Three new families of large cyclic subspace codes via high-dimensional Sidon spaces

Abstract

k-dimensional subspace codes, also called as constant dimension codes, play a crucial role in error correction of random network coding for ten years. Especially, large cyclic constant dimension codes with the optimal minimum distance are possible candidates owing to efficient encoding and decoding algorithms. The seminal work of Ben-Sasson et al. was using the roots of subspace polynomials to construct cyclic subspace codes. Shortly after, Roth et al. introduced the definitions of max-span and min-span Sidon spaces and designed a kind of new cyclic subspace codes which contain multiple orbits from Sidon spaces. By exploring Roth's ideas, Niu et al. and Li et al. further investigated high-dimensional Sidon spaces employing the direct sum to extend the previous results. However, we find that they only constructed high-dimensional Sidon spaces and presented several examples of cyclic subspace codes from Sidon spaces with dimension k. Combining the orbit of Sidon spaces with dimension k or more than k respectively, we will provide three families of larger cyclic subspace codes. Additionally, our constructions of cyclic subspace codes can yet maintain their the optimality of minimum distance.