Math 5410 Additional Homework Assignment 3

1. Find all the primitive binary polynomials of degree 4.

2. Prove that all the irreducible binary polynomials of degree 5 are primitive.

3. What is the period of the Linear Feedback Shift Register which has the characteristic polynomial

   x⁵ + x⁴ + x² + x + 1?

4. Find a non-primitive irreducible binary polynomial of degree 6. Construct a LFSR with this polynomial as its characteristic polynomial and determine the statistics of the runs of this LFSR (i.e., how many runs of what lengths are there?)

5. Find the linear equivalence and an LFSR which produces the period 7 sequence that starts 1 0 1 0 0 0 1 .... What starting state will give this sequence?

6. Construct a multiplexed FSR by the Jennings method with (m,n)1 and determine its periods (i.e., the periods of the sequences for each of its starting states).