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The card security number is not contained in the magnetic stripe information, nor does it appear on sales receipts. Using the card security scheme helps prevent merchants from receiving counterfeit cards or being a victim of fraud. For transactions conducted over the Internet, you may ask cardholders for their CVV2 online. Their Internet screen might include elements shown in Exhibit 5.2. Authorization requests must include at least: Account number Expiration date CVV2 value Transaction dollar amount

document all of the required elements vendors ask for. It is your choice to accept or reject with most of these services, but if you accept when vendors said no, you are on your own.

This research was supported by grants from NSERC, INRIA, and NSF. The authors are grateful to Roger Wattenhoffer for a fruitful e-mail discussion that contributed to clari cation of major aspects of dominating sets and subsequently to an improved presentation of this chapter. We are also grateful to Stefano Basagni for a careful reading of the manuscript and several useful suggestions.

. . . . . . . . . . . .

where the inequality follows from Jensen s inequality and the concavity of the logarithm. Now since the yi are distinct, we have i:|yi |=l,si =s P (yi |si ) 1. Thus, log Qk (x1 , x2 , . . . , xn |s1 )

Tables 8.4 and 8.5 show the calculated values of coef cients Q i0j and Q i1j , respectively. In the second iteration, the updated values of coef cients Q i0j and Q i1j allow us to determine the updated values of coef cients Ri0j and Ri1j . This iteration is different from the rst iteration in the sense that coef cients Q i0j and Q i1j now contain updated information, rather than simply channel information. In this second iteration the calculation of values of coef cients Ri0j and Ri1j leads to a new estimated decoded vector d, which again does not satisfy the syndrome condition, and so the decoding process continues. In the example under analysis, the decoder is able to nd the correct code vector after three iterations, and is also able to correct the two errors that the hard-decision received vector contained. In this particular case the errors are in the message part of the code vector, that is, in two of the four bits that are nally taken as the message bits, after truncating the redundancy. The iterative decoding algorithm is able to correct these two errors. Tables 8.6 and 8.7 illustrate the evolution of the decoding algorithm by presenting the values of the coef cients involved, until arriving at the nal solution. These tables show values that are truncated to four decimal places, though actual values were determined in a more accurate way by using MATLAB R Program 5.3. The updated values of coef cients Ri0j and Ri1j allow us to determine a new estimated decoded vector. This second estimate of the decoded vector is shown in Table 8.8.