Weight (strings)

From Wikipedia, the free encyclopedia

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), $a\in A$ 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 $wt a (c)$ , 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 $wt(c)$ of $c$ , often simply referred to as "weight", is the number of nonzero letters in $c$ .

[edit] Examples

Let $A = {x, y, z}$ . In the string $c = y x x z y y z x y z z y x$ , $y$ occurs 5

times, so the $y$ -weight of $c$ is $wt y (c) = 5$ .

Let $A=\mathbf{Z}_3=\{0,1,2\}$ (an abelian group) and $c = 002001200$ . Then $wt 0 (c) = 6$ , $wt 1 (c) = 1$ , $wt 2 (c) = 2$ and $wt(c) = wt 1 (c) + wt 2 (c) = 3$ .

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Category: Semigroup theory

Weight (strings)

From Wikipedia, the free encyclopedia

[edit] Examples

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