Question

Problem 1. In this problem we work in the finite field 25, i.e. the numbers (mod 5). 1. Show that 2 is a primitive 4-th root of 1. 2. Show that X1-1= (x - 2)(x - 22)(x - 2)(X – 24). 3. Show that g(x) = (x - 2)(X - 4) generates a cyclic code C with d>3. (Hint: invoke a property that we have shown in class.) 4. What is the generating matrix G of the code C given above? 5. What are the parity-check polynomial h(x) and the parity-check ma- trix H for the code C? 6. Show that c, where c = (1,1,1,1), is a codeword in C. 7. Consider et = (0,0,2,0). What is the syndrome of e? 8. A codeword c is transmitted, but there is an error and y=(3, 2, 2, 1) is received. Find c. Problem 2. Write the elements of GF(24), obtained using the irre- ducible polynomial f(x) = 1 + X + X", using the 3 representations: poly- nomnial, binary, powers of primitive element « (where we take a = X). (In my notes on "Finite Fields", there is something similar for GF(23).) I have written in the table below the first 4 elements, you need to add the remaining 12 elements. polynomial binary o powers 0,0,0,0 0,0,0,1 X 0.0.1,0 1+X 0,0,1,1 10

Answer 1