midterm-exam - University of California San Diego Fall 2008...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
University of California San Diego Fall 2008 ECE 259A: Midterm Exam Instructions: There are four problems, weighted as shown below. The exam is open-book: you may use any auxiliary material that you like. Good luck! Problem 1. (25 points) Suppose that M 2 binary vectors of length n are all at Hamming distance d from each other. a. Prove that either all the vectors have even weight, or they all have odd weight. b. Show that d must be even. c. If M 3 , show that there exists a unique vector at distance d 2 from the three vectors. Problem 2. (35 points) Suppose that we wish to communicate over a binary channel that either transmits the codeword at its input as is, or inverts the ±rst i bits in the codeword for some i 1, 2, . . . , n . Thus the set of all possible error-patterns in the channel is n 1 i 0 n i : i 0, 1, . . . n , where denotes concatenation. We wish to construct a binary linear code of length n 2 m 1 that corrects all the possible error- patterns: for any c and any e 2 m 1 the codeword
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Ask a homework question - tutors are online