This preview shows page 1. Sign up to view the full content.
Unformatted text preview: C A n be an ( n , M , d )code. (i) Prove that the projection : C A n d + 1 , de f ned by ( 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 ( c ) = ( 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 ( c , c ) d 1 < d . Since d is the minimum distance, c = c and 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 is the Hamming distance, prove that w A n : ( u , w) = i = n i ( q 1 ) i ....
View Full
Document
 Fall '11
 KeithCornell

Click to edit the document details