1252646243 - ECE 635 Dr. N. El Naga Page 2 3.6 Suppose the...

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ECE 635 Dr. N. El Naga Homework # 3 3.1 Consider a (7,4) code whose generator matrix is: G = 0 1 1 1 0 0 0 1 1 0 0 1 0 0 1 0 1 0 0 1 0 1 1 1 0 0 0 1 (a) Find all the code vectors of this code. (b) Find the parity check matrix H of this code. 3.2 Consider a systematic (8,4) code whose parity equations are given below: u 5 = m 1 + m 2 + m 4 u 6 = m 1 + m 3 + m 4 u 7 = m 1 + m 2 + m 3 u 8 = m 2 + m 3 + m 4 Where m 1 , m 2 , m 3 , m 4 are message digits. Find the parity and generator matrices for this code. Show that the minimum weight of this code is 4. 3.3 Construct a standard array for the code given in Problem 3.1. 3.4 Use the standard array which you obtained in Problem 3.3 to construct a decoding table for the code given in Problem 3.1. 3.5 Let C be an (n,k) code with minimum weight d = 2t + 1. Prove that at least one error pattern of weight t + 1 cannot be used as a coset leader.
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Unformatted text preview: ECE 635 Dr. N. El Naga Page 2 3.6 Suppose the minimum weight of an (n,k) code C is d. Prove that every combination of d - 1 or fewer columns of the parity check matrix H of C is linearly independent. Also prove that there exists at least one combination of d columns of H which is linearly dependent. 3.7 Let H be the parity check matrix for an (n,k) linear code C which has odd minimum weight d. Construct a new code C 1 whose parity check matrix is: | 0 | 0 | 0 H 1 = H | . | . | 0 1 1 1 1. . . | 1 (a) Prove that C 1 is an (n + 1,k) code. (b) Prove that every code vector in C 1 has even weight. (c) Prove that the minimum weight of C 1 is d + 1....
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1252646243 - ECE 635 Dr. N. El Naga Page 2 3.6 Suppose the...

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