New constructions of large cyclic subspace codes and Sidon spaces

Abstract Let n be a positive integer with a factor k such that n ≥ 3 k . 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 q n . In this paper, we construct cyclic subspace codes in G q ( n , k ) with minimum distance 2 k − 2 and size ( ⌈ n 2 k ⌉ − 1 ) ⋅ ( q n − 1 ) q k q − 1 . In the case n = 3 k , 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 = 2 k . We also establish the existence of Sidon spaces of G q ( 7 k , 2 k ) , and thus we resolve part of the conjecture about the existence of cyclic subspace codes in G q ( n , k ) with minimum distance 2 k − 2 and size q n − 1 q − 1 .