New constructions of large cyclic subspace codes and Sidon spaces

Abstract

Let n be a positive integer with a factor k such that n≥3k. Let q be a prime power, and let Gq(n,k) be the set of all k-dimensional Fq-subspaces of the field Fqn. In this paper, we construct cyclic subspace codes in Gq(n,k) with minimum distance 2k−2 and size (⌈n2k⌉−1)⋅(qn−1)qkq−1. In the case n=3k, their sizes differ from the sphere-packing bound for subspace codes by a factor of 1q−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 Gq(7k,2k), and thus we resolve part of the conjecture about the existence of cyclic subspace codes in Gq(n,k) with minimum distance 2k−2 and size qn−1q−1.