Bh Sets as a Generalization of Golomb Rulers

Abstract

A set of positive integers $A$ is called a Golomb ruler if the difference between two distinct elements of $A$ are different, equivalently if the sums of two elements are different ( $B_{2}$ set, Sidon set). An extension of this concept is to consider that the sum of $h $ elements in $A $ are all different, except for permutation of the summands, with $h \geq 2 $ , in this case it is said that $A $ is a set $B_{h} $ , the length of $A $ is given by $\ell (A)\,\,= \max A- \min A $. One problem associated with this type of set is that of the optimal dense $B_{h} $ sets, that is, determining the greatest cardinal of a set $B_{h} $ contained in the integer interval $\left [{1, N }\right] $ , for this defines the function $F_{h} (N)~$. Another problem that can be associated is the optimally short $B_{h} $ sets, that is, finding a shorter $B_{h} $ set with $m $ elements, for which the $G_{h} (m)~$ function is defined. In this paper we are going to prove that these two problems are inverse, that is, that the functions $G_{h} (m)~$ and $F_{h} (N)~$ have inverse relationships. Furthermore, the asymptotic behavior of the $G_{h} (m)~$ function is studied, obtaining some upper and lower bounds, we also obtain tables of $B_{3}$ and $B_{4}$ near-optimal up to $m = 31$.