Q2. (10 MARKS TOTAL FOR ALL PARTS-3+3+4) A. (3 Marks) Three codes shown below ar
ID: 674818 • Letter: Q
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Q2. (10 MARKS TOTAL FOR ALL PARTS-3+3+4) A. (3 Marks) Three codes shown below are each defined by a list of their codewords. For each code, provide the numerical value of r, n, k, and dmin t- the number of errors that the code is guaranteed to correct in an n bit codeworkd, e- the number of errors that the code is guaranteed to detect in an n bit codeword For each code, determine t and e (i) Code 1 (Hint: Code 1 is not a linear code) Codewords 100] [001] [010] (i) Code 2 Codewords [00000] [10100] [11011] (ii) Code 3 Codewords [00000]Explanation / Answer
Solution :
A) i) n=3, k=2 (there are 4 codewords), d = 2. The code rate is 2/3.
ii) n=5, k=2 (there are 4 codewords), d = 2. The code rate is 2/5.
iii) n=5, k=0, d = undefined. The code rate is 0.
n=5, k=1, d = undefined. The code rate is 1.
B) An (n, k) block code can represent in its parity bits at most 2n-k patterns, and these must cover all the error cases we wish to correct, as well as the one case with no errors. When the minimum Hamming distance is 2t + 1, the code can correct up to t errors. The number of ways in which the transmission can experience 0,1,2,...,t errors is equal to 1 + (n choose 1) + (n choose 2) + ... + (n choose t), and clearly this number must not exceed 2n-k.
C) L = 8.
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