Homework 4

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Reading Assignment:

• Berlekamp(Ref#1), pp87-90
• MacWilliams(Ref#6), pp93-112
• Peterson(Ref#7), pp19-25, pp144-154
• Pless(Ref#8), pp44-50

Reference numbers refer to the numbered references given in the syllabus. 

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Problem 1.

Let  ksi  be the primitive element of  GF(2⁶)  defined by  ksi = x mod x⁶ + x + 1.  Compute the orders of the elements of  ksiⁱ  for  i=0,1, ... , 62.  Summarize your results in a log/order table.  For which i's are the ksiⁱ's primitive?  Do you see a pattern?  Make a conjecture about this pattern.

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Problem 2.

Compute the addition and multiplication tables for the ring

R₃ = GF(2)[x]/(x³+1)

Also express each of the following ideals in the ring  R₃ as a set of elements of R₃ .

                        (0),  (1+x), (x²+x+1), (1), (x²+1), (x³+1), (x⁵+x+1).

For example,

                        (0) = { 0 }   and   ( x⁴ + x² +1 ) = { 0, x² + x+1 }

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Problem 3.  Let  V  denote the cyclic code of length  15 in  R₁₅  given by the generator polynomial:

g(x) = x⁸ + x⁴ + x² + x + 1.

• (a) Use  g(x)  to compute the generator matrix  G  of  V.  What is the dimension of  V ?
• (b) Compute the parity check polynomial   h(x)  of  V.
• (c) Use  h(x)  to compute a generator matrix   of V^! .
• (d) From  h(x),  compute the generator polynomial  of  V^PERP.
• (e) Use the polynomial computed in (d)   to compute the parity check matrix  H   of  V.

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Last Modified: March 14, 2001