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cycliccodes

# cycliccodes - Assignment on Cyclic Codes EE512 Error...

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Assignment on Cyclic Codes EE512: Error Control Coding Questions marked (Q) or (F) are questions from previous quizzes or final exams, respectively. 1. What is the ideal describing the cyclic code { 0000 , 0101 , 1010 , 1111 } ? 2. Describe the smallest cyclic code containing the vector 0011010. 3. Show that in an ( n, k ) cyclic code any k consecutive bits can be taken to be the message bits. 4. Consider the n = 7 binary cyclic code generated by g ( x ) = 1 + x + x 3 . (a) Find all codewords of the code. (b) The allzero codeword c ( x ) = 0 is obtained uniquely by multiplying g ( x ) by m ( x ) = 0 in GF(2)[ x ]. Find all f ( x ) GF(2)[ x ] / ( x 7 + 1) such that f ( x ) g ( x ) = 0 in GF(2)[ x ] / ( x 7 + 1). 5. A binary cyclic code of length 15 has generator polynomial g ( x ) = ( x 4 + x +1)( x 4 + x 3 + x 2 + x +1). Give a generator matrix and parity-check matrix for the code. Find the generator matrix for the dual of the code. 6. Find the dimension and generator polynomial for every binary cyclic code of length 15, 17, 21, 31, 51, 73, 85. 7. Let C be the n = 3 cyclic code over GF(4)= { 0 , 1 , α, α 2 } ( α 3 = 1, α 2 = 1 + α ) generated by g ( x ) = x + α .

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