![]() However, this is not the case with the bounds on maximum likelihood hard decision decoding of codes. Using these bounds, we see that as the SNR per bit becomes large the performance of the codes can be completely determined by the lower bound. The union bound can be expressed as both an upper and lower bound. The union bound is a simple and well-known bound for the performance of codes under maximum likelihood soft decision decoding. For maximum likelihood soft decision decoding, the bounds are computed using the weight distribution of the codes. For hard decision decoding, lower and upper bounds on maximum likelihood decoding are computed using information on the coset weight leader distribution. These bounds are in lower and upper form and the expected performance of the code is within the region bounded by the two. In practice, maximum likelihood decoding of codes is computationally difficult, and as such, theoretical bounds on the performance of codes are used instead. Maximum likelihood decoding gives the best performance possible for a code and is therefore used to assess the quality of the code. This chapter is concerned with the performance of binary codes under maximum likelihood soft decision decoding and maximum likelihood hard decision decoding. It is shown that the binomial weight distribution provides a good indicative performance for codes whose weight distribution is difficult to obtain. Numerical performance comparisons are made for a wide range of different codes. The implications of this observation is that the soft decision decoding performance may be determined for a linear code by the number of minimum Hamming weight codewords without the need to determine the weight distribution of the code. An analysis of the upper and lower union bounds on the probability of error for maximum likelihood soft decision decoding shows that contrary to hard decision decoding above relative low values of SNR per information bit, the two bounds coincide. For hard decision decoding, evaluation of the performance of specific codes shows that full decoding produces better performance than the usual bounded distance decoder. Upper and lower bounds on hard and soft decision decoding are discussed. In this chapter, we discuss the performance of codes under soft and hard decision decoding.
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