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Binary codes with XOR operation forms an abelian group.

${\displaystyle (\{T,F\},\oplus )}$ or $⟨B^n,⨁⟩$ is an abelian group.

• associativity : $(a\oplus b)\oplus c=a\oplus (b\oplus c)$
• identity element : $0\oplus a=a$
• $a\oplus a=0$ every element is its own inverse
• commutativity : $a\oplus b=b\oplus a$

binary words: strings of 1s and 0s.

the set of all binary words of length n called $B^n$

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• Exclusive or
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