In the rest of this problem we will take p = 2. This is particular natural for applications to computer science, because computers like to communicate using an alphabet with 2 symbols. Other primes also occur in applications but would take us a bit too far afield.? The Hamming code. For example, take N = 8, p=2, n = 7. Alice could choose the following N codewords to represent the messages she might send: Code Word Message 0000000 Hello 1001101 Goodbye. 0101011 The water is fine. 1100110 I have met an unusual animal. 0010111 Bring an umbrella. 1011010 I just saw a rainbow. 0111100 I'm ready to return. 1110001 Send help immediately! A code is called linear when the list of code words is a linear subspace of F. Let VHumming be the list of codewords above. Problem 4 Give a basis of Vamming, and give a matrix Hamming whose image is VHamming The matrix G is called the generator matrix. Let V be a linear code and let d be the smallest number of nonzero bits of any nonzero vector i in V. For Vamming, we have d = 4. The number d is called the distance of the code. In the rest of this problem we will take p = 2. This is particular natural for applications to computer science, because computers like to communicate using an alphabet with 2 symbols. Other primes also occur in applications but would take us a bit too far afield.? The Hamming code. For example, take N = 8, p=2, n = 7. Alice could choose the following N codewords to represent the messages she might send: Code Word Message 0000000 Hello 1001101 Goodbye. 0101011 The water is fine. 1100110 I have met an unusual animal. 0010111 Bring an umbrella. 1011010 I just saw a rainbow. 0111100 I'm ready to return. 1110001 Send help immediately! A code is called linear when the list of code words is a linear subspace of F. Let VHumming be the list of codewords above. Problem 4 Give a basis of Vamming, and give a matrix Hamming whose image is VHamming The matrix G is called the generator matrix. Let V be a linear code and let d be the smallest number of nonzero bits of any nonzero vector i in V. For Vamming, we have d = 4. The number d is called the distance of the code