Summary: Let \(n\) be a positive integer with a factor \(k\) such that \(n \geq 3 k\). Let \(q\) be a prime power, and let \(\mathcal{G}_q ( n , k )\) be the set of all \(k\)-dimensional \(\mathbb{F}_q\)-subspaces of the field \(\mathbb{F}_{q^n} \). In this paper, we construct cyclic subspace codes in \(\mathcal{G}_q ( n , k )\) with minimum distance \(2 k - 2\) and size \(( \lceil \frac{ n}{ 2 k} \rceil - 1 ) \cdot \frac{ ( q^n - 1 ) q^k}{ q - 1} \). In the case \(n = 3 k\), their sizes differ from the sphere-packing bound for subspace codes by a factor of \(\frac{ 1}{ q - 1}\) asymptotically as \(k\) goes to infinity. Our construction makes use of variants of the Sidon spaces constructed by R. M. Roth et al. [IEEE Trans. Inf. Theory 64, No. 6, 4412–4422 (2018; Zbl 1395.94237)] and analogous to the results they attained for the case \(n = 2 k\). We also establish the existence of Sidon spaces of \(\mathcal{G}_q ( 7 k , 2 k )\), and thus we resolve part of the conjecture about the existence of cyclic subspace codes in \(\mathcal{G}_q ( n , k )\) with minimum distance \(2 k - 2\) and size \(\frac{ q^n - 1}{ q - 1} \).