Construction of Sidon spaces with applications to coding

A subspace of a finite extension field is called a Sidon space if the product of any two of its elements is unique up to a scalar multiplier from the base field. Sidon spaces were recently introduced by Bachoc et al. as a means to characterize multiplicative properties of subspaces, and yet no explicit constructions were given. In this paper, several constructions of Sidon spaces are provided. In particular, in some of the constructions the relation between $k$, the dimension of the Sidon space, and $n$, the dimension of the ambient extension field, is optimal. These constructions are shown to provide cyclic subspace codes, which are useful tools in network coding schemes. To the best of the authors' knowledge, this constitutes the first set of constructions of non-trivial cyclic subspace codes in which the relation between $k$ and $n$ is polynomial, and in particular, linear. As a result, a conjecture by Trautmann et al. regarding the existence of non-trivial cyclic subspace codes is resolved for most parameters, and multi-orbit cyclic subspace codes are attained, whose cardinality is within a constant factor (close to $1/2$) from the sphere-packing bound for subspace codes.