hw-1-2009 - measure” 3 Problem 1.2 of Ron Roth’s...

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University of Illinois, Urbana-Champaign ECEN 556 – Fall 2009 Homework 1 Issued: September 8 th ; Due: September 15 th “A- Grade Problems” 1. Binary error-correcting codes are usually designed for communication system modeled by the binary symmetric channel. However, in certain communication systems, such as optical communication systems, a class of computer memories, and nano-storage devices (such as the Millipede device), the probability of changing 1 to 0 is significantly higher than the probability of changing 0 to 1 (these communication systems are often modeled by the so-called Z-channel). Let N ( x , y ) be as defined in class. Define the asymmetric distance as: { } ( , ) max ( , ), ( , ) as d x y N x y N y x = . For a binary code C of length n , the minimum asymmetric distance is defined as: { } ( ) min ( , ): , as C d x y x y C and x y = Show that if ( ) 1 C t ≥ + , then the code C can correct up to t asymmetric errors. 2. Describe a channel model for which the Lee distance is the appropriate “error-correcting
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Unformatted text preview: measure”. 3. Problem 1.2 of Ron Roth’s text (page 19) 4. Problem 1.3 of Ron Roth’s text (page 19) 5. Problem 1.12 of Ron Roth’s text (page 22) “A Grade Problems”: additional question 6. GKOF_UJF_IFPFTBVPT_NCUTJZ_PH_B_[8,4,4]_EPFF. “A+ Grade Problems”: additional question 7. Let C be a circular sequence of zeros and ones of length 2 n , such that all the binary sequences of length n appear as subsequences of C (here, subsequence implies that the elements are taken in consecutive order). Here is an example: 0000111100101101. The sequence C is usually called a deBruijn sequence. Come up with an example of a deBrujin sequence of length 32. Do you see a way to construct these sequences? Can you find a lower or upper bound on their size? Remark: these sequences are closely tied to a well-known class of ECC, and they have applications for data scrambling as well....
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