Let $A$ be an alphabet, $a\in A$ a letter from $A$ and $c\in A^*$ a string over $a$ . Then the $a$ -weight of $c$ , denoted by $\wt_a(c)$ , is the number of times $a$ occurs in $c$ .

If $A$ is an abelian group, the Hamming weight $\wt(c)$ of $c$ (no index), 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 $\wt_y(c)=5$ .
• Let $A=\mbb{Z}_3=\{0,1,2\}$ (an abelian group) and $c:=002001200$ . Then $\wt_0(c)=6$ , $\wt_1(c)=1$ , $\wt_2(c)=2$ and $\wt(c)=\wt_1(c)+\wt_2(c)=3$ .