Introduction to Coding Theory 89-662 Final Exam, Moed Aleph 2008 Summary: Introduction to Coding Theory 89-662 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, 2k-1]). 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} Collections: Mathematics