Gray Codes and Enumerative Coding for Vector Spaces
Moshe Schwartz Seniormember Ieee, Gray Codes, Frank Gray · 2014
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 nonexistence 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. Index Terms—Enumerative coding, Grassmannian, Gray codes, projective-space graph.