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Suggested poblems Problem 1 1. Generate the G and H matrix for the dual code of the Hamming code with n=15. 2. Determine a code basis for the dual code 3. Evaluate the minimum distance 4. Evaluate the error correction capability t and the error detection capability e Solution 1. The code parameters will be: n=15, 2 r =n+1=16, r=4, k=n-r=11. H is built with columns that contain all the combinations of 1, 2, …, r ones (which is 1,2,3,4 ones) (see HW3). Possible H , and G are: = 1 1 1 1 0 1 1 0 1 0 0 1 0 0 0 1 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 1 0 1 1 0 1 0 0 1 0 0 0 1 H = 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 G and then G dual =H and H dual =G. 2. The rows of G dual (i.e. of H ) form a possible basis for the dual code Basis = {(100010010110111), (010011001011011), (001001110011101), (000100101101111)} 3. Minimum distance d min =8 4. 7 1 d e 3 2 1 min min = = = = d t Problem 2
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