Some New Optimal Generalized Sidon Sequences

Abstract

A sequence $A$ is a $B_h^{\ast }[g]$ sequence if the coefficients of $(\sum _{a\in A}(z{)}^a{)}^h$ are bounded by $g$. The standard Sidon sequence is a $B[2]$ sequence. Finite Sidon sequences are called Golomb rulers, which are found to have many applications such as error correcting codes, radio frequency selection, and radio antennae placement. Let $R_h(g,n)$ be the largest cardinality of a $B[g]$ sequence contained in $\{1,2,\dots ,n\}$, and $F(h,g,k)=min\{n:R_h(g,n)\ge k\}$. In this paper, computational techniques are applied to construct optimal generalized Sidon sequences, and $49$ new exact values of $F(2,g,k)$ are found.