On the length of Golomb Ruler: A function construction approach based on difference triangle

Abstract

Since the significance of Golomb Ruler Problem in some context, we proposes a function construction approach based on difference triangle to generate near-optimal Golomb rulers. Let x1, x2, ..., xn be an increasing sequence of integers, where x1 = 0, which satisfies the following conditions: if abs(xi-xj) = abs(xp-xq) then {i, j} = {p,q}. Our objective is to find the order of minimum xn for any given n. In this paper, the two results in a paper are both improved. In addition, it will be shown that the length of Golomb Ruler have been shortened to a half, and that the satisfying sequence can not be generated by such a quadratic formula as $xi = ai^2+bni+ci+dn^2+en+f$ for any rational a, b, c, d, e and f.