By Robert H. Morelos-Zaragoza
The identify given to this ebook is particularly deceptive, one may possibly think its the seminal comparable to TAOCP for mistakes correcting codes, but it basically falls dismally short.
The 2d variation of this publication nonetheless comprises the blunders and misprints from the 1st variation. The assurance and intensity of the themes handy are abysmal at most sensible. the most important subject matters are skimmed over, key proof and figures are by no means effectively defined. An instance of this can be the bankruptcy on reed-solomon codes, it is easy to simply locate greater fabric at the topic just by looking out wikipedia.
The accompanying code examples on hand at the website revolve round a long time outdated c code snippets which provide no perception into the subject matter and are reason for extra confusion and misdirection (as such a lot of them do not functionality thoroughly to start with). a much better e-book will be both Todd Moon's "Error Correction Coding: Mathematical tools and Algorithms" or the normal textual content by means of Lin and Costello.
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Additional info for The Art of Error Correcting Coding
A generator matrix for this code is 1 0 G= 0 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 1 0 0 0 . 0 1 Alternatively, the parity submatrix of a generator matrix of a cyclic code can be constructed with the vectors associated with the following polynomials: x n−1 mod g(x), ¯ .. ¯ x n−k+1 mod g(x), ¯ x n−k mod g(x), and a systematic encoding is obtained, as illustrated in the example below. 4 Let C be the cyclic Hamming (7, 4, 3) code. Then g(x) ¯ = x 3 + x + 1, and x6 x5 x4 x3 mod(x 3 + x + 1) = x 2 + 1, mod(x 3 + x + 1) = x 2 + x + 1, mod(x 3 + x + 1) = x 2 + x, mod(x 3 + x + 1) = x + 1.
Clearly, the code associated with H is equivalent to the original Hamming code with the parity-check matrix H , up to a permutation (or exchange) of positions. 18). Clearly, when expressing H with columns equal to the binary representation of the (integer) column number, the identity matrix is contained in those columns of H that correspond to even powers of 2, that is, of the form 2 , 0 ≤ < m. This is illustrated in the example below. 2 Let m = 3. Then a systematic parity-check matrix is 1 1 0 1 1 0 0 H = 1 0 1 1 0 1 0 , 0 1 1 1 0 0 1 Matrix H is given by the binary representation of integers 1 to 7 (in the following, the topmost part corresponds to the least-significant bit (LSB) in the binary representation of an integer): 1 0 1 0 1 0 1 H = 0 1 1 0 0 1 1 , 0 0 0 1 1 1 1 and matrix In−k is contained in columns 1, 2 and 4.
10 Probability of a decoding error for HDD and SDD of a binary (3,1,3) code with binary transmission over an AWGN channel. INTRODUCTION 19 In Fossorier et al. (1998), the authors show that for systematic binary linear codes with binary transmission over an AWGN channel, the probability of a bit error, denoted Pb (C), has the following upper bound: n Pb (C) ≤ w=dmin wAw Q n 2wR Eb N0 . 37) Interestingly, besides the fact that the above bound holds only for systematic encoding, the results in Fossorier et al.
The Art of Error Correcting Coding by Robert H. Morelos-Zaragoza