Weight (strings)

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

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

• Let ${\displaystyle A=\{x,y,z\}}$. In the string ${\displaystyle c=yxxzyyzxyzzyx}$, ${\displaystyle y}$ occurs 5 times, so the ${\displaystyle y}$-weight of ${\displaystyle c}$ is ${\displaystyle {\mathrm{w}\mathrm{t}}_y(c)=5}$.
• Let ${\displaystyle A={\mathbf{𝐙}}_3=\{0,1,2\}}$ (an abelian group) and ${\displaystyle c=002001200}$. Then ${\displaystyle {\mathrm{w}\mathrm{t}}_0(c)=6}$, ${\displaystyle {\mathrm{w}\mathrm{t}}_1(c)=1}$, ${\displaystyle {\mathrm{w}\mathrm{t}}_2(c)=2}$ and ${\displaystyle \mathrm{w}\mathrm{t}(c)={\mathrm{w}\mathrm{t}}_1(c)+{\mathrm{w}\mathrm{t}}_2(c)=3}$.

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