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Unformatted text preview: 30 Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length n over the field F is a subspace of F n . Thus the words of linear code the codespace F n are vectors, and we often refer to codewords as codevectors . codevectors In the first section we develop the basics of linear codes, in particular we introduce the crucial concept of the dual of a code. The second and third sections then discuss the general principles behind encoding and decoding linear codes. We encounter the important concept of a syndrome. 3.1 Basics If C is a linear code that, as a vector space over the field F , has dimension k , then we say that C is an [ n,k ] linear code over F , or an [ n,k ] code, for short. [ n,k ] linear code There is no conflict with our definition of the dimension of C as a code, since  C  =  F  k . (Indeed the choice of general terminology was motivated by the special case of linear codes.) In particular the rate of an [ n,k ] linear code is k/n . If C has minimum distance d , then C is an [ n,k,d ] linear code over F . The number n k is again the redundancy of C . redundancy We begin to use F 2 in preference to { , 1 } to denote our binary alphabet, since we wish to emphasize that the alphabet carries with it an arithmetic structure. Similar remarks apply to ternary codes. Examples. (i) The repetition code of length n over F is an [ n, 1 ,n ] linear code. (ii) The binary parity check code of length n is an [ n,n 1 , 2] linear code. (iii) The [7 , 4], [8 , 4], and [4 , 2] Hamming codes of the introduction were all defined by parity considerations or similar equations. We shall see below that this forces them to be linear. (iv) The real ReedSolomon code of our example is a [27 , 7 , 21] linear code over the real numbers R . 31 32 CHAPTER 3. LINEAR CODES (3.1.1) Theorem. (Shannons theorem for linear codes.) Let F be a field with m elements, and consider a mSC ( p ) with p < 1 /m . Set L = { linear codes over F with rate at least } . Then L is a Shannon family provided < C m ( p ) . 2 Forney (1966) proved a strong version of this theorem which says that we need only consider those linear codes of length n with encoder/decoder complexity on the order of n 4 (but at the expense of using very long codes). Thus there are Shannon families whose members have rate approaching capacity and are, in a theoretical sense, practical 1 . The Hamming weight (for short, weight ) of a vector v is the number of its Hamming weight nonzero entries and is denoted w H ( v ). We have w H ( x ) = d H ( x , ). The mini mum weight of the code C is the minimum nonzero weight among all codewords minimum weight of C , w min ( C ) = min 6 = x C (w H ( x )) ....
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 Spring '11
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