Summary
Gallager [1] and Gallager, Shannon, Berlekamp [2] establish exponentially decreasing upper and lower bounds, respectively, on the error of the best codes for fixed code rates R smaller than the capacity for the standard channels (stationary finite alphabet channels without memory). These bounds happen to coincide up to the first order for rates near to the capacity.
The authors of [2] regret that their proof of the lower bound cannot be extended to infinite alphabet channels or nonstationary channels because of the use of “fixed composition codes” (while the proofs of the upper estimate can be easily transferred to those channels).
Changing parts of the proof in [2], we automatically obtain estimates of the same type as in [2] for the latter channels.
Furthermore, as a matter of minor importance, we show that the order of coincidence of upper and lower bound for the high rates R can be rigorously improved.
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References
Gallager, R.: A simple derivation of the coding theorem. IEEE Trans. Inform. Theory IT-11, 3–18 (1965).
—, Shannon, C. E., Berlekamp, E. R.: Lower bounds to error probability for coding. Inform. and Control, 10, 65–103 (1967).
Augustin, U.: GedÄchtnisfreie KanÄle für diskrete Zeit. Z. Wahrscheinlichkeitstheorie verw. Geb. 6, 10–61 (1966).
Wolfowitz, J.: Coding theorems of information theory. Ergebnisse d. Math. u. Grenzgebiete, Vol. 31. Berlin-Göttingen-Heidelberg: Springer 1964.
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I would like to thank Johan H. B. Kemperman for a very stimulating discussion and for pointing out a gap in my original proof.
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Augustin, U. Error estimates for low rate codes. Z. Wahrscheinlichkeitstheorie verw Gebiete 14, 61–88 (1969). https://doi.org/10.1007/BF00534118
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DOI: https://doi.org/10.1007/BF00534118