Some New Optimal Generalized Sidon Sequences

Abstract

A sequence \(A\) is a \(B_h^*[g]\) 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 \(\{1,2,\ldots,n\}\), and \(F(h,g,k) = \min\{n : R_h(g,n) \geq k\}\). 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.