lmin 2 in .NET Generating qr codes in .NET lmin 2

lmin 2 using vs .net toconnect denso qr bar code with web,windows application Developing with Visual Studio .NET (3 ). + O( 2 ).. For lmin = 2 this specializes .NET qrcode to Gcw/ss ( ) = ln( (0) (1)) + O( 2 ). Note: If lmin 3, then G( ) is always negative for su ciently small (due to the term log( )).

e situation is more interesting if lmin = 2. In this case G( ) has a derivative at zero and this derivative is negative/positive if (0) (1) is smaller/bigger than 1. For example, for the growth rate of the (2, 4) ensemble shown in Figure .

we have (0) (1) = 3 so that G(2,4) ( ) = log2 (3) + O( 2 ). At this point it is tempting to conjecture that the minimum distance grows linearly if and only if (0) (1) < 1. Unfortunately the situation is more involved.

First, note that the growth rate, as de ned earlier, corresponds to. 1 log E[A(G, n)]. n is quantity has the advantage QR-Code for .NET of being relatively easy to compute. But we have not shown (and there is no reason to believe that this is in fact correct) that the growth rate of typical elements of the ensemble behaves like this average.

To answer this 1 question one should determine instead limn E[ n log A(G, n)]. Unfortunately this quantity is much harder to compute. Using Jensen s inequality we know that ( .

) 1 lim E[ log A(G, n)] n n. 1 log E[A(G, n)] = G( ). n is implies that the computed QR-Code for .NET growth rate G( ) is in general an upper bound on the typical growth rate. Fortunately, this issue does not play a role if we are interested in the question whether typical codes exhibit a linear minimum distance or not.

e second point is important in our current context. e growth rate G( ) only captures the behavior of codewords/ss of weight linear in n. It remains to investigate the behavior of sublinear-sized constellations.

For what follows it is convenient to recall from De nition . the notion of a minimal codeword. (E N C F W A L ).

Consider the ensemble LDPC (n, , ) and de ne = (0) (1). Let E[Acw ss (G, w)] (E[Acw ss (G, w)]) denote the expected number of (minimal) codewords/ss. Let Pcw ss (x) = w 0 pcw ss (w)xw (Pcw ss (x) = w 1 pcw ss (w)xw ) denote the asymptotic generating function counting the number of (minimal) cw/ss, i.

e., pcw ss (w) = lim E[Acw ss (G, w)],. pcw ss (w) = lim E[Acw ss (G, w)].. en ( . ) Pcw ss (x) = 1 , 1 visual .net Quick Response Code x 1 Pcw ss (x) = ln(1 x).

2. 2w w Proof. From (D. ) we know t QR Code JIS X 0510 for .

NET hat pcw ss (w) = w Pcw ss (x) =. 4 w , so that 1 , 1 x 2w w w 4 x = w and we know that pcw ss (w) =. w 2w ,. so that w xw 1 = ln(1 x). w 2 1 Pcw ss (x) = 2w L . (A E F ). Consider the en semble LDPC (n, , ) and de ne = (0) (1).

Let s denote the expurgation parameter introduced in De nition . . en for < BP.

lim EELDPC(n,s, , ) [PB MAP BP (G, )] = 1 e (G, )] =. ( )w 2w lim n EELDPC(n,s, , ) [Pb MAP BP 1 ( )w . 2w s For s = 1 these expressions specialize to lim ELDPC(n, , ) [PB MAP BP (G, )] = 1 (G, )] =. 1 ,. lim n ELDPC(n, , ) [Pb MAP BP 1 . 2 1 . Discussion: For both the BP d ecoder and the MAP decoder we only consider the erasure oor below BP . We conjecture that under MAP decoding the expression for the erasure oor stays valid in the regime BP < MAP . Proof.

We start with the unexpurgated case. Consider BP decoding. We claim that for < BP the erasure probability due to large-weight stopping sets is negligible.

Let be any strictly positive constant. Since < BP , there exists an so that x , the expected fraction of erasures le in the -th iteration, is strictly less than . By (the Concentration) eorem .

this implies that the contribution to the block erasure probability due to erasures of size exceeding n decays exponentially in the length n. Since this is true under BP decoding this is also true under MAP decoding..

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