problem4

problem4 - University of California San Diego ECE 259A...

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University of California San Diego ECE 259A: Problem Set #4 1 . How many binary linear cyclic codes of length 15 are there? 2 . Let g x be a generator polynomial of a cyclic code of length n over GF q , where n and q are relatively prime. (a) Prove that all the codewords c 0 , c 1 , . . . c n 1 satisfy c 0 c 1 c n 1 0 if and only if x 1 is a factor of g x . In particular, for binary cyclic codes, all the codewords are of even weight if and only if g x is divisible by x 1 . (b) Prove that the all-one vector 111 111 is a codeword if and only if x 1 is not a factor of g x . 3 . Consider the polynomial g x x 3 x 1 . (a) Show that g x is a generator polynomial of a cyclic code of length 8 over GF 3 . (b) Find the generator polynomial of . 4 . Suppose that g 1 x and g 2 x are generator polynomials of two cyclic codes 1 and 2 of the same block length n over GF q . Prove that if all the roots of g 1 x are also the roots of g 2 x then 2 is a subcode of 1 . 5 . (a) Suppose that a binary linear code generated by g x has the property that whenever c x

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• Fall '08
• Prof.AlexanderVardy
• Coding theory, Polynomial code, CYCLIC CODES, University of California San Diego, Cyclic code, narrow-sense BCH

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