problem2

# problem2 - University of California San Diego ECE 259A...

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University of California San Diego ECE 259A: Problem Set #2 1 . Recall that the syndrome s is a linear function of the error-pattern e . A linear decoder finds its estimate e of the error-pattern as a linear function of s . That is e L s , where L satisfies L s s L s L s . Show that a linear decoder for a binary linear code can correct at most n k of the n single erorrs (and, hence, it is not a particularly useful decoder). 2 . Another proof of the Gilbert-Varshamov bound (due to Gilbert): (a) Given an n , 2 k , d binary code – not necessarily linear – show that for any integer r , the average number of codewords in a Hamming sphere of radius r is given by r def x n 2 x , r 2 n V n , r 2 n k where V n , r r i 0 n i is the volume of the Hamming sphere of radius r . (b) Use the above relation with r d to show that there exist binary codes of rate R and mini- mum distance d , which satisfy R 1 1 n log 2 V n , d . 3

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problem2 - University of California San Diego ECE 259A...

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