Unformatted text preview: C ⊆ A n be an ( n , M , d )code. (i) Prove that the projection π : C → A n − d + 1 , de f ned by π( c 1 , . . . , c n ) = ( c d , . . . , c n ), is an injection. Solution. Let c = ( c 1 , . . . , c n ) and c = ( c 1 , . . . , c n ) lie in C . If π( c ) = π( c ) , then ( c d , . . . , c n ) = ( c d , . . . , c n ) ; that is, c and c agree in at least n − d + 1 positions, and so δ( c , c ) ≤ d − 1 < d . Since d is the minimum distance, c = c and π is an injection. (ii) ( Singleton bound . ) Prove that M ≤ q n − d + 1 . Solution. M =  C  , and part (i) gives  C  ≤ q n − d + 1 . 4.69 (i) If A is an alphabet with  A  = q and δ is the Hamming distance, prove that ± ± © w ∈ A n : δ( u , w) = i ª ± ± = ³ n i ´ ( q − 1 ) i ....
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 Fall '11
 KeithCornell
 Linear Algebra, Characteristic polynomial, Coding theory, π, Cayley–Hamilton theorem, −d +1

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