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 $$\mathrm{wt}(c)=\mathrm{wt}_1(c)+\mathrm{wt}_2(c)=3$$.