State of Art in Subspace Coding

Abstract

The problems of subspace coding of large cardinality are considered. The main most interesting results which were published to this time are presented. A few directions of the development of this scientific area are mentioned. One of the direction was stated by Koetter, Kschischang and Silva and simultaneously was continued by Gabidulin and Bossert. It is based on so called lifting constructions of Gabidulin rank codes. Another direction is prepared by Etzion and Silberstein. It uses Ferrera diagramme to creat right position in the code matrix to lifted MRD codes. The second branch of this direction represents the subspace codes which use uncomplete balanced block designs. There exist codes that were constructed by computing search, for example Schischkin codes. There is a direction which uses others approaches such as using Steiner systems, Honold's method of removing codewords from lifting constructions and adding much more new words obtained by special way. It gives some examples of the subspace code that surpassed cardinality of some known codes. There were shown open problems in this area.