a) Which of these are valid Hamming code parameters in the (n,k,d) format: (10,3,3), (15,11,4), (7,4,3), (1023,1000,23)? Explain your answer.
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For a q-ary Hamming code with parameter m (m >= 1): n = (q^m - 1)/(q - 1), k = n - m, and the classical Hamming minimum distance is d = 3. For the binary case (q = 2) this specializes to n = 2^m - 1, k = 2^m - 1 - m, d = 3. The binary extended Hamming code (adding Show more…
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(a) Is it possible to create a code with 4 codewords such that each codeword is a binary string of length 3, and the code can detect any single error? If it is possible, give the 4 codewords. If it is not, explain why not. (b) The following message is received using the 15-digit Hamming Code. Correct any single error that may have occurred during transmission. Explain how you know where the error is. (c) Using the same table as in part (a), what 15-digit codeword would you send if you wanted to communicate the message 11000101010?
Adi S.
2. a) The (7,4) Hamming code has the following generator matrix. G = 1 0 0 0 1 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 1 i) What is the minimum Hamming distance of the code? How many errors the Hamming code can be detected and how many errors can be corrected using the code. ii) Write down the parity check matrix H for the Hamming code. iii) If the information bits are 1010, determine the code word. iv) If the received word is 0100011, what is the output of the decoder? b) A (7,4) cyclic code has the generator polynomial g(x) = x^3 + x^2 + 1, find the generator matrix in the systematic format (Standard Echelon Format).
Consider a systematic (n, k) linear block code with parity check equations: v0 = m0 + m1 + m3 v1 = m0 + m2 + m3 v2 = m0 + m1 + m2 v3 = m1 + m2 + m3 where mi is the message bits (i = 0, 1, 2, 3). Determine the following: a. The parameters n and k. Construct the generator matrix for this code. b. Show that this code is capable of correcting all single-bit errors in any received r. c. Are the vectors r1 = [10101010] and r2 = [01011100] valid code-words?
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