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W w=1 in .NET Integration qrcode in .NET W w=1

W w=1 generate, create qr code iso/iec18004 none with .net projects Visual C# w w = 1 e . By using a simila QR-Code for .NET r argument as for the bit erasure probability one can show that words of large weight (larger than a constant) have a negligible in uence on the block erasure probability. We get the promised expression by letting W tend to innity and observing that e w=1 2w = 1 .

e expurgated case follows in essentially the same manner by taking into account the following key fact: the distribution of the number of cw/ss of size w, w s, in the expurgated ensemble is the same as the corresponding distribution in the unexpurgated ensemble. erefore the only change in the above derivations is the lower summation index..

( )w Regular LDPC code qr barcode for .NET s were de ned by Gallager in his ground breaking thesis [ ]. Rather than looking at ensembles, Gallager showed that one can construct speci c instances so that the corresponding computation graph is tree-like for c log(n) iterations, where c is a suitable positive constant.

He then argued that for that number of iterations the message-passing decoder can be applied and analyzed in terms of what would now be called density evolution. Although the limited computational resources available at that time did not allow him to compute actual thresholds, explicit computations on some examples for a few iterations gave an idea of the large potential of this scheme. Historically it is interesting to note that LDPC codes were never patented (neither by Gallager himself, nor by Codex for whom Gallager consulted, nor by Motorola which later acquired Codex).

According to Forney, Codex did not foresee a market for LDPC codes except the government level. For the latter, patents are of no use. In Elias introduced the erasure channel as a toy example [ ].

With the advent of the Internet this channel was thrown into the limelight. For a general historical outline of message-passing decoding see the Notes at the end of starting on page . We limit our present discussion to the BEC.

. Although Gallager did not explicitly consider transmission over the BEC, the general (class of) message-passing decoder(s) that he introduced specializes to the BP decoder discussed in this chapter. Zyablov and Pinsker were the rst authors to consider the decoding problem for sparse graph codes on the erasure channel [ ]. ey introduced a particular type of regular code ensemble and a decoding algorithm that is equivalent to the peeling decoder (and therefore also equivalent to the message-passing decoder): at each decoding round the decoder recovers all erasures that are connected to checks which are otherwise connected only to known bits.

ey showed that in the asymptotic case and for a suitable choice of parameters such a decoder has a positive erasure correcting radius and that the number of decoding rounds is of the order log(n). Translating their results into modern language, they accomplished this by bounding the probability that a submatrix does not contain checks of degree 1 (i.e.

, by bounding the probability of containing a stopping set). ey also computed the average weight distribution of the ensemble. Unfortunately, the paper was hidden in the Russian literature and was essentially forgotten until recently.

It therefore had little impact on the ensuing development. Without doubt one of most important post-Gallager developments in the realm of the analysis of message-passing systems was the series of papers [ , , , ] by Luby, Mitzenmacher, Shokrollahi, Spielman, and Stemann. A good portion of the material presented in this chapter is taken from these papers.

In particular, they introduced the notion of irregular ensembles together with the elegant and compact description of these ensembles in terms of degree distributions discussed in Section . . It is only through this added degree of freedom introduced by degree distributions that capacity can be approached arbitrarily closely.

ese papers also contain the complete asymptotic analysis, which we have presented. In particular, the fact that the decoding performance is independent of the transmitted codeword (the all-zero codeword assumption discussed in Section . .

), the notion of concentration and the proof technique to show concentration ( eorem . ), the convergence to the tree ensemble ( eorem . ), the density evolution equations ( eorem .

), the stability condition ( eorem . ), the proof that the heavy-tail Poisson distribution (which the authors called Tornado sequence) gives rise to capacity-achieving degree distributions (Example . ), and, nally, the idea of using linear programming to nd good degree distributions (Section .

) are contained in these papers. e original analysis was based on the peeling decoder discussed in Section . .

It is fair to say that almost all we know about message-passing coding we rst learned in the context of message-passing coding for the BEC, and a good portion of the fundamental ideas was developed in this sequence of papers. e fact that there can be global constraints in addition to the local constraints described by the computation tree (see discussion on page ) was pointed out in-. dependently by Ma Visual Studio .NET Quick Response Code cris, Montanari, and Xu as well. e lower bound on the gap to capacity expressed in eorem .

is due to Shokrollahi [ ]. It was the rst bound in which the Shannon capacity was derived explicitly from the density evolution equations. One can therefore think of it as a precursor to the (asymptotic) area theorem.

A systematic study of capacityachieving degree distributions was undertaken by Oswald and Shokrollahi [ ]. ey showed that the check-concentrated (which was called right-concentrated in their paper) degree distribution has a better complexity-versus-performance tradeo than the Tornado sequence. Further properties of the class of capacity-achieving degree distribution pairs were discussed by Orlitsky, Viswanathan, and Zhang [ ].

Sason and Urbanke [ ] presented the so-called Gallager lower bound on the density of codes (Section . as well as eorem . ).

It is a variant of an argument originally put forth by Gallager in his thesis [ ]. e same paper also contained the material of Section . , which proves that check-concentrated ensembles are essentially optimal.

Upper bounds on achievable rates of LDPC ensembles under message-passing decoding were derived by Barak, Burshtein, and Feder [ ]. EXIT charts were introduced by ten Brink [ ] as an e cient visualization of BP decoding. For the BEC, EXIT charts are equivalent to the density evolution analysis and they are exact.

e basic properties of EXIT charts for transmission over the BEC, in particular the duality result ( eorem . ) as well as the area theorem ( eorem . ) are due to Ashikhmin, Kramer, and ten Brink [ , ].

Rather than following one of the original proofs of the area theorem as given by these authors (one of which is discussed in Problem . ), we have taken the point of view of M asson, Montanari, Richardson, and Urbanke [ ], which uses characterization (iii) in Lemma . as a starting point.

From this vantage point, the area theorem is just an application of the fundamental theorem of calculus (see eorem . ). e MAP decoding threshold for transmission over the BEC was rst determined by Montanari [ ] essentially by the method on page which we use to prove claim (i) of eorem .

. e realization that the EXIT curve can be derived from the BP curve via the area theorem is due to M asson and Urbanke [ ]. e material in Section .

is taken from the papers of M asson, Montanari, and Urbanke [ , ]. Several authors have considered ways of improving the BP decoder. Although one can phrase this in many ways, the essential idea is to use the BP decoder to reduce the originally large linear system to a (hopefully) small one (think of the Maxwell decoder that introduces some symbolic variables and expresses the remaining bits in terms of these unknowns).

e latter system can then be decoded directly by Gaussian elimination with manageable complexity. A patent application that contains this idea was led by Shokrollahi, Lassen, and Karp [ ]. A variation on this theme was independently suggested by Pishro-Nik and Fekri [ ].

e idea just described is reminiscent of the e cient encoding method described in Appendix A.. e Maxwell decoder (see Section . ) as the hidden bridge between MAP and BP decoding is due to M asson, Montanari, and Urbanke [ , ]. Further connections relating to the upper bound on the MAP threshold can be found at the end of .

e concept of stopping sets (Section . . ) was introduced by Richardson and Urbanke in the context of e cient encoding algorithms for LDPC ensembles [ ].

Stopping sets play the same role for BP decoding over the BEC that codewords play under MAP decoding. e exact nite-length analysis (Section . ) for regular ensembles was initiated by Di, Proietti, Richardson, Telatar, and Urbanke in [ ].

is paper contained recursions to compute the exact ensemble average block erasure probability for the regular case and more e cient recursions for the special case of le degree 2 and 3. is was quickly followed by extensions to irregular ensembles and the development of more e cient recursions by Zhang and Orlitsky [ ]. E cient expressions for the general case which allowed the determination of the bit as well as block erasure probability (with equal complexity) as well as the determination of the nite-length performance for a xed number of iterations and expurgated cases were given by Richardson and Urbanke [ ].

Our exposition follows closely this paper. An alternative approach to the nite-length analysis was put forth by Yedidia, Sudderth, and Bouchaud [ ] (see also Wang, Kulkarni, and Poor [ ]). Scaling laws have a long and successful history in statistical physics.

We refer the reader to the books by Fisher [ ] and Privman [ ]. e idea of scaling was introduced into the coding literature by Montanari [ ]. e scaling law presented in Section .

is due to Amraoui, Montanari, Richardson, and Urbanke [ ]. e explicit determination of the scaling parameters for the irregular case as well as the optimization of nite-length ensembles is due to Amraoui, Montanari, and Urbanke [ ] (for a description of the Airy functions see [ ]). e re ned scaling law stated in Conjecture .

was shown to be correct for le -regular right-Poisson ensembles by Dembo and Montanari [ ]. Miller and Cohen [ ] proved that the rate of a regular LDPC code converges to the design rate (Lemma . ).

Lemma . , which gives a general condition of convergence of the code rate to the design rate, is due to M asson, Montanari, and Urbanke [ ]. e rst investigations into the weight distribution of regular LDPC ensembles were already done by Gallager [ ].

e combinatorial expressions and expressions for the growth rate were extended to the irregular case simultaneously for various avors of LDPC codes by Burshtein and Miller [ ], Litsyn and Shevelev [ , ], as well as Di, Richardson, and Urbanke [ , ]. e related weight distribution of stopping sets was investigated by Orlitsky, Viswanathan, and Zhang [ ]. Asymptotic expansions for the weight distribution (not only its growth rate) using the Hayman.

method were rst g QRCode for .NET iven by Di [ ]. Rathi [ ] used the Hayman method to prove concentration results for the weight distribution of regular codes (see also Barak and Burshtein [ ]).

e error oor under BP decoding was investigated by Shokrollahi, Richardson, and Urbanke [ ]. e weight distribution problem also received considerable attention in the statistical physics literature. Sourlas pointed out [ , ] that a code can be considered a spin-glass system, opening the way for applying the powerful methods of statistical physics to the investigation of the performance of codes.

e weight distribution of regular LDPC code ensembles was considered by Kabashima, Sazuka, Nakamura, and Saad [ ] and it was shown both by Condamin [ ] and by van Mourik, Saad, 1 and Kabashima [ ] that in this case the limit of n log of the expected number of 1 codewords equals the expected value of n log of the number of codewords, i.e., that inequality ( .

) is in fact an equality. e general weight distribution was investigated by Condamin [ ] and by van Mourik, Saad, and Kabashima [ , ]..

. (P E ). Conside r the in nite degree distribution Ri = c i i!, i 0, where is strictly positive.

Find the constant c so that i Ri = 1. Express the generating function R(x) = i Ri x i in a compact form. What is the average degree R (1) Find the corresponding degree distribution from an edge perspective (x).

Consider the ensemble LDPC (n, , ). How should we choose so that the ensemble has rate r . (E P ).

Let (L(x), R(x)) denote a degree distribution pair from the node perspective. Let ( (x), (x)) be the corresponding degree distribution pair from an edge perspective as de ned in ( . ).

Prove that i ( i ) is the probability that a randomly chosen edge is connected to a variable (check) node of degree i. . (A V C N D ).

Prove that ( ) 1 is the average variable-node degree and, similarly, that ( ) 1 is the average check-node degree. . (D R ).

Prove that 1 ensemble LDPC (n, , ). is equal to the design rate r( , ) of the. . (Y F H M F T Y qr barcode for .NET ).

e following standard exercise in graph theory has a priori no connection to coding. But it demonstrates very well the di erence between node and edge perspective. Consider a graph (not bipartite) with node degree distribution (x) = i i x i .

is means that there are i nodes of degree i. e total number of nodes is (1). and, since every edge has two ends, in order that such a graph exists we need (1) to be even. ink of the nodes as people and assume that each edge represents the relationship of friendship, i.e, two nodes are connected if and only if the two respective people are friends.

Express the average number of friends, call it a, in terms of (x). Now express the average number of friends of a friend in terms of (x). Denote this quantity by b.

Show that in average a friend has more friends than the average person; i.e., show that b a 0.

What is the condition on (x) so that this inequality becomes an equality Don t take it personally. is applies to everyone. .

(R V N D 1 A B ). Consider an ensemble LDPC (n, , ) LDPC (n, L, R) with (0) = 1 0. Prove that in the limit of large blocklengths, the fraction of variable nodes of degree 1 which are connected to check nodes which in turn are connected to at least two variable nodes of degree 1 converges to = 1 (1 1 ).

Use this result to conclude that if transmission takes place over the BEC( ), then under any decoder the resulting bit erasure probability is lower bounded by L1 2 , which is strictly positive for any 0. Discussion: is is the reason why we did not include variable nodes of degree 1 in our de nition of LDPC ensembles. e picture changes if edges are placed in a structured way as discussed in more detail in .

. (T C T [ ]). e de nition of the ensemble can be generalized in the following way.

Consider a set of n variable nodes and m generalized check nodes. e check nodes are de ned in terms of a code C, where C has length r and dimension k. Assume that a check node is satis ed if and only if the connected variable nodes form a codeword in C.

What is the design rate of the code as a function of the parameters . (P P ). Prove Lemma .

for general . Also, what happens if you consider a projection onto a set of k randomly chosen components Hint: Take G and remove from it the computation tree T except for the leaf variable nodes. Compute the expected number of codewords in such a graph assuming that the leaf nodes take on the value 0.

Lemma D. might come in handy for this purpose. en proceed as in the case = 0.

. (E H(n, k) H H D ). e purpose of this exercise is to show that almost no binary linear code has a low-density representation.

Consider binary linear codes C[n, k], where k is the design dimension. (i) Write down the total number of binary (n k) n parity-check matrices H..

(ii) Prove that e VS .NET QR Code JIS X 0510 ach code C[n, k] is represented by at most 2(n k) distinct (n k) n parity-check matrices H..

(iii) Determine a n upper bound on the number of binary (n k) n parity-check matrices H with at most na non-zero entries, where a R. (iv) Conclude from (i), (ii), and (iii) and using Problem . that if we pick a binary linear code C[n, k] uniformly at random from Gallager s parity-check ensemble then the probability that one of its parity-check matrices has at most na ones tends to zero.

More precisely, show that if k = rn, where r (0, 1), then the probability is upper bounded by an expression that asymptotically (as n 2 grows) reads 2 n for some constant that is positive (calculate ). . (E D ).

Show that the message-passing decoder of Section . and the iterative decoder introduced in Section . (more precisely, the generalization of this decoder working on the Tanner graph) lead to identical results.

For any graph and any erasure pattern, the remaining set of erasures at the end of the decoding process is identical. . (S I D ).

Find a simple graph and an erasure pattern so that message-passing decoding fails on the BEC, but on the other hand the ML decoder recovers the transmitted codeword. is shows that in general the message-passing decoder is suboptimal. What is the smallest example you can nd .

(R E ). Consider the regular degree distribution pair ( (x) = x2 , (x) = x6 ). Determine the rate, the threshold BP (both graphically and analytically), the gap to capacity, and the stability condition.

Further, determine the gap to capacity that is predicted by eorem . and eorem . .

. (D E R E ). Consider the regular degree 2 5 distribution pair ( (x) = x , (x) = x ).

For = 4 10, determine PBP ( ) and PBP ( ) T T for [10]. D T B , R , U [ ]). Although one can use the graphical method discussed in Section .

to determine the threshold to any degree of accuracy, it is nevertheless pleasing (and not too hard) to derive analytic expressions. Solving ( . ) for we get ( .

) (x) = x . (1 (1 x)) . (A.

In words, for a g .NET qr barcode iven positive real number x there is a unique value of such that x ful lls the xed point equation f( , x) = x. erefore, if x then the threshold is upper bounded by .

Consider the regular degree distribution pair (l, r). Let xBP denote the unique positive real root of the polynomial p(x) = ((l 1)(r 1) 1)xr 2 r 3 x i . i=0.

Prove that the threshold BP (l, r) is equal to BP (l, r) = (1 xBP ), where (x) is the function de ned in ( . ). 3125 Show that BP (3, 4) = 3672+252 21 0.

647426. . (O T G V ).

Let C be a binary linear code with dual C . Let H be the parity-check matrix of C whose rows consist of all elements of C and let G be the corresponding Tanner graph. Assume that transmission takes place over the BEC( ) and that the BP decoder continues until it no longer makes progress.

Show that in this case the BP decoder performs bit MAP decoding. What is the drawback of this scheme ..

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