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# perfect ruler

A *perfect ruler* of length $n$ is a ruler with a subset of the integer markings $\{0,a_{2},\ldots,n\}\subset\{0,1,2,\ldots,n\}$ that appear on a regular ruler. The defining criterion of this subset is that there exists an $m$ such that any positive integer $k\leq m$ can be expresses uniquely as a difference $k=a_{i}-a_{j}$ for some $i,j$. This is referred to as an *$m$-perfect ruler*.

A 4-perfect ruler of length $7$ is given by $\{0,1,3,7\}$. To verify this, we need to show that every number $1,2,\ldots,4$ can be expressed as a difference of two numbers in the above set:

$\displaystyle 1$ | $\displaystyle=1-0$ | ||

$\displaystyle 2$ | $\displaystyle=3-1$ | ||

$\displaystyle 3$ | $\displaystyle=3-0$ | ||

$\displaystyle 4$ | $\displaystyle=7-3$ |

An optimal perfect ruler is one where for a fixed value of $n$ the value of $a_{n}$ is minimized.

Synonym:

Golomb ruler

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

03E02*no label found*05A17

*no label found*

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new correction: Closure properties on languages: DCFL not closed under reversal by babou

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