dimension codes.

Findings

New formula for subspace distance computation

Representation of lifted MRD codes as transversal designs

Construction of codes attaining upper bounds

Abstract

The projective space of order $n$ over a finite field $\F_q$ is a set of all subspaces of the vector space $\F_q^{n}$. In this work, we consider error-correcting codes in the projective space, focusing mainly on constant dimension codes. We start with the different representations of subspaces in the projective space. These representations involve matrices in reduced row echelon form, associated binary vectors, and Ferrers diagrams. Based on these representations, we provide a new formula for the computation of the distance between any two subspaces in the projective space. We examine lifted maximum rank distance (MRD) codes, which are nearly optimal constant dimension codes. We prove that a lifted MRD code can be represented in such a way that it forms a block design known as a transversal design. The incidence matrix of the transversal design derived from a lifted MRD code can be viewed as a parity-check matrix of a linear code in the Hamming space. We find the properties of these codes which can be viewed also as LDPC codes. We present new bounds and constructions for constant dimension codes. First, we present a multilevel construction for constant dimension codes, which can be viewed as a generalization of a lifted MRD codes construction. This construction is based on a new type of rank-metric codes, called Ferrers diagram rank-metric codes. Then we derive upper bounds on the size of constant dimension codes which contain the lifted MRD code, and provide a construction for two families of codes, that attain these upper bounds. We generalize the well-known concept of a punctured code for a code in the projective space to obtain large codes which are not constant dimension. We present efficient enumerative encoding and decoding techniques for the Grassmannian. Finally we describe a search method for constant dimension lexicodes.