Editing Golomb ruler (section)

==Definitions==

===Golomb rulers as sets===
A set of integers <math>A = \{a_1,a_2,...,a_m\}</math> where <math>a_1 < a_2 < ... < a_m</math> is a Golomb ruler if and only if

:<math>\text{for all } i,j,k,l \in \left\{1,2,...,m\right\} \text{such that } i \neq j \text{ and } k \neq l,\ a_i - a_j = a_k - a_l \iff i=k \text{ and } j=l.</math><ref>{{cite web
  | last = Dimitromanolakis
  | first = Apostolos
  | title = Analysis of the Golomb Ruler and the Sidon Set Problems, and Determination of Large, Near-Optimal Golomb Rulers
  | url = http://www.cs.toronto.edu/%7Eapostol/golomb/main.pdf
  | access-date = 2009-12-20
  }}</ref>

The ''order'' of such a Golomb ruler is <math>m</math> and its ''length'' is <math>a_m - a_1</math>. The [[canonical form]] has <math>a_1 = 0</math> and, if <math>m>2</math>, <math>a_2 - a_1 < a_m - a_{m-1}</math>. Such a form can be achieved through translation and reflection.

===Golomb rulers as functions===
An [[injective function]] <math>f:\left\{1,2,...,m\right\} \to \left\{0,1,...,n\right\}</math> with <math>f(1) = 0</math> and <math>f(m) = n</math> is a Golomb ruler if and only if

:<math>\text{for all } i,j,k,l \in \left\{1,2,...,m\right\} \text{such that } i \neq j \text{ and } k \neq l, f(i)-f(j) = f(k)-f(l) \iff i=k \text{ and } j=l.</math><ref name="Drakakis">{{Cite journal
  | last = Drakakis
  | first = Konstantinos
  | title = A Review Of The Available Construction Methods For Golomb Rulers
  | journal = Advances in Mathematics of Communications
  | volume = 3
  | issue = 3
  | pages = 235–250
  | year = 2009
  | doi = 10.3934/amc.2009.3.235| doi-access = 
  }}</ref>{{rp|236}}

The ''order'' of such a Golomb ruler is <math>m</math> and its ''length'' is <math>n</math>. The canonical form has
:<math>f(2)<f(m)-f(m-1)</math> if <math>m>2</math>.

===Optimality===
A Golomb ruler of order <var>m</var> with length <var>n</var> may be [[optimization (mathematics)|optimal]] in either of two respects:<ref name="Drakakis"/>{{rp|237}}

* It may be ''optimally dense'', exhibiting maximal <var>m</var> for the specific value of <var>n</var>,
* It may be ''optimally short'', exhibiting minimal <var>n</var> for the specific value of <var>m</var>.

The general term ''optimal Golomb ruler'' is used to refer to the second type of optimality.