g 6 4 g 8 4 in .NET Generate Denso QR Bar Code in .NET g 6 4 g 8 4

g 6 4 g 8 4 generate, create qr codes none on .net projects Recommended GS1 barcodes for mobile apps g 2 is odd, g 2 is even Conversely, can you sh ow that (l, g). g+4 4 (l 1) 2 l 2 and (2, g) g 2 Note: If l = 2 the bounds on g grow linearly in n and, therefore, the size of the smallest stopping set is at most O(log n). is suggests that if in a graph a signi cant fraction of variable nodes has degree 2 then small stopping sets are likely to exist..

. (F G C K ,L , F [ ] Denso QR Bar Code for .NET ).

Consider the following construction of an (l, r)-regular LDPC code of length n. Pick a prime p. Let n = p2 and arrange the n variable nodes in a p p grid.

Choose a slope and a shi and consider a line, i.e., the set of all points on this line.

Each such line contains exactly p points. Associate to each such line a check node, which is connected to all the variable nodes on this line. If we pick all p shi s for a given slope then we get pcheck nodes and every variable node participates in exactly one check.

Show that if we pick l distinct slopes then we get an LDPC code (i) of length p2 , (ii) with lp check nodes, (iii) with variable node degree l, and (iv) with check-node degree p. Show further that the resulting graph has girth exactly 6. .

(V C EXIT F of the various characterizations in Lemma . . ).

Prove the equivalence. . (C C ). Consider a c ycle code ensemble LDPC (n, x, (x)).

Show that the BP threshold and the MAP threshold are identical. . (EXIT F H C ).

Derive the EXIT function given in Example . for the [7, 4] Hamming code. .

(A P A T A ,K , B [ , ]). For the BEC there are many alternative proofs of the area theorem. Let us consider one such alternative here.

Let C be a binary linear code of rate r and 1 length n. We want to show that 0 h( )d = r. Let (n) denote the set of permutations on n letters.

Assume that the 2nr codewords of the code C are equally likely. Let K denote the index set of known bits. Justify each of the following steps:.

1 1 n. h( )d = = 0 n i=1 hi ( )d = n i=1 0 H(Xi Y[n]. i=1 1 0 K [n] i (1 ) K n 1 K H(Xi XK )d (n 1 K )!( K )! H(Xi XK ) n! H(X ( j) X ([ j 1]) ). n i=1 K [n] i (ii). 1 n! 1 n! . n j=1 [n] n [n] j=1 H(X ( j) X ([ j 1]) ). (iv) (v). (iii). 1 n! . H(X1 , . . . , Xn ) = H(X) = nr. [n]. . (I A T ). In the gen .

net vs 2010 QR Code eral case where bits are sent over di erent channels (the Area) eorem . has the following interpretation: if we change the set of all channels from some starting state to some nal state (change the individual channel parameters) then by doing so we change the conditional entropy H(X Y). Assume that we connect the initial and the nal state by some smooth path; i.

e., i = i ( ) is a piecewise di erentiable function of for i [n]. en the (average) EXIT function h( ) measures the change per d of H(X Y).

1 More interestingly, since h( ) = n n hi ( ), every bit position i contributes loi=1 cally to this change according to hi ( ). For di erent curves that connect the same nal and initial state the total change of H(X Y) along the path is the same but the individual contributions according to hi ( ) are in general di erent. is is best seen by a simple example.

Consider the [2, 1, 2] repetition code. Assume rst that the two channels are parameterized by 1 = = 2 , where goes from 0 to 1. Consider next the alternative parameterization 1 ( ) = min 1, and 2 ( ) = max 0, 1 , where [0, 2].

For both cases compute the individual contributions of the two EXIT functions. (EXIT R E ). Prove inequality ( .

). Hint: e proof is conceptually simple but on the lengthy side. Either pick a particular l and prove the assertion for this l or prove the statement for some sufciently large l.

Bon courage! . . (S PB H).

Consider Gallager s parity-check ensemble H(n, r) and transmission over the BEC( ). Fix z = n( ) and let n tend to in nity. Show that in this limit the average block error probability under MAP decoding behaves like EH(n,r) [PB (H, z)] = Q (1 ) z (1 + O(1 n)) .

. Hint: Make use of the results of Problem . . More precisely, let A be a k m random binary matrix where each entry is chosen independently uniformly at random from 0, 1 .

Use the fact that P rank (A) = k = 0,.
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