Download Error Control Coding

Error Control Coding
Midterm
Wednesday, April 25, 2007.
1.
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6.
Close book.
Theorems of the textbook are provided.
170 minutes (9:10-12:00).
Answers without detail derivations or explanations earn no points.
100 points.
Good Luck.
Problem 1. [15] Order
Let 
and

be elements in GF(q) such that
   i . If the ord    t , then
ord     t GCD  i, t  , where ord   is the order of  and GCD stands for the greatest
common divisor.
Problem 2. [15] The Multiplicative Structure of Galois Fields
Prove the following properties.
a.
[10] Consider the Galois field GF(q). If t | q  1 , then there are   t  elements of order t in
GF(q).
b.
[5] In every finite field GF(q), there are exactly   q  1 primitive elements.
Problem 3. [15] Galois Field
a.
[9] Construct a table for GF  25  based on the primitive polynomial p  X   1  X 2  X 5 .
b.
[6] Let  be a primitive element of GF  25  . Find the minimal polynomials of  3 .
Problem 4. [10] Irreducible Polynomial
Let f  X  be a polynomial of degree n over GF  2 . The reciprocal of f  X  is defined as
1
f *X   Xn f 
X

.

Prove that f *  X  is irreducible over GF  2 if and only if f  X  is irreducible over GF  2 .
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Problem 5.
[10] Error probability
Let Αi denote the weight distribution of a linear block code C and
 i denote the weight
distribution of the coset leaders.
I. [5] Derive the probability of an undetected error. Please explain it.
II. [5] Derive the probability of a decoding error for a binary symmetric channel with transition
probability p. Please explain it.
Problem 6. [20] Encoding and Decoding Circuits
Consider a systematic (8,4) code whose parity-check equations are given by:
v0  u1  u2  u3 ,
v1  u0  u1  u2 ,
v2  u0  u1  u3 ,
v3  u0  u2  u3 ,
where u0 , u1 , u2 , and u3 are message digits, and v0 , v1 , v2 , and v3 are parity-check digits.
a.
b.
c.
[5] Find the generator and parity-check matrices for this code.
[5] Construct an encoding circuit for this code.
[10] Suppose that this code is used for a binary symmetric channel and this decoder is designed
to correct the 16 most probable error patterns. Devise a decoding circuit for this code based on
the table-lookup decoding scheme.
Problem 7. [15]
Consider an (n, k) linear code C whose generator matrix G contains no zero column. Arrange all the
codewords of C as rows of a 2k -by-n array.
a.
b.
c.
[5] Show that no column of the array contains only zeros.
[5] Show that each column of the array consists of 2k 1 zeros and 2k 1 ones.
[5] Show that the set of all codewords with zeros in a particular component position forms a
subspace of C.
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