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iOS Let us de ne the Maxwell ( M) decoder: Given the received word, which was transmitted over the BEC( ), the M decoder proceeds like the standard peeling decoder described in Section . . At each time step a parity-check equation involving a single undetermined variable is chosen and used to determine the value of the variable.

is value is substituted in any parity-check equation involving the same variable. If at any time the peeling decoder gets stuck in a non-empty stopping set, a position i [n] is chosen uniformly at random from the set of yet undetermined bits and a binary (symbolic) variable vi representing the value of bit i is associated with this position. In what follows, the decoder proceeds as if position i was known and whenever the value of bit i is called for it is referred to as vi .

is means that messages consist not only of numbers 0 or 1 but in general contain (combinations of) symbolic variables vi . In other words, the messages are really equations that state how some quantities can be expressed in terms of other quantities. It can happen that during the decoding process of the peeling decoder a yet undetermined variable is connected to several nodes of degree 1.

It will then receive a message describing its value from each of these connected check nodes of degree 1. Of course, all these messages describe the same value (recall that, over the BEC, no errors occur). erefore, if and only if at least one of these messages contains a symbolic variable, then the condition that all these messages describe the same value gives rise to linear equations which have to be ful lled.

Whenever this happens, the decoder resolves this set of equations with respect to some of the previously introduced variables vi and eliminates those resolved variables in the whole system. e decoding process nishes once the residual graph is empty. By de nition of the process, the decoder always terminates.

At this point there are two possibilities. e rst is that all introduced variables vi i I , I [n], were resolved at some later stage of the decoding process (a special case of this being that no such variables ever had to be introduced). In this case each bit has an associated value (either 0 or 1) and this is the only solution compatible with the received information).

In other words, the decoded word is the MAP estimate. e other possibility is that there are some undetermined variables vi i I remaining. In this case each variable node either already has a speci c value (0 or 1) or by de nition of the decoder can be expressed as a linear combination of the variables vi i I .

In such a case each realization (choice) of vi i I 0, 1 I gives rise to a valid codeword and all codewords compatible with the received information are the result of a particular such choice. In other words, we have accomplished a complete list decoding, so that I equals the conditional entropy H(X Y). All this is probably best understood by an example.

E . (M D A (l = 3, r = 6) C ). Figure .

shows. (i) Decoding bit from check (ii) Decoding bit from check (iii) Decoding bit v2 v2 from check (iv) Introducing variable v2 v2 +v6 v6 v2 v2 v6 (v) Introducing variable v6 v2 +v6 v6 v2 v2 +v6 v2 v6 (vi) Decoding bit v2 +v6 v6 v2 (v2 + v6 ) from check v2 +v6 v6 v2 (vii) Decoding bit v6 +v12 v2 +v6 v2 (v6 ) from check v +v v2 +v6 6 12 v2 (viii) Introducing variable v12 v2 v12 v2 (ix) Decoding bit (v6 + v12 = v12 ) from checks and v6 = 0 v2 +v12 v2 v2 +v12 v12 v2 (x) Decoding bit (v12 ) from check v2 +v12v2 +v12 v2 (xi) Decoding bit (v2 + v12 ) from check v2 +v12 v12 v2 v12 (xii) Decoding bit (v2 + v12 ) from check 0 v12 v12 (xiii) Decoding bit (v12 = v2 + v12 ) from checks and v2 = 0 (xiv) Decoding bit (v12 = 0) from checks , , and v12 = 0 Figure . : M decoder applied to a (l = 3, r = 6)-regular code of length n = 30. the workings of the M deco none none der applied to a simple code of length n = 30. Assume that the all-zero codeword has been transmitted. In the initial decoding step the received (i.

e., known and equal to 0) bits are removed from the bipartite graph. e remaining graph is shown in (i).

e rst phase is equivalent to the standard peeling algorithm: in the rst three steps, the decoder determines the bits 1 (from check node 1), 10 (from check node 5), and 11 (from check node 8). At this point the peeling decoder is stuck in the constellation shown in (iv). e second phase is distinct to the M decoder: the decoder assigns the variable v2 to the (randomly chosen) bit 2, which is now considered to be known.

e decoder then proceeds again as the standard peeling algorithm. Any time it gets stuck, it assigns a new variable vi to a yet undetermined and randomly chosen position i. is process continues until some of the previously introduced variables can be eliminated.

For example, consider step (ix): the variable node at position 30 receives the messages v6 + v12 as well as the message v12 . is means that the decoder has deduced from the received word that the only compatible codewords are the ones for which the value of bit 30 is equal to the value of bit 12 and also equal to the sum of the values of bit 6 and bit 12. e decoder can now deduce from this that v6 = 0, i.

e., the previously introduced variable v6 is eliminated from the system. Phases in which new variables are introduced, phases during which some previously introduced variables are resolved, and regular BP decoding phases might alternate.

Decoding is successful (in the sense that a MAP decoder would have succeeded) if at the end of the decoding process, all introduced variables have been resolved. is is the case for the shown example. e following lemma, whose proof we skip, explains why at the MAP threshold the two dark gray areas are in balance.

In short, the area on the le is proportional to the total number of variables which the decoder introduces and the area on the right is proportional to the total number of equations which are generated and which are used to resolve those variables. Further, as long as the number of generated equations is no larger than the number of introduced variables then these equations are linearly independent with high probability. erefore, when these two areas are equal then the number of unresolved variables at the end of the decoding process is (essentially) zero, which means that MAP decoding is possible.

L . (A N U V ). Consider the (l, r)-regular degree distribution pair.

Let (x) = x (1 (1 x)) and let P(x) denote the trial entropy of De nition . . Let G be chosen uniformly at random from LDPC n, (x) = xl 1 , (x) = xr 1 .

Assume that transmission takes place over the BEC( ) where MAP and that we apply the M decoder. Let S(G, ) denote the number of variable nodes in the residual graph a er the -th decoding step and let V(G, ) denote the number of unre-. solved variables vi i I at this point, i.e., V(G, ) = I .

en, as n tends to in nity, (s(x), v(x)) = limn (EG [S(G, xn ) n], EG [V(G, nx ) n]) exists and is given by s(x) = (x)hEBP (x), v(x) = P(x) P(x) + ( BP (x))hEBP (x)1.
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