Problem4

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Unformatted text preview: eld (not necessarily finite). Prove that m1 m αi αj £ ¨¨ ¨ £ £ m α1 1 m α2 1 m αm 1 $ . . . "   Vm "" 2 α1 3 α1 . . . 2 α2 3 α2 . . . 2 αm 3 αm . . . "" "" ¨¨ ¨ ¨¨ ¨ ¨¨ ¨ ¨¨ ¨ 1 α1 1 α2 1 αm "" #!  6. A Vandermonde matrix is a square m m matrix of the form ¡ ¡ £ ¡£ ¡ ¡  ¡ ¡ £ ¢ 5. (a) Suppose that a binary linear code generated by g x has the property that whenever c x is a codeword, then so is x n 1 c x 1 — which is the same codeword with the bits written in reverse order. Such a code is called reversible. Prove that g x x r g x 1 in this case, where r deg g x . (b) What is the corresponding statement for reversible nonbinary cyclic codes? (c) Show that a cyclic code is reversible if and only if g γ 0 implies g γ 1 0 for any n-th root of unity γ . 1 mod n, then every cyclic code of (d) Show that if there exists an integer m such that q m length n over GF q is reversible. ¡ ¢ ¢ ¡...
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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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