Introduction to Coding Theory 89-662 Final Exam, Moed Aleph 2009 Summary: Introduction to Coding Theory 89-662 Final Exam, Moed Aleph 2009 Exam instructions: The exam is closed book: no material is allowed! Answer all questions and formally prove all of your answers. The exam time is 2.5 hours. Question 1 (25 points): 1. Formally define the notion of local decodability, and show that the Walsh-Hadamard code is 2-locally decodable with 1 4 . 2. Prove that if C is an [n, k] code such that C has distance d +1, then C is not ( -1)-locally decodable. 3. Prove that there exists a code of length n that is 1-locally decodable for < 1 2 . Question 2 (15 points): Show that if there exists a linear code C with parameters [n, k, d] where d is even, then there exists a linear code C with parameters [n, k, d] such that every codeword has even weight. Question 3 (30 points): Let C be a binary linear code and denote by C the code derived by taking the complement of all words in C. 1. Show that if the word (1, . . . , 1) C then C = C. 2. Prove or refute: C is a linear code. 3. Prove or refute: C C is a linear code. Collections: Mathematics