strings

Weight (strings)

The a-weight of a string, for a a letter, is the number of times that letter occurs in the string. More precisely, let A be a finite set (called the alphabet), ain A a letter of A, and cin 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.

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