(10 points) Construct a finite field of 16 elements as F2(a)/4 + 1. Show that is a primitive element in the field, and express every element of the field as a power Of . . Identify all the primitive elements in the field . Identify all the roots of the polynomial x3 -1. 3 . Find the multiplicative inverse of 2 ++ 1. 2. (10 points) We use the field of the previous problem to construct a conventional, primitive (n -15, k 9) Reed Solomon code (a) Write the generator and parity check matrix of the Reed Solomon code. Express entries as powers of the primitive element (b) Write the generator polynomial for the code (c) Write out any non-zero codeword of the Reed Solomon code. Erase the 11th and 12th co-ordinates Then use Lagrange interpolation to reconstruct the erased co-ordinates. from the remaining surviving co-ordinates (d) (10 points) Consider a (7,2) Reed Solomon code over F8-F2(a)(a" ++1). Assume that the evaluation points are ,--1, 2-1, 2, 3, 4, 5, 6, Suppose that the receiver receives which is a codeword corrupted by at most 2 errors. Using any of the algorithms discussed in class, find the original message 3. (5 points) Consider a finite field F with characteristic q. Let S = {0, 1, 1+1, . . . ,1 + 1+ + 1). Show q-1 times 4. (10 points) Let G-lk Pl be the k n generator matrix of an MDS code, where Ik is the k k identity matrix (a) Show that every square sub-matrix of P is full rank. (b) Show that the dual code is also an MDS code of dimension n-k (10 points) Construct a finite field of 16 elements as F2(a)/4 + 1. Show that is a primitive element in the field, and express every element of the field as a power Of . . Identify all the primitive elements in the field . Identify all the roots of the polynomial x3 -1. 3 . Find the multiplicative inverse of 2 ++ 1. 2. (10 points) We use the field of the previous problem to construct a conventional, primitive (n -15, k 9) Reed Solomon code (a) Write the generator and parity check matrix of the Reed Solomon code. Express entries as powers of the primitive element (b) Write the generator polynomial for the code (c) Write out any non-zero codeword of the Reed Solomon code. Erase the 11th and 12th co-ordinates Then use Lagrange interpolation to reconstruct the erased co-ordinates. from the remaining surviving co-ordinates (d) (10 points) Consider a (7,2) Reed Solomon code over F8-F2(a)(a" ++1). Assume that the evaluation points are ,--1, 2-1, 2, 3, 4, 5, 6, Suppose that the receiver receives which is a codeword corrupted by at most 2 errors. Using any of the algorithms discussed in class, find the original message 3. (5 points) Consider a finite field F with characteristic q. Let S = {0, 1, 1+1, . . . ,1 + 1+ + 1). Show q-1 times 4. (10 points) Let G-lk Pl be the k n generator matrix of an MDS code, where Ik is the k k identity matrix (a) Show that every square sub-matrix of P is full rank. (b) Show that the dual code is also an MDS code of dimension n-k
