 
Summary: Introduction to Coding Theory 89662
Final Exam, Moed Aleph 2008
Exam instructions:
1. Closed book: no material is allowed
2. Answer all questions
3. Time: 2.5 hours
4. Good luck!
Question 1 (15 points): Describe a binary linear code with parameters [n, log2(n + 1), n+1
2 ]
(equivalently, with parameters [2k  1, k, 2k1]). Provide a full proof that your construction is a
binary linear code and that it meets these parameters.
Question 2 (30 points): Prove the following. Let > 0 and > 0 be any constants and let d
be any natural number. Then, for large enough k and n fulfilling k
n = 1  H(d/n)  , there exist a
pair of functions (E, D) where E : {0, 1}k {0, 1}n and D : {0, 1}n {0, 1}k such that for every
vector e {0, 1}n with wt(e) d it holds that
Prx{0,1}k [D(E(x) + e) = x]
In what way does the above differ from Shannon's theorem?
Question 3 (20 points): Let Ci be an [n, ki, di] linear code over Fq for i = 1, 2. Define
C = {(a + c, b + c, a + b + c)  a, b C1, c C2}
