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MATH 245 CH10

MATH 245 CH10 - DISCRETE MATHEMATICS W W L CHEN c W W L...

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DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1991, 2008. This chapter is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 10 INTRODUCTION TO CODING THEORY 10.1. Introduction The purpose of this chapter and the next two is to give an introduction to algebraic coding theory, which was inspired by the work of Golay and Hamming. The study will involve elements of algebra, probability and combinatorics. Consider the transmission of a message in the form of a string of 0’s and 1’s. There may be interference (“noise”), and a different message may be received, so we need to address the problem of accuracy. On the other hand, certain information may be extremely sensitive, so we need to address the problem of security. We shall be concerned with the problem of accuracy in this chapter and the next. In Chapter 12, we shall discuss a simple version of a security code. Here we begin by looking at an example. Example 10.1.1. Suppose that we send the string w = 0101100. Then we can identify this string with the element w = (0 , 1 , 0 , 1 , 1 , 0 , 0) of the cartesian product Z 7 2 = Z 2 × . . . × Z 2 | {z } 7 . Suppose now that the message received is the string v = 0111101. We can now identify this string with the element v = (0 , 1 , 1 , 1 , 1 , 0 , 1) Z 7 2 . Also, we can think of the “error” as e = (0 , 0 , 1 , 0 , 0 , 0 , 1) Z 7 2 , where an entry 1 will indicate an error in transmission (so we know that the 3rd and 7th entries have been incorrectly received while all other entries have been correctly received). Note that if we interpret w , v and e as elements of the group Z 7 2 with coordinate-wise addition modulo 2, then we have w + v = e , w + e = v , v + e = w . Chapter 10 : Introduction to Coding Theory page 1 of 5

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Discrete Mathematics c W W L Chen, 1991, 2008 Suppose now that for each digit of w , there is a probability p of incorrect transmission. Suppose we also assume that the transmission of any signal does not in any way depend on the transmission of prior signals. Then the probability of having the error e = (0 , 0 , 1 , 0 , 0 , 0 , 1) is (1 - p ) 2 p (1 - p ) 3 p = p 2 (1 - p ) 5 . We now formalize the above. Suppose that we send the string w = w 1 . . . w n ∈ { 0 , 1 } n . We identify this string with the element w = ( w 1 , . . . , w n ) of the cartesian product Z n 2 = Z 2 × . . . × Z 2 | {z } n .
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