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 ${\mathrm{w}\mathrm{t}}_a(c)$, 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 $\mathrm{w}\mathrm{t}(c)$ 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 ${\mathrm{w}\mathrm{t}}_y(c)=5$.
• Let $A={𝐙}_3=\{0,1,2\}$ (an abelian group) and $c=002001200$. Then ${\mathrm{w}\mathrm{t}}_0(c)=6$, ${\mathrm{w}\mathrm{t}}_1(c)=1$, ${\mathrm{w}\mathrm{t}}_2(c)=2$ and $\mathrm{w}\mathrm{t}(c)={\mathrm{w}\mathrm{t}}_1(c)+{\mathrm{w}\mathrm{t}}_2(c)=3$.