New Constant-Dimension Subspace Codes from Maximum Rank Distance Codes

摘要

The main problem of constant-dimension subspace coding is to determine the maximal possible sizen${mathbf{A}}_{q}(n,d,k)$nof a set ofn$k$n-dimensional subspaces inn${mathbf{F}}_{q}^{n}$nsuch that the subspace distance satisfiesn$d(U,V) geq d$nfor any two different subspacesn$U$nandn$V$nin this set. In this paper, we give a direct construction of constant-dimension subspace codes from two parallel versions of maximum rank-distance codes. The problem about the sizes of our constructed constant-dimension subspace codes is transformed into finding a suitable sufficient condition to restrict number of the roots ofn$L_{1}(L_{2}(x))-x$nwheren$L_{1}$nandn$L_{2}$naren$q$n-polynomials over the extension fieldn${mathbf{F}}_{q^{n}}$n. New lower bounds forn${mathbf{A}}_{q}(4k,2k,2k)$n,n${mathbf{A}}_{q}(4k+2,2k,2k+1)$n, andn${mathbf{A}}_{q}(4k+2,2(k-1),2k+1)$nare presented. Many new constant-dimension subspace codes better than previously best known codes with small parameters are constructed.