# Homework 4

Due: Monday, March 2, 1998

• 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.

• Construct the multiplication table of the group G given by the presentation:
( r, s | r4 = s2 = 1, sr = r3s )
You may assume that the distinct group elements are:
1, r, r2,r3, s, rs, r2s, r3s

Also compute in GF(2)G the product
(1+r3+r2s)(r+s+r3s)

• Problem 2.

• The polynomial
p(x) = x4 + x3 + x2 + x + 1
is irreducible over GF(2), and therefore the algebra of polynomials modulo p(x) is.
GF(24).  Let
ksi = x  mod  p(x).
Show that ksi is not a primitive element, and therefore p(x) is not a primitive polynomial.  Show that
alpha = 1 + ksi
is primitive.

• Problem 3.

• Let  R  be a commutative ring with non-zero elements  and  b such that
ab = 0

Prove that  R  is not a field.

• Problem 4.

• 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 5.

• 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 }