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Problem 1. In this problem we work in the finite field 25, i.e. the numbers (mod...

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

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