Design and analysis of codes with distance 4 and 6 minimizing the probability of decoder error

The problem of minimization of the decoder error probability is considered for shortened codes of dimension 2 m with distance 4 and 6. We prove that shortened Panchenko codes with distance 4 achieve the minimal probability of decoder error under special form of shortening. This shows that Hamming codes are not the best. In the paper, the rules for shortening Panchenko codes are defined and a combinatorial method to minimize the number of words of weight 4 and 5 is developed. There are obtained exact lower bounds on the probability of decoder error and the full solution of the problem of minimization of the decoder error probability for [39,32,4] and [72,64,4] codes. For shortened BCH codes with distance 6, upper and lower bounds on the number of minimal weight codewords are derived. There are constructed [45,32,6] and [79,64,6] BCH codes with the number of weight 6 codewords close to the lower bound and the decoder error probabilities are calculated for these codes. The results are intended for use in memory devices.