New Constant-Dimension Subspace Codes from Maximum Rank Distance Codes

Abstract

The main problem of constant-dimension subspace coding is to determine the maximal possible size A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (n, d, k) of a set of k-dimensional subspaces in Fnq such that the subspace distance satisfies d(U, V) ≥ d for any two different subspaces U and V in 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 of L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> (L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (x)) - x where L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> are q-polynomials over the extension field Fqn. New lower bounds for A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (4k, 2k, 2k), Aq(4k t 2, 2k, 2k t 1), and A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (4k t 2, 2(k - 1), 2k t 1) are presented. Many new constantdimension subspace codes better than previously best known codes with small parameters are constructed.