Summary: Introduction to Coding Theory 89-662
Final Exam, Moed Bet 2008
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