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170 IEEE TRANSACTIONS ON lNFOkMATION THEORY, VOL. IT-18, NO. 1, JANUARY 1912 A Class of Algorithms for Decoding Block Codes With Channel Measurement Information DAVID CHASE, MEMBER, IEEE Abstract-A class of decoding algorithms that utilizes channel measure- ment information, in addition to the conventional use of the algebraic properties of the code, is presented. The maximum number of errors that can, with high probability, be corrected is equal to one less than d, the minimum Hamming distance of the code. This two-fold increase over the error-correcting capability of a conventional binary decoder is achieved by using channel measurement (soft-decision) information to provide a measure of the relative reliability of each of the received binary digits. An upper bound on these decoding algorithms is derived, which is pro- portional to the probability of an error for dth order diversity, an expres- sion that has been evaluated for a wide range of communication channels and modulation techhiques. With the aid of a lower hound on these algorithms, which is also a lower hound on a correlation (maximum- likelihood) decoder, we show for both the Gaussian and Rayleigh fading channels, that as the signal-to-noise ratio (SNR) increases, the asynip- totic behavior of these decoding algorithms cannot be improved. Com- puter simulations indicate that even for low SNR the performance of a correlation decoder can be approached by relatively simple decoding procedures. In addition, we study the effect on the performance of these decoding algorithms when a threshold is used to simplify the decoding process. I. INTRODUCTION B LOCK codes have been the subject of a considerable amount of research in information theory. General coding theorems [I], [2] and sophisticated implementation techniques [3]-[5] have been developed for these codes. In general, though, when dealing with binary codes, the decoding developed assume a channei whose output is also binary. For many communication applica- tions this assumption is not necessary, and furthermore, as indicated by the coding theorems developed for input channels with J(> 2) outputs, ’ there is a significant loss in performance when just a binary output is assumed. The subject of this paper is to extend the binary decoding techniques previously developed to channels where more than just a binary output is available. A block diagram of the communication system of interest is shown in Fig. 1. Each sequence of K binary information digits is assumed to be encoded into a block of N binary digits denoted by X = X, ,X2,. . .,X,. These binary digits are fed into a data modulator, which determines the trans- mitted waveform x(t). When a binary channel is assumed, the data demodulator produces a sequence of N binary Manuscript received January 15, 1971 ; revised April 30, 1971. Some of the results in this paper were presented at the IEEE International Symposium on Information Theory, Noordwijk, the Netherlands, June 1970. The author is with the General Atronics Corporation, a subsidiary of the Magnavox Company, Philadelphia, Pa. 19118. 1 For example. if we consider the additive noise Gaussian channel
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