Download Coding Theory - Midterm Exam 1 Question Paper | ECE 556 and more Exams Electrical and Electronics Engineering in PDF only on Docsity! University of Illinois Fall 2004 ECE 556: First MidSemester Exam Tuesday October 12, 2004, 8:30 a.m. – 9:50 a.m. This is a open-text open-handwritten-notes examination. No printed materials other than the textbook and Problem Sets and Solutions distributed in class are permitted. Calculators, laptop computers, PDAs, iPods, cellphones, e-mail pagers, etc. are neither needed nor permitted. This Examination contains three problems 1. (a) Either construct a (10, 1) linear binary code with minimum distance 7 or prove that such a code cannot exist. Either construct a (10, 2) linear binary code with minimum distance 7 or prove that such a code cannot exist. Either construct a (16, 12) linear binary code with minimum distance 3 or prove that such a code cannot exist. (b) Let C denote an (n, k) linear binary code with the property that the Hamming weight of every codeword is a multiple of 4. Prove that C is a subcode of its dual code C⊥, and hence has rate ≤ 12 . (c) Let C denote an (n, k) linear binary code with the property that the columns of its parity- check matrix H are distinct vectors of odd Hamming weight. Show that the minimum distance of the code is at least 4. (d) In Problem 2 of Problem Set 3, you considered the ensemble S of systematic linear binary codes whose generator matrices are of the form G = [I|P ] where I is a k × k identity matrix and P is some k × (n− k) binary matrix, and proved that any nonzero binary vector x of length n either belongs to none of the codes in S, or it belongs to exactly 2(k−1)(n−k) codes in S. Use this result to show that if d is an integer, 0 < d < n, such that 2(k−1)(n−k) d−1∑ i=1 ( n i ) < 2k(n−k), that is, d−1∑ i=1 ( n i ) < 2n−k, then there exists a n (n, k) systematic linear binary code with minimum distance at least d. This result is a version of the Gilbert bound for linear codes (cf. Chapter 12 of the textbook).
