A set of positive integers A is called a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula> -Golomb ruler if the difference between two distinct elements of A is repeated up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula> times. This definition is a generalization of the Golomb ruler <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(g = 1)$ </tex-math></inline-formula> . In this paper, we obtain new constructions for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula> -Golomb rulers from Golomb rulers, using these constructions we find some suboptimal 2 and 3-Golomb rulers with up to 124 marks and we prove two theorems related to extremal functions associated with this sets improving already known results.