Bh Sets as a Generalization of Golomb Rulers

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
$m = 31$
.