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Improvements of Winograd's result on computation in the presence of noise (Corresp.)

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Winograd's result concerning Elias' model of computation in the presence of noise can be stated without reference to computation. If a code \varphi : {0,1}^{k} \rightarrow {0,1}^{n} is min-preserving (\varphi (a \wedge b) = \varphi (a) \wedge \varphi (b) for a,b \in {0,1}^{k}) and \epsilon n -error correcting, then the rate k/n \rightarrow 0 as k \rightarrow \infty . This result is improved and extended in two directions. begin{enumerate} item For min-preserving codes with {em fixed} maximal (and also average) error probability on a binary symmetric channel again k/n \rightarrow 0 as k \rightarrow \infty (strong converses). item Second, codes with lattice properties without reference to computing are studied for their own sake. Already for monotone codes ( \varphi (a) \leq \varphi (b) for a \leq b) the results in direction 1) hold for maximal errors. end{enumerate} These results provide examples of coding theorems in which entropy plays no role, and they can be reconsidered from the viewpoint of multiuser information theory.

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Information Theory, IEEE Transactions on  (Volume:30 ,  Issue: 6 )