‎"SHANNON, C. E. (+) R. G. GALLAGER (+) E. R. BERLEKAMP.‎
‎Lower Bounds to Error Probability for Coding on Discrete Memoryless Channels. II.‎

‎New 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].‎

Reference : 44958

‎First printing 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. ‎