A new construction of cyclic subspace codes

Subspace codes have attracted a lot of attention in the last few decades due to their applications in noncoherent linear network coding, in particular cyclic subspace codes can be encoded and decoded more efficiently because of their special algebraic structure. In this paper, we present a family of cyclic subspace codes with minimum distance \(\varvec{2k-2}\) $\mathbf{2}\mathit{k}\mathbf{-}\mathbf{2}$ and size \(\varvec{seq^{k}(q^k-1)^{s-1}(q^n-1)+\frac{q^n-1}{q^k-1}}\) $\mathit{s}\mathit{e}{\mathit{q}}^{\mathit{k}}{\mathbf{(}{\mathit{q}}^{\mathit{k}}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathit{s}\mathbf{-}\mathbf{1}}\mathbf{(}{\mathit{q}}^{\mathit{n}}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{+}\frac{{\mathit{q}}^{\mathit{n}}\mathbf{-}\mathbf{1}}{{\mathit{q}}^{\mathit{k}}\mathbf{-}\mathbf{1}}$ , where \(\varvec{k|n}\) $\mathit{k}\mathbf{|}\mathit{n}$ , \(\varvec{\frac{n}{k}\ge 2s+1}\) $\frac{\mathit{n}}{\mathit{k}}\mathbf{\ge }\mathbf{2}\mathit{s}\mathbf{+}\mathbf{1}$ , \(\varvec{s\ge 1, e=\lceil \frac{n}{2sk} \rceil -1}\) $\mathit{s}\mathbf{\ge }\mathbf{1}\mathbf{,}\mathit{e}\mathbf{=}\mathbf{\lceil }\frac{\mathit{n}}{\mathbf{2}\mathit{s}\mathit{k}}\mathbf{\rceil }\mathbf{-}\mathbf{1}$ . In the case of \(\varvec{n=(2s+1)k}\) $\mathit{n}\mathbf{=}\mathbf{(}\mathbf{2}\mathit{s}\mathbf{+}\mathbf{1}\mathbf{)}\mathit{k}$ with \(\varvec{2\le s <q^k}\) $\mathbf{2}\mathbf{\le }\mathit{s}\mathbf{<}{\mathit{q}}^{\mathit{k}}$ , our cyclic subspace codes have larger size than the known ones in the literature.