Weight (strings)

The $a$ -weight of a string, for a letter $a$ , 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 $\mathrm {wt} (c)=\mathrm {wt} _{1}(c)+\mathrm {wt} _{2}(c)=3$ .

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