Celebratio Mathematica

D. Blackwell: “The entropy of functions of finite-state Markov chains,” pp. 13–20 in Information theory, statistical decision functions, random processes (Liblice, Czech Republic, 28 November–30 November 1956), vol. 1. Edited by J. Kožešnik. Czechoslovak Academy of Sciences (Prague), 1957. MR 0100297 Zbl 0085.12401 incollection

D. Blackwell, L. Breiman, and A. J. Thomasian: “Proof of Shannon’s transmission theorem for finite-state indecomposable channels,” Ann. Math. Stat. 29 : 4 (1958), pp. 1209–1220. MR 0118570 Zbl 0096.10901 article

Aram John Thomasian	Related
Leo Breiman	Related

D. Blackwell: “Infinite codes for memoryless channels,” Ann. Math. Stat. 30 : 4 (1959), pp. 1242–1244. MR 0127448 Zbl 0104.11701 article

For any finite set \( S \), we denote by \( S^{(N)} \) the set of all sequences \( (s_1, \dots \), \( s_N) \), where \( s_n \in S \) for \( n = 1 \), \( 2, \dots \), \( N \). For a memoryless channel with finite input alphabet \( A \), finite output alphabet \( B \), an infinite code (for transmitting at rate 1) is defined as consisting of

a sequence \( \{f_n\} \) of functions, where \( f_n \) maps \( I^{(n)} \) into \( A \), and \( I \) consists of the two elements 0 and 1,
a nondecreasing sequence \( \{M(n)\} \) of positive integers such that \( M(n)/n \to 1 \) as \( n \to \infty \), and
a sequence \( \{g_n\} \) of functions, where \( \{g_n\} \) maps \( B^{(n)} \) into \( I^{(M(n))} \).

An infinite sequence \( x = (x_1 \), \( x_2, \dots) \) of 0s and 1s, together with an infinite code, defines a sequence of independent output variables \( y_1 \), \( y_2, \dots \), with \[ \Pr \{y_n = b\} = p\bigl(b \mid f_n(x_1, \dots, x_n)\bigr), \] where \( p(b \mid a) \) is the probability that the output symbol of the channel is \( b \), given that the corresponding input symbol is \( a \), and defines a sequence of estimated messages \( t_1 \), \( t_2, \dots \), where \[ t_n = g_n(y_1, \dots\( , \)y_n) .\] We shall say that the code is effective at \( x \) if, with probability 1, \[ t_n = (x_1, \dots,x_{M(n)}) \] for all sufficiently large \( n \), and shall say that the code is effective if it is effective for every \( x \). The result of this note is the

For any memoryless channel with capacity \( C > 1 \), there is an effective code.

D. Blackwell, L. Breiman, and A. J. Thomasian: “The capacity of a class of channels,” Ann. Math. Stat. 30 : 4 (1959), pp. 1229–1241. MR 0127449 Zbl 0104.11604 article

Aram John Thomasian	Related
Leo Breiman	Related

D. Blackwell, L. Breiman, and A. J. Thomasian: “The capacities of certain channel classes under random coding,” Ann. Math. Stat. 31 : 3 (1960), pp. 558–567. MR 0127450 Zbl 0119.13805 article

Aram John Thomasian	Related
Leo Breiman	Related

D. Blackwell: “Exponential error bounds for finite state channels,” pp. 57–63 in Proceedings of the fourth Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 20 June–30 July 1960), vol. I. Edited by J. Neyman. University of California Press (Berkeley and Los Angeles), 1961. MR 0135199 Zbl 0104.11603 incollection

D. Blackwell: “Information theory,” pp. 182–193 in Modern mathematics for the engineer. Edited by E. F. Beckenbach. McGraw-Hill (New York), 1961. MR 0129161 incollection

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David H. Blackwell

Information theory

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