Gray codes and Enumerative Coding for vector spaces

TL;DR

This paper explores Gray codes for vector spaces within the Grassmann and projective-space graphs, providing new constructions, non-existence results, and efficient encoding/decoding algorithms with improved performance in some cases.

Contribution

It introduces new cyclic optimal Gray code constructions for the Grassmann graph and specific constructions for the projective-space graph, along with efficient algorithms for encoding and decoding.

Findings

Constructed cyclic optimal codes for all parameters in the Grassmann graph.

Provided new constructions and non-existence results for the projective-space graph.

Developed encoding and decoding algorithms with competitive or improved computational complexity.

Abstract

Gray codes for vector spaces are considered in two graphs: the Grassmann graph, and the projective-space graph, both of which have recently found applications in network coding. For the Grassmann graph, constructions of cyclic optimal codes are given for all parameters. As for the projective-space graph, two constructions for specific parameters are provided, as well some non-existence results. Furthermore, encoding and decoding algorithms are given for the Grassmannian Gray code, which induce an enumerative-coding scheme. The computational complexity of the algorithms is at least as low as known schemes, and for certain parameter ranges, the new scheme outperforms previously-known ones.