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midterm-1-2009 - distance Show that if C is maximal then...

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Midterm I: Issued October 22 nd , 2009 1. Suppose that instead of using f( v )= v 3 as the second row of the parity-check matrix of a double-error correcting BCH code, we use f( v )= v -1 . For the case m =4, does this structure work as a double-error correcting code? 2. The International Standard Book Number (ISBN) system works as follows. The alphabet consists of the symbols {0,1,2,…,9,X}, where X denotes 10. The “information part” of the book number consists of nine digits. These are added with weights 10,9,…,2 and reduced mod 11 to give the final digit check. In this application, the common errors are for the digits to be wrong, and also transpositions, in which two adjacent symbols are interchanged. Quantify the performance of the ISBN code against these types of errors. 3. A [n,k,d] code C is maximal if the addition of any codeword to C reduces its minimal
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Unformatted text preview: distance. Show that if C is maximal, then r<d (r denotes the covering radius). Let C be with the same parameters as above, over a field F, such that any lengthening of C obtained by adding one column to a [n-k,n] parity-check matrix H of C generates d-1 dependent columns in H. Show that in this case r<d-1. 4. We saw in class that each power of a primitive element a of a field F(2^m) has a representation in terms of a polynomial of degree less than m (the field element 0 excluded). If you write these polynomial expansions out for all possible consecutive powers, show that the free coefficients in the expansion form a deBruijn sequence! Start with a small example of length 8, just to convince yourself that this is true!...
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  • Fall '08
  • Milenkovic,O
  • ISBN, International Standard Book Number, book Number, nine digits, Parity-check matrix, double-error correcting code

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