Problem4

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

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.

Ask a homework question - tutors are online