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weight (strings) (Definition)

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 $ {\mathrm{wt}}_a(c)$, is the number of times $ a$ occurs in $ c$.

If $ A$ is an abelian group, the Hamming weight $ {\mathrm{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 $ {\mathrm{wt}}_y(c)=5$.
  • Let $ A=\mathbb{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$.



"weight (strings)" is owned by GrafZahl.
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See Also: Kleene star

Other names:  weight
Also defines:  Hamming weight
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Cross-references: abelian group, occur ins, number, string, alphabet
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This is version 3 of weight (strings), born on 2005-04-30, modified 2005-05-01.
Object id is 6985, canonical name is WeightStrings.
Accessed 3452 times total.

Classification:
AMS MSC: 94A55 (Information and communication, circuits :: Communication, information :: Shift register sequences and sequences over finite alphabets)
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