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Random Generation of Optimal Codes

Abstract

THE problem dealt with in this communication is that of generating an optimal code for the transmission of a long sequence of binary digits over a noisy channel. The channel is supposed to produce errors at random, p (< ½) being the probability that a particular digit is wrong, and q (= 1 − p) being the probability that it is right. The capacity of such a channel is where X = − (p log p + q log q) lies between 0 and 1, and all logarithms are referred to the base 2. Consequently, if it is desired to transmit B bits of information reliably, a total of N digits must be sent, where N > B/ (1 − X). One way of doing this is to begin by sending B message digits, and to follow them with C checking digits, where C = N (X + δ) and δ is a positive finite number. Then B/N = 1 − C/N = 1 − X − δ < 1 − X as required. The checking digits are made to depend on the message digits so as to enable the recipient, in spite of the transmission errors, to reconstruct the complete set of N digits with virtual certainty. The question is: can this be done with a number of checking digits which is close to the theoretical minimum ? In other words, is it possible to specify an actual coding procedure such that when N is sufficiently large δ can be made arbitrarily small ? It is possible, and I now describe such a procedure.

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LoNGUET-HIGGINS, H. Random Generation of Optimal Codes. Nature 213, 1160–1161 (1967). https://doi.org/10.1038/2131160a0

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