Introduction to Coding Theory 89-662 Final Exam, Moed Bet 2008 Summary: Introduction to Coding Theory 89-662 Final Exam, Moed Bet 2008 Exam instructions: 1. Closed book: no material is allowed 2. Answer all questions 3. Time: 2.5 hours 4. Good luck! Question 1 (20 points): Prove the Gilbert-Varshamov lower bound: Let n, k and d be natural numbers such that 2 d n and 1 k n. If V n-1 q (d - 2) < qn-k then there exists a linear code [n, k] over Fq with distance at least d. Question 2 (25 points): The heaviest codeword problem is defined as follows: Upon receiving a parity check matrix H that fully defines a binary linear code C, find the codeword c C with the maximum weight (i.e., find c such that wt(c) wt(c ) for all c C). Give an efficient (polynomial-time) algorithm for this problem or show that it is NP-complete. Question 3 (25 points): 1. Show that there exists no binary linear code with parameters [2m, 2m - m, 3] for any m 2. 2. Let C be a binary linear code with parameters [2m, k, 4] for some m 2. Show that k 2m - m - 1. 3. Let and R be such that R = 1 - H(). Is it possible to construct a code with rate R = k Collections: Mathematics