Almost Difference Sets From Singer Type Golomb Rulers

Let
$G$
be an additive group of order
$v$
. A
$k$
-element subset
$D$
of
$G$
is called a
$(v, k, \lambda, t)$
-almost difference set if the expressions
$g-h$
, for
$g$
and
$h$
in
$D$
, represent
$t$
of the non-identity elements in
$G$
exactly
$\lambda $
times and every other non-identity element
$\lambda + 1$
times. Almost difference sets are highly sought after as they can be used to produce functions with optimal nonlinearity, cyclic codes, and sequences with three-level autocorrelation. A set of positive integers
$A$
is called a Golomb ruler if the difference between two distinct elements of
$A$
are different. In this paper, we use Singer type Golomb rulers to construct new families of almost difference sets. Additionally, we constructed 2-adesigns from these almost difference sets.