Constructions of Cyclic Subspace Codes and Maximum Rank Distance Codes

This chapter is a survey of the recent results on the constructions of cyclicsubspace codes and maximum rank distance codes. Linearized polynomials are themain tools used to introduce both constructions in this chapter. In the constructionof cyclic subspace codes, codewords are considered as the root spaces of somesubspace polynomials (which are a particular type of linearized polynomials). Inthis set up, some algebraic manipulations on the coefficients and degrees of suchpolynomials are applied to provide a systematic construction of cyclic subspacecodes. In constructions of maximum rank distance codes, linearized polynomialsare used as codewords again, but in a different way. Codewords of rank metriccodes are considered as the linear maps that these polynomials represent. All knownconstructions of maximum rank distance codes in the literature are summarized usingthis linearized polynomial representation. Connections among the constructions andfurther explanations are also provided