Homework 4

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.

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.

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 }

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  VPERP.
• (e) Use the polynomial computed in (d)   to compute the parity check matrix  H   of  V.

Last Modified: March 14, 2001