On generalized Sidon spaces

TL;DR

This paper explores the properties and constructions of Sidon and r-Sidon spaces, their bounds, and applications to cyclic subspace codes, with implications for network coding and combinatorial design.

Contribution

It provides new bounds, explicit constructions, and algebraic and combinatorial methods for Sidon and r-Sidon spaces, advancing their theoretical understanding and practical applications.

Findings

Upper and lower bounds on the dimension of r-span of Sidon spaces

Explicit constructions of r-Sidon spaces achieving bounds

Examples of B_r-sets derived from these constructions

Abstract

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.