Constructions of Orbit Codes Based on Unitary Spaces Over Finite Fields

Orbit codes, as special constant dimension codes, have attracted much attention due to their applications for error correction in random network coding. This paper is devoted to constructing large orbit codes by making full use of unitary space. Firstly, we construct a cyclic unitary group of order
$q^{2n}-1$
by means of the companion matrix of a primitive polynomial over finite fields
$\mathbb {F}_{q^{2}}$
, and so the corresponding code is unitary cyclic orbit code. As a special application, a new quaternary orbit code
$(6,63,4,3)$
is given. Secondly, we obtain orbit codes with large size using the external direct product of unitary groups acting on the direct sum of subspaces. Finally, a table is given for illustrating our codes improve upon those constructed by Trautmann
et al.
and Poroch
et al.