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Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.572 47 — Low-Density Parity-Check Codesand Soljanin, 2001); and for background reading on this topic see (Hartmannand Rudolph, 1976; Terras, 1999). There is a growing literature on the practicaldesign of low-density parity-check codes (Mao and Banihashemi, 2000;Mao and Banihashemi, 2001; ten Brink et al., 2002); they are now beingadopted for applications from hard drives to satellite communications.For low-density parity-check codes applicable to quantum error-correction,see MacKay et al. (2004).47.9 ExercisesExercise 47.1. [2 ] The ‘hyperbolic tangent’ version of the decoding algorithm. Insection 47.3, the sum–product decoding algorithm for low-density paritycheckcodes was presented first in terms of quantities qmn 0/1 and rmn, 0/1 thenin terms of quantities δq and δr. There is a third description, in whichthe {q} are replaced by log probability-ratios,l mn ≡ ln q0 mnqmn1 . (47.26)Show thatδq mn ≡ q 0 mn − q1 mn = tanh(l mn/2). (47.27)Derive the update rules for {r} and {l}.Exercise 47.2. [2, p.572] I am sometimes asked ‘why not decode other linearcodes, for example algebraic codes, by transforming their parity-checkmatrices so that they are low-density, and applying the sum–productalgorithm?’ [Recall that any linear combination of rows of H, H ′ = PH,is a valid parity-check matrix for a code, as long as the matrix P isinvertible; so there are many parity check matrices for any one code.]Explain why a random linear code does not have a low-density paritycheckmatrix. [Here, low-density means ‘having row-weight at most k’,where k is some small constant ≪ N.]Exercise 47.3. [3 ] Show that if a low-density parity-check code has more thanM columns of weight 2 – say αM columns, where α > 1 – then the codewill have words with weight of order log M.Exercise 47.4. [5 ] In section 13.5 we found the expected value of the weightenumerator function A(w), averaging over the ensemble of all randomlinear codes. This calculation can also be carried out for the ensemble oflow-density parity-check codes (Gallager, 1963; MacKay, 1999b; Litsynand Shevelev, 2002). It is plausible, however, that the mean value ofA(w) is not always a good indicator of the typical value of A(w) in theensemble. For example, if, at a particular value of w, 99% of codes haveA(w) = 0, and 1% have A(w) = 100 000, then while we might say thetypical value of A(w) is zero, the mean is found to be 1000. Find thetypical weight enumerator function of low-density parity-check codes.47.10 SolutionsSolution to exercise 47.2 (p.572). Consider codes of rate R and blocklengthN, having K = RN source bits and M = (1−R)N parity-check bits. Let all
Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.47.10: Solutions 573the codes have their bits ordered so that the first K bits are independent, sothat we could if we wish put the code in systematic form,G = [1 K |P T ]; H = [P|1 M ]. (47.28)The number of distinct linear codes is the number of matrices P, which isN 1 = 2 MK = 2 N 2 R(1−R) . Can these all be expressed as distinct low-density log N 1 ≃ N 2 R(1 − R)parity-check codes?The number of low-density parity-check matrices with row-weight k is( Nk) M(47.29)and the number of distinct codes that they define is at most( ) /NMN 2 =M!, (47.30)kwhich is much smaller than N 1 , so, by the pigeon-hole principle, it is notpossible for every random linear code to map on to a low-density H.log N 2 < Nk log N
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