You can solve Exercise 7 below, that asks for a computer implementation, for 25 marks or Exercise 8 below that is also worth 25 Marks. If you give solutions for both Exercise 7 and also for Exercise 8, your mark is the maximum of your grades in Questions 7 and 8. 7. (25 Marks) This exercise requires computer implementation. We want to implement the decoding algorithm of the binary 2-error correcting BCH code. The following information is given to your program . an irreducible polynomial f e 3(x] of degree n that defines the extension (assume that this polynomial is indeed irreducible; no need to check this); . a primitive element corresponding to a polynomial in F2 (again, assume that this is a primitive element; no need to check this); and vectors to be decoded (alternatively, receive the number (, and in the program ask (a) Construct the table of elements in the field containing the elements as powers of y and (b) Compute the parity-check matrix H (the columns should be ordered in terms of powers (c) Decode the & given vectors providing the corrected messages if possible. For that follow the l vectors to be decoded). the associated n-tuple. of y). For this task, you can use the function f(i) 3 these steps: (i) compute the syndrome of the received word; (ii) separate 21 and 2; (iii) check whether = 0 and z= 0; (iv) if z1 0, check whether z2(use y for this); (v) form the quadratic, and use trial and error to check whether there were two errors. I strongly suggest you use a modular approach. For instance, several times it will be needed to take the sum of two elements and return the power of that represents that sum. Thus, prepare one procedure that receives the two elements and compute the power of y corre- sponding to that sum. Test your program with the data in Assignment 1, and with another example of your choice maybe from lectures. You can solve Exercise 7 below, that asks for a computer implementation, for 25 marks or Exercise 8 below that is also worth 25 Marks. If you give solutions for both Exercise 7 and also for Exercise 8, your mark is the maximum of your grades in Questions 7 and 8. 7. (25 Marks) This exercise requires computer implementation. We want to implement the decoding algorithm of the binary 2-error correcting BCH code. The following information is given to your program . an irreducible polynomial f e 3(x] of degree n that defines the extension (assume that this polynomial is indeed irreducible; no need to check this); . a primitive element corresponding to a polynomial in F2 (again, assume that this is a primitive element; no need to check this); and vectors to be decoded (alternatively, receive the number (, and in the program ask (a) Construct the table of elements in the field containing the elements as powers of y and (b) Compute the parity-check matrix H (the columns should be ordered in terms of powers (c) Decode the & given vectors providing the corrected messages if possible. For that follow the l vectors to be decoded). the associated n-tuple. of y). For this task, you can use the function f(i) 3 these steps: (i) compute the syndrome of the received word; (ii) separate 21 and 2; (iii) check whether = 0 and z= 0; (iv) if z1 0, check whether z2(use y for this); (v) form the quadratic, and use trial and error to check whether there were two errors. I strongly suggest you use a modular approach. For instance, several times it will be needed to take the sum of two elements and return the power of that represents that sum. Thus, prepare one procedure that receives the two elements and compute the power of y corre- sponding to that sum. Test your program with the data in Assignment 1, and with another example of your choice maybe from lectures