weight (strings)

Let $A$ be an alphabet, $a\in A$ a letter from $A$ and $c\in A^{*}$ astring over $a$. Then the $a$-weight of $c$, denoted by${\mathrm{wt}}_a(c)$, is the number of times $a$ occurs in $c$.

If $A$ is an abelian group, the Hamming weight $\mathrm{wt}(c)$ of $c$ (no ),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 ${\mathrm{wt}}_y(c)=5$.

• Let $A={\mathbb{Z}}_3=\{0,1,2\}$ (an abelian group) and$c:=002001200$. Then ${\mathrm{wt}}_0(c)=6$, ${\mathrm{wt}}_1(c)=1$, ${\mathrm{wt}}_2(c)=2$ and$\mathrm{wt}(c)={\mathrm{wt}}_1(c)+{\mathrm{wt}}_2(c)=3$.