5682
1-22-07
Theory ECEN and Practice of Error Control Codes
Spring 2007
P. Mathys
Problem Set 1
(Solutions are due Mon. 1-29-07)
1) The 10-digit ISBN (International Standard Book Number) system assigns a unique number of the form ISBN = d10 d9 d8 d7 d6 d5 d4 d3 d2 d1 , to each commercially traded book. The rst nine digits (d10 . . . d2 ) are all decimal and identify the language, the publisher, and the title and specic edition of the book. The last digit is a check digit that can be either 0 . . . 9 or X (=10). It is used to check the integrity of the ISBN number as follows:
10
v=
i=1
i di .
If v is divisible by 11 (without remainder), then the ISBN number is assumed to be correct, otherwise an error has been detected. (a) Compute the check digit d1 for the 9-digit (d10 . . . d2 ) book numbers (i) 2-253-00950 and (ii) 3-442-07553. (b) One of the most frequent errors that humans make is the transposition of two letters or digits, e.g., 478 becomes 487. Claim: The ISBN-10 code can detect any single transposition of digits. Prove or disprove this claim. (c) Claim: The ISBN-10 code can detect every double (i.e., error any two digits in error, but not necessarily transpositions). Prove or disprove this claim. 2) What is the shortest single-error correcting binary code with at least 8 codewords that you can nd by trial and error? Assume the memoryless BSC error model (i.e., errors 0 1 and 1 0 are equally likely and independent). Is an exhaustive search feasible? 3) The simplest model of an asymmetric binary channel is the memoryless Z-channel shown in the following gure.
1 Input X 0 p 1p 1 1 Output Y 0
Using trial and error, nd a binary single error correcting code of length n = 5 for the Z-channel with as many codewords as possible.
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4) Problem 3.1 in the book. Consider a systematic binary (8, 4) code whose parity-check equations (modulo 2) are v 0 = u1 + u 2 + u 3 , v 1 = u0 + u 1 + u 2 , v 2 = u0 + u 1 + u 3 , v 3 = u0 + u 2 + u 3 , where u0 , u1 , u2 , and u3 are message digits and v0 , v1 , v2 , and v3 are parity-check digits. Find the generator and parity-check matrices for this code. Show analytically that the minimum distance of this code is 4.
c 19942007, P. Mathys.
Last revised: 1-21-07, PM.
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