# Homework 7

Due: Wednesday, April 16, 1998

Problem 1.  Let  x  be the primitive element of  GF(26)  which is the zero of the primitive polynomial:

p(x) = x6 + x + 1 .

Let  g(x)  be the polynomial of smallest degree having the following zeros:

x, x2, x3, x4, x5, x6, x7, x8

Let  V = ( g(x) ) be the corresponding cyclic code of shortest length.

• (a) Write  g(x)  as a product of minimal polynomials  mi(x), where  mi(x)  is the minimum polynomial of  xi.  (Do not explicitly compute the  mi(x)'s.)
• (b) Find the degree of  g(x).
• (c) Find the length  n  of  V.
• (d) What is the dimension  k  of  V?
Note:  If you have not installed the symbolic fonts on your web browser, then the above ksi's will look like x's.

Problem 2.
• (a) Draw the linear sequential circuit (LSC) that multiplies by the polynomial
h(x) = 1 + x3 +x6
• (b) Draw the linear sequential circuit (LSC) that divides by the polynomial
g(x) = 1 + x2 + x4 + x6 + x7
• (c) Draw the linear sequential circuit (LSC) that simultaneously multiplies by h(x)  and divides by  g(x).

Problem 3.  Draw an LSC which takes as inputs polynomials  a(x)  and  b(x)  and then produces the output  h(x)a(x) + k(x)b(x), where  h(x)  and  k(x)  are the polynomials:

h(x) = 1 + x4 + x10       and       k(x) = x + x2 + x4 + x7 + x9

Last Modified: April 4, 1998