The authors study the intersection distribution rather of cyclic orbit codes. The intersection distribution is the number of codeword pairs whose intersection attains a given dimension and this is equivalent to studying the distance distribution.
A cyclic orbit code is a subspace code of the form \(\mathrm{Orb}(U)=\{\alpha U \mid \alpha\in F_{q^n}^\ast\},\) where \(U\) is an \(F_q\)-subspace of the field extension \(F_{q^n}.\) The first major result shows that the distance distribution of optimal full-length orbit codes is fully determined by the parameters \(q,\) \(n,\) and \(k,\) regardless of the choice of the Sidon space. The authors show the minimum and maximum possible value of \(f(U)\) (the number of fractions inside the field \(F_{q^n}\)) over all \(k\)-dimensional subspaces. It is also proved that \(f(U)\) is minimal if and only if \(\text{Orb}(U)\) is a spread code and maximal if and only if \(\text{Orb}(U)\) is an optimal full-length orbit code.
Further on, an investigation in the distance distribution of full-length orbit codes with distance less than \(2k-2\) is performed. distance distribution is not fully determined by \(q, n, k\) and the distance. A description of the distance distribution for the case where the distance is \(2k-4\) is given. Alternatively, the distance distribution is fully determined by \(q, n, k\) and the above mentioned parameter \(f(U).\)
Lastly, the authors consider codes that are the union of optimal full-length orbit codes such that the entire code has distance \(2k-2.\) In the case \((q, n, k) = (2, 13, 3)\), a Steiner system of this form has been found by computer search and here a generalization to unions of optimal full-length orbit codes is proved, that is, the distance distribution is fully determined by \(q, n, k,\) and the number of orbits.