IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 7, NOVEMBER 1998 3083 Bit-Error Probability for Maximum-Likelihood Decoding of Linear Block Codes and Related Soft-Decision Decoding Methods Marc P. C. Fossorier, Member, IEEE, Shu Lin, Fellow, IEEE, and Dojun Rhee Abstract— In this correspondence, the bit-error probability for maximum-likelihood decoding of binary linear block codes is investigated. The contribution of each information bit to is considered and an upper bound on is derived. For randomly generated codes, it is shown that the conventional approximation at high SNR , where represents the block error probability, holds for systematic encoding only. Also systematic encoding provides the minimum when the inverse mapping corresponding to the generator matrix of the code is used to retrieve the information sequence. The bit- error performances corresponding to other generator matrix forms are also evaluated. Although derived for codes with a generator matrix ran- domly generated, these results are shown to provide good approximations for codes used in practice. Finally, for soft-decision decoding methods which require a generator matrix with a particular structure such as trellis decoding, multistage decoding, or algebraic-based soft-decision decoding, equivalent schemes that reduce the bit-error probability are discussed. Although the gains achieved at practical bit-error rates are only a fraction of a decibel, they remain meaningful as they are of the same orders as the error performance differences between optimum and suboptimum decodings. Most importantly, these gains are free as they are achieved with no or little additional circuitry which is transparent to the conventional implementation. Index Terms—Bit-error probability, block codes, maximum-likelihood decoding, soft-decision decoding, weight distribution. I. INTRODUCTION For equally likely binary phase-shift keying (BPSK) transmission over the additive white Gaussian noise (AWGN) channel, maximum- likelihood decoding (MLD) of a linear block code provides the optimum decoding strategy to minimize the probability that a decoded block is in error [1, p. 26]. However, MLD does not guarantee that the probability of a bit being in error is minimized. Although this fact has long been recognized, the optimal decoding strategy that minimizes the bit-error probability associated with MLD for the AWGN is still unknown due to the high dependency on the code structure which makes the analysis very difficult. Most of the research on this subject is related to the binary- symmetric channel (BSC) and mostly consists of improving the traditional decoding rule using the standard array [2, p. 68]. Therefore, these works assume hard-decision decoding of the received sequence. A concise overview of these decoding schemes is available in [3, Part II ]. For the AWGN, asymptotic bounds on the error performance have been derived in [4]. In [4], the probability of a bit error in the decoded sequence of length is minimized. However, in many applications, the main goal is the minimization of a bit error in the corresponding Manuscript received June 8, 1986; revised February 5, 1998. This work was supported by NSF under Grant NCR-94-15374, NASA under Grant NAG 5-931, and LSI Logic Corporation. The material in this correspondence was presented in part at the IEEE International Symposium on Information Theory and Its Applications, Victoria, BC, Canada, September 1996. M. P. C. Fossorier and S. Lin are with the Department of Electrical Engineering, University of Hawaii at Manoa, Honolulu, HI 96822 USA. D. Rhee is with the LSI Logic Corporation, Milpitas, CA 95035 USA. Publisher Item Identifier S 0018-9448(98)06742-X. sequence of information bits. The two corresponding bit-error probabilities are the same for a systematic code, but they differ for other encoding methods not in systematic form because of error propagation effects when retrieving the information bits from the decoded binary sequence. In this correspondence, we consider the minimization of the bit-error probability for MLD of linear block codes and other soft- decision decoding methods. Although not optimum, this minimization remains important as MLD has been widely used in practical ap- plications. We assume that the information sequence of length is recovered from the decoded codeword based on the inverse mapping defined from the generator matrix of the code. For block codes, the large error coefficients can justify this strategy which is explicitly or implicitly used in many decoding methods such as conventional trellis decoding, multistage decoding, or majority-logic decoding. Therefore, for a particular code and the same optimal block error probability, we determine the best encoding method for delivering as few erroneous information bits as possible when- ever a block is in error at the decoder output. We first derive a general upper bound on the bit-error probability which applies to any generator matrix and is tight at medium to high signal-to- noise ratio (SNR). This bound considers the individual contribution of each information bit separately. For randomly generated codes, we then show that the systematic generator matrix (SGM) pro- vides the minimum bit-error probability. To this end, a submatrix of the generator matrix defining an equivalent code for the bit considered is introduced. A similar general result holds for the optimum bit-error probability related to the BSC [3]. We finally discuss how to achieve this performance whenever the systematic encoding is not the natural choice, as for trellis decoding [5], multistage decoding [6]–[8], or MLD in conjunction with algebraic decoding [9]–[13]. Minimizing the bit-error probability associated with MLD becomes also important whenever the considered block code is used as the inner code of a concatenated coding system [14]. In Section II of this correspondence, general definitions and bounds for the bit-error probability associated with MLD are briefly reviewed. The individual contribution of each information bit to these bounds is then evaluated in Section III. Results for randomly generated codes are derived in Section IV. In Section V, we discuss how to improve the bit-error probability corresponding to a generator matrix not in systematic form, as for trellis decoding or algebraic-based soft- decision decoding. Finally, concluding remarks are given in Section VI. II. REVIEW OF THE BIT-ERROR PROBABILITY FOR MLD Suppose an binary linear code with generator matrix and minimum Hamming distance is used for error control over the AWGN channel. The matrix consists of the rows for . Let be a codeword in . For BPSK transmission, the codeword is mapped into a normalized bipolar sequence with . After transmission, the received sequence at the output of the sampler in the demodulator is with , where for , ’s are statistically independent Gaussian random variables with zero mean and variance . Since is a linear code, we can assume that the all-zero codeword is transmitted. If represents the number of codewords of weight in , the block error probability associated with MLD is upper- 0018–9448/98$10.00  1998 IEEE