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Unformatted text preview: University of California San Diego ECE 259A: Problem Set #4 1. How many binary linear cyclic codes of length 15 are there? (a) Prove that all the codewords c 0 , c1 , . . . cn 1 satisfy c0 c1 cn 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 . 4. Suppose that g1 x and g2 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 . j 1i j 1 and hence Vm is nonsingular if and only if α 1 , α2 , . . . αm are all distinct. Research Problem: The Vandermonde matrix is defined by the property that the rows are consecutive powers of α1 , α2 , . . . αm . Find a simple expression for the determinant of a square ¡  '& & ¡ det Vm ∏∏ £ % where α1 , α2 , . . . αm are elements in a ...
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This note was uploaded on 10/12/2009 for the course COMM ECE259A taught by Professor Prof.alexandervardy during the Fall '08 term at San Diego.

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