Homework 7

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Reading Assignment.

• (a) MacWilliams & Sloane, Chapters 3 & 9.
• (b) Peterson & Weldon, Chapter 9
• (c) Berlekamp, Chapter 7

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Problem 1. Let  GF(2⁴) = GF(2)[X]/(X⁴+X+1)  and let  a = X mod X⁴+X+1.   Let   V = (g(x)) be the  (15,5)  BCH code given by 

g(X) = LCM( m₁(X), m₃(X), m₅(X) ),

where  m_i(X)  denotes the minimum polynomial of  aⁱ.

• (a) Find the maximum designed distance of  V.
• (b) Let  r(X)    denote a received vector.  If   r(a)= a⁵, r(a³) = a⁹, and r(a⁵) = a⁵, then use the BCH decoding algorithm to find the most probable error pattern  e(X).

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Last Modified: April 2, 2001