7. Given an (n, k binary linear code, the dual code is a (n, n-k) binary, linear code. This problem considers the (7,3) code that is dual to a (7,4) Hamming code. Call this (7,3) code C Find a generator matrix for C and a parity check matrix for the code. (Begin with a (7,4) Hamming code of your choice, then construct the dual code.) a. b. Determine the minimum distance of the code. c. The standard array is formed with the 8 codewords as the first row, beginning with the all-zero codeword, and the seven 1-bit error patterns as the coset leaders of the next seven rows. Without actually constructing the standard array and checking, prove that no 7-tuple of weight 2 appears in the first 8 rows of the standard array Construct the first column of the standard array, and find the corresponding syndromes. Show how the code can be used to correct all 1-bit error patterns and detect all 2-bit errors patterns. Can the code be used to detect any other error patterns? d. 7. Given an (n, k binary linear code, the dual code is a (n, n-k) binary, linear code. This problem considers the (7,3) code that is dual to a (7,4) Hamming code. Call this (7,3) code C Find a generator matrix for C and a parity check matrix for the code. (Begin with a (7,4) Hamming code of your choice, then construct the dual code.) a. b. Determine the minimum distance of the code. c. The standard array is formed with the 8 codewords as the first row, beginning with the all-zero codeword, and the seven 1-bit error patterns as the coset leaders of the next seven rows. Without actually constructing the standard array and checking, prove that no 7-tuple of weight 2 appears in the first 8 rows of the standard array Construct the first column of the standard array, and find the corresponding syndromes. Show how the code can be used to correct all 1-bit error patterns and detect all 2-bit errors patterns. Can the code be used to detect any other error patterns? d