**18.310
Exam #1: Take Home Part**

Due Tuesday, October 16, 2007, on Stellar

Problem 1. 10 points

Find a primitive polynomial p(x) of degree 6 which is not
1 + x + x^{6}, and construct its remainder table.

Problem 2. 10 points

Find the polynomial p_{3}(x) associated with the polynomial
you chose in problem 1.

Have a place to enter the appropriate length message.

Create an encoded message from it.

Allow introduction of errors.

Compute t

_{1}and t_{3}from the encoded message by using the appropriate remainder tables.Compute the error locator polynomial from t

_{1}and t_{3}and the appropriate tables.Compute the locations of the errors and correct up to two errors.

Find the original message by dividing the corrected message by the encoding polynomial.

Problem 3. 40 points

Construct an encoder and decoder spreadsheet for the two-error-correcting BCH code associated with this polynomial. It should: