1. (a) What is meant by a cyclic error-control code? Explain your answer by suitable examples (at least 2). (b) Can cyclic codes be non-linear? Explain your answer by suitable examples (at least 2). 2. Determine whether the polynomial 1+X+X3+X4 is a generator polynomial of a linear cyclic block code with code length n=7. Explain your answer. 3. Verify that the generator polynomial g(X)=1+X+X2+X2 generates a (8,5) linear cyclic code and determine the code polynomial for the message vector m=(10101) in systematic form. 4. A linear cyclic code (n,k) has code length n=7 and generator polynomial g(X)=1+X2+X3+X4 (a) Find the code rate, the generator and parity check matrices of the code in systematic form, and its Hamming distance. (b) If all the information symbols are ' 1 's, what is the corresponding code vector? (c) Find the syndrome corresponding to an error in the first information symbol, and show that the code is capable of correcting this error. 5. A linear cyclic block code has code length n=14 and generator polynomial g(X)=1+X3+X4+X5 (a) If all the information symbols are ' 1 's, what is the corresponding code vector? (b) Find the syndrome corresponding to an error in the last information symbol. Is this code capable of correcting this error? 6. (a) Determine the table of code vectors of the (6,2) linear cyclic block code generated by the polynomial g(X)=1+X+X3+X4. (b) Calculate the minimum Hamming distance of the code, and its error correction capability. 7. A linear cyclic block code with a code length of n=14 has the generator polynomial g(X)=1+X2+X6 (a) Determine the number of information and parity check bits in each code vector. (b) Determine the number of code vectors in the code. (c) Determine the generator and parity check matrices of the code. (d) Determine the minimum Hamming distance of the code. (e) Determine the burst error-correction capability of the code. (f) Describe briefly how to encode and decode this code. 8. Consider a (15,11) linear cyclic block code generated by the polynomial (a) Determine the code vector in systematic form of the message vector (b) Decode the received vector r=(00001000110