solution2 - University of California San Diego ECE 259A...

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University of California San Diego ECE 259A: Solutions to Problem Set #2 1 . The set of all the syndromes s is equal to n k 2 which is a vector space of dimension n k . Thus the image of n k 2 under any linear mapping L s cannot have dimension larger than n k . On the other hand, since the n vectors of weight 1 form a basis for n 2 , any subset of n 2 which contains more than n k such vectors (single error-patterns) must have dimension greater than n k . 2 . (a) Count in two ways the total number 2 n r of codewords contained in Hamming spheres of radius r about every point in the space n 2 . The Frst way to count this number is the def- inition of r . The second way is as follows. Because any codeword has exactly V n , r points x n 2 within distance r from it, it lies in exactly V n , r such spheres. Therefore the total number of codewords in all such spheres must equal V n , r . Hence, we have r V n , r 2 n V n , r 2 n k . (b) We may always assume w.l.o.g. that d 1 . Otherwise, there must exist at least one point x n 2 which has no codeword within distance d from it. We could then declare x to be a codeword, thereby enlarging the code and increasing the rate. Hence 2 k n V n , d 1 or k n log 2 V n , d , from which the required result easily follows. 3 . Exactly half of the codewords of have even weight and half have odd weight. Hence the num- ber of codewords of weight d is at at most 2 k 1 , and we have: d 2 k 1 d 1 2 k 1 1 x wt x n 2 k 1 where the second inequality is essentially the Plotkin bound. If d is even, then the number of non- zero codewords of weight d is at most 2 k 1 1 , and we have d 2 k 1 1 d 1 2 k 1 n 2 k 1 which is a slightly better bound. 4 . (a) Extending: Clearly n n 1 and k k . Since the weight of all the codewords in is even by construction, we have d d 1 if d is odd, and d d if d is even. A generator matrix for is given by G G G 1 t , where 1 denotes the all-one vector. A parity-check matrix for may be obtained by appending to
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This note was uploaded on 10/12/2009 for the course COMM ECE259A taught by Professor Prof.alexandervardy during the Fall '08 term at San Diego.

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solution2 - University of California San Diego ECE 259A...

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