Weight (strings)

From Wikipedia, the free encyclopedia

The $a$ -weight of a string, for $a$ a letter, is the number of times that letter occurs in the string. More precisely, let $A$ be a finite set (called the alphabet), $a\in A$ a letter of $A$ , and $c\in A^{*}$ a string (where $A^{*}$ is the free monoid generated by the elements of $A$ , equivalently the set of strings, including the empty string, whose letters are from $A$ ). Then the $a$ -weight of $c$ , denoted by ${\mathrm {wt}}_{a}(c)$ , is the number of times the generator $a$ occurs in the unique expression for $c$ as a product (concatenation) of letters in $A$ .

If $A$ is an abelian group, the Hamming weight ${\mathrm {wt}}(c)$ of $c$ , often simply referred to as "weight", is the number of nonzero letters in $c$ .

Examples

Let $A=\{x,y,z\}$ . In the string $c=yxxzyyzxyzzyx$ , $y$ occurs 5 times, so the $y$ -weight of $c$ is ${\mathrm {wt}}_{y}(c)=5$ .
Let $A={\mathbf {Z}}_{3}=\{0,1,2\}$ (an abelian group) and $c=002001200$ . Then ${\mathrm {wt}}_{0}(c)=6$ , ${\mathrm {wt}}_{1}(c)=1$ , ${\mathrm {wt}}_{2}(c)=2$ and ${\mathrm {wt}}(c)={\mathrm {wt}}_{1}(c)+{\mathrm {wt}}_{2}(c)=3$ .

This article incorporates material from Weight (strings) on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.