Variations on a theme by Schalkwijk and Kailath

Schalkwijk and Kailath (1966) developed a class of block codes for Gaussian channels with ideal feedback for which the probability of decoding error decreases as a second-order exponent in block length for rates below capacity. This well-known but surprising result is explained and simply derived here in terms of a result by Elias (1956) concerning the minimum mean-square distortion achievable in transmitting a single Gaussian random variable over multiple uses of the same Gaussian channel. A simple modification of the Schalkwijk-Kailath scheme is then shown to have an error probability that decreases with an exponential order which is linearly increasing with block length. In the infinite bandwidth limit, this scheme produces zero error probability using bounded expected energy at all rates below capacity. A lower bound on error probability for the finite bandwidth case is then derived in which the error probability decreases with an exponential order which is linearly increasing in block length at the same rate as the upper bound.Comment: 18 Pages, 4 figures (added reference

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Contains research objectives, summary of research and reports on one research project.Joint Services Electronics Programs (U. S. Army, U.S. Navy, and U. S. Air Force) under Contract DA 28-043-AMC-02536(E)National Aeronautics and Space Administration (Grant NGL-22-009-013

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Contains research objectives and reports on three research projects.National Aeronautics and Space Administration (Grant NsG-334)Joint Services Electronics Programs (U. S. Army, U.S. Navy, and U.S. Air Force) under Contract DA 28-043-AMC-02536(E

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Contains research objectives and reports on two research projects.National Aeronautics and Space Administration (Grant NsG-334)National Aeronautics and Space Administration (Grant NsG-496)Joint Services Electronics Programs (U. S. Army, U.S. Navy, and U.S. Air Force) under Contract DA 36-039-AMC-03200(E)National Science Foundation (Grant GP-835)Sloan Fund for Basic Research (M.I.T. Grant

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Contains research objectives and summary of research.National Aeronautics and Space Administration (Grant NGL 22-009-013)National Aeronautics and Space Administration (Contract NAS8-27733)Joint Services Electronics Programs (U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DAAB07-71-C-0300U.S. Army Research Office - Durham (Contract DAHC04-71-C-0039)U.S. Army Research Office - Durham (Contract DAHC04-69-C-0042

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Contains research objectives and reports on three research projects.National Science Foundation (Grant GP-2495)National Institutes of Health (Grant MH-04737-04)National Aeronautics and Space Administration (Grant NsG-334)National Aeronautics and Space Administration (Grant NsG-496

Processing and Transmission of Information

Contains research objectives, summary of research and reports on two research projects.National Aeronautics and Space Administration (Grant NGL 22-009-013)Joint Services Electronics Programs (U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DA 28-043-AMC-02536(E)U. S. Army Research Office - Durham (Contract DAHCO4-69-C-0042

Processing and Transmission of Information

Contains research objectives and reports on two research projects.National Aeronautics and Space Administration under Grant NsG-334Joint Services Electronics Programs (U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DA 36-039-AMC-03200(E)National Science Foundation (Grant GP-2495)National Aeronautics and Space Administration (Grant NsG-496