Lower Bounds Error Probability Coding by Shannon Gallager Berlekamp (3 results)
More imagesLower Bounds to Error Probability for Coding on Discrete Memoryless Channels. [Part] I [offprint]
Shannon, C. E. [Claude Elwood] ; Gallager, R. G. [Robert] ; Berlekamp, E. R. [Elwyn]
Published by Academic Press, Inc, [New York, NY], 1967
- Softcover
Seller: Kuenzig Books ( ABAA / ILAB ), Topsfield, MA, U.S.A.Kuenzig Books ( ABAA / ILAB )
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Add to basketWraps. Condition: Very Good. 65-103, [1-blank] pages. 8 15/16 x 6 inches. Self-wrappers stapled at the spine. Wraps. "Information and Control," Vol 10, No 1, February 1967 (pp. 65-103) first published this paper, here offered in offprint form. This offprint does not have separate wrappers that we are aware of - the reprint state…ment is printed upper left on the first page of the paper. "The noisy channel coding theorem (Shannon, 1948) states that for a broad class of communication channels, data can be transmitted over the channel in appropriately coded form at any rate less than channel capacity with arbitrarily small error probability. Naturally, there is a rub in such a delightful sounding theorem, and the rub here is that the error probability can, in general, be made small only by making the coding constraint length large; this, in turn, introduces complexity into the encoder and decoder. Thus, if one wishes to employ coding on a particular channel, it is of interest to know not only the capacity but also how quickly the error probability can be made to approach zero with increasing constraint length." (pp.65-66) "New lower bounds are presented for the minimum error probability that can be achieved through the use of block coding on noisy discrete memoryless channels. Like previous upper bounds, these lower bounds decrease exponentially with the block length N. The coefficient of N in the exponent is a convex function of the rate. From a certain rate of transmission up to channel capacity, the exponents of the upper and lower bounds coincide. Below this particular rate, the exponents of the upper and lower bounds differ, although they approach the same limit as the rate approaches zero. Examples are given, and various incidental results and techniques relating to coding theory are developed. The paper is presented in two parts: the first, appearing here, summarizes the major results and treats the case of high transmission rates in detail; the second, to appear in the subsequent issue, treats the case of low transmission rate." (abstract) PROVENANCE: The personal files of Claude E. Shannon (unmarked). There were multiple examples of this item in Shannon's files. REFERENCES: Sloane and Wyner, "Claude Elwood Shannon Collected Papers," #122 Reprinted in D. Slepian, editor," Key Papers in the Development of Information Theory," IEEE Press, NY, 1974, pp 194-204.
More imagesLower Bounds to Error Probability for Coding on Discrete Memoryless Channels. [Part] II [offprint]
Shannon, C. E. [Claude Elwood]; Gallager, R. G.; Berlekamp, E. R.
Published by Academic Press, Inc, [New York, NY], 1967
- Softcover
Seller: Kuenzig Books ( ABAA / ILAB ), Topsfield, MA, U.S.A.Kuenzig Books ( ABAA / ILAB )
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Add to basketWraps. Condition: Very Good. [1-blank], 522-552 pages. 8 15/16 x 6 inches. Self-wrappers stapled at the spine. Wraps. "Information and Control," Vol 10, No 5, May 1967 (pp. 522-552) first published this paper, here offered in offprint form. Not issued with separate wrappers that we are aware of - the reprint statement is at the…top of the first page. "New lower bounds are presented for the minimum error probability that can be achieved through the use of block coding on noisy discrete memoryless channels. Like previous upper bounds, these lower bounds decrease exponentially with the block length N. The coefficient of N in the exponent is a convex function of the rate. From a certain rate of transmission up to channel capacity, the exponents of the upper and lower bounds coincide. Below this particular rate, the exponents of the upper and lower bounds differ, although they approach the same limit as the rate approaches zero. Examples are given, and various incidental results and techniques relating to coding theory are developed. The paper is presented in two parts: the first, appearing in the January issue, summarizes the major results and treats the case of high transmission rates in detail; the second, appearing here, treats the case of low transmission rates." (abstract) This paper is the follow-up to Part I, this time dealing with low transmission rates. PROVENANCE: The personal files of Claude E. Shannon (unmarked). One of two examples found in Shannon's files. REFERENCES: Sloane and Wyner, "Claude Elwood Shannon Collected Papers," #123 D. Slepian, editor, "Key Papers in the Development of Information Theory," IEEE Press, NY, 1974, pp 205-213.
- Softcover
- First Edition
Seller: Herman H. J. Lynge & Søn ILAB-ABF, Copenhagen, DenmarkHerman H. J. Lynge & Søn ILAB-ABF
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Add to basketNew York, Academic Press Inc, 1967. 8vo. In the original orange printed wrappers. One library stamp and two library labels pasted on to front wrapper. A white label stating the issue's name and date pasted on to back wrapper. Internally very fine and clean. Pp. 522-52. [Entire issue: 447-552 + 6 pages with commercials]. First pr…inting of Shannon's paper on lower bounds to error probability. This was his "final effort to establish tight upper and lower bounds on error probability for the DMC. Earlier, Robert Fano [22] had discovered, but not completely proved, the sphere-packing lower bound on error probability. In [20], [21], the sphere-packing bound was proven rigorously, and another lower bound on error probability was established which was stronger at low data rates. The proof of the sphere-packing bound given here was quite complicated" it was later proven in a simpler way." (Gallager, Claude E. Shannon: A Retrospective on His Life, Work, and Impact, 2001, Pp. 2691-2)."Claude E. Shannon invented information theory and provided the concepts, insights, and mathematical formulations that now form the basis for modern communication technology. In a surprisingly large number of ways, he enabled the information age."(Ibid.).The issue contain the following papers:Gold, E. Mark, Language Identification in the Limit, Pp. 447-474.Cohen, Joel M, The Equivalence of Two Concepts of Categorical Grammar, Pp. 475-484.Mulholland, R. G., Aravind K. Joshi, J. T. Chu. Optimal Decision Functions for Two Noise States, Pp. 485-498.Harary, Frank, Edgar M. Palmer. Enumeration of Finite Automata, Pp. 499-508.Kramer, Anthony J. Use of Orthogonal Signaling in Sequential Decision Feedback. Pp. 509-521.
