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.