problem4

# 3 consider the polynomial g x x3 x 1

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Unformatted text preview:  ¢ (b) Find the generator polynomial of . ¡ ¢ (a) Show that g x is a generator polynomial of a cyclic code of length 8 over GF 3 . §     § £ ¡ ¡ 3. Consider the polynomial g x x3 x 1. ¡   £ ¡ §¨¨¨ ©§ § ¡ § ¢¤ ¦¥¡ £ ¡ ¢ 2. Let g x be a generator polynomial of a cyclic code are relatively prime. of length n over GF q , where n and q ¨¨¨ ¡ ¡ ¡ ¡ ¡ ¢ ¡ ¡ ¡ ¡   ¡ ¢ matrix whose rows are non-consecutive powers of α 1 , α2 , . . . αm . Use this to determine the necessary and sufﬁcient conditions for such a matrix to be nonsingular. For known results, see T. Muir, A Treatise on the Theory of Determinants, Dover, 1960, and W. Ledermann, Introduction to Group Characters, Cambridge University Press, 1977. 8. Let g x be a cyclic code of length n, and suppose that α b , α b r , α b 2r , . . . , α b δ 2 r are zeros of g x . Here, α is a primitive n-th root of unity, and b, δ , r are positive integers with δ 2 and gcd r, n 1. Prove that the minimum distance of is at least δ. 4 £ 3' 2  7. Let be a narrow-sense BCH code of length n ab and designed distance δ the true minimum distance of is equal to the designed minimum distance. a. Prove that ' ' ¢  ¢ ¡ ¡ 0 1¡ ( ) ¢ ¢ 5...
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