New constructions of large cyclic subspace codes and Sidon spaces

Let n be a positive integer with a factor k such that n >= 3k. Let q be a prime power, and let G(q)(n, k) be the set of all k-dimensional F-q-subspaces of the field F-qn. In this paper, we construct cyclic subspace codes in G(q)(n, k) with minimum distance 2k - 2 and size (left ceiling pi/2k right ceiling - 1) . (q(n)-1)q(k)/q-1 . In the case n = 3k, their sizes differ from the sphere-packing bound for subspace codes by a factor of 1/q-1 asymptotically as k goes to infinity. Our construction makes use of variants of the Sidon spaces constructed by Roth et al. (2018) and analogous to the results they attained for the case n = 2k. We also establish the existence of Sidon spaces of G(q)(7k, 2k), and thus we resolve part of the conjecture about the existence of cyclic subspace codes in G(q)(n, k) with minimum distance 2k - 2 and size q(n)-1/q-1. (C) 2020 Elsevier B.V. All rights reserved.