Sidon spaces have been introduced by Bachoc, Serra and Z\'emor as the $q$-analogue of Sidon sets, classical combinatorial objects introduced by Simon Szidon. In 2018 Roth, Raviv and Tamo introduced the notion of $r$-Sidon spaces, as an extension of Sidon spaces, which may be seen as the $q$-analogue of $B_r$-sets, a generalization of classical Sidon sets. Thanks to their work, the interest on Sidon spaces has increased quickly because of their connection with cyclic subspace codes they pointed out. This class of codes turned out to be of interest since they can be used in random linear network coding. In this work we focus on a particular class of them, the one-orbit cyclic subspace codes, through the investigation of some properties of Sidon spaces and $r$-Sidon spaces, providing some upper and lower bounds on the possible dimension of their \textit{r-span} and showing explicit constructions in the case in which the upper bound is achieved. Moreover, we provide further constructions of $r$-Sidon spaces, arising from algebraic and combinatorial objects, and we show examples of $B_r$-sets constructed by means of them.

翻译：Sidon空间由Bachoc、Serra和Zémo引入，作为经典组合对象Sidon集（由Simon Szidon提出）的q-模拟。2018年，Roth、Raviv和Tamo将Sidon空间推广为r- Sidon空间的概念，可视为经典Sidon集推广形式B_r集的q-模拟。由于他们揭示了Sidon空间与循环子空间码的关联，学界对Sidon空间的兴趣迅速增长。这类码因可用于随机线性网络编码而备受关注。本文聚焦于一类特殊的循环子空间码——单轨道循环子空间码，通过研究Sidon空间和r-Sidon空间的若干性质，给出了其r-张成空间可能维数的上下界，并展示了达到上界时的显式构造。此外，我们还利用代数与组合对象给出了r-Sidon空间的更多构造方法，并展示了通过它们构造B_r集的实例。