“It from Bit” and the Quantum Probability Rule

Bruce L. Montgomery, B. V. K. Vijaya Kumar

Decoding rules to minimize the probability of information bit errors in binary linear block codes are investigated. A novel decoding rule is proposed and compared to a recently proposed rule (see M. Elia and G. Prati, 1985) and to the standard rule, with respect to both the information bit error rate and the information word error rate. It is concluded that the proposed rule not only can be implemented much more efficiently that the standard rule, but also is superior with respect to either error rate. Some particular codes have been presented to show that the authors' rule can coincide or can differ from Elia and Prati's rule when applied to a given code.>

Ali Barzegar, Afshin Shafiee, Mostafa Taqavi

J.T. Coffey

Decoding rules for linear block codes used over binary symmetric channels with crossover probability p when the goal is to minimize the probability that sub-blocks of information bits are in error are considered, with application to the (24,12,8) Golay code. (The optimal rules to minimize the probability of individual bits being in error are known for all p.) The results require detailed knowledge of the weight enumerators of given sub codes, which can be calculated from knowledge of the orbit and coset structure of the code. It is interesting that in all cases examined, the rule in decoding a block given the coset received is to select the symbol that is optimal as p → 0 or the symbol that is optimal as p → 1/2, depending on p.

Makoto Ohya

M. Ohya

Aaron B. Kiely, J.T. Coffey, Mark R. Bell

We consider the problem of decoding a linear block code used over a binary symmetric channel with crossover probability p when the goal of the decoding is to minimize the probability of an information bit error. In general, there will be several different decoding strategies, each corresponding to the optimal rule over some range p/spl isin/[p/sub i-1/, p/sub i/]. Each of these strategies can be implemented by standard array with the appropriate choice of coset leaders that are not in general of minimum weight. We give optimal rules for selecting coset leaders in the first region [0, p/sub 1/] and the region [p/sub max/, 1/2] for any linear mapping between information bits and codewords. The determination of the optimal rule has been an open problem for small p. The rule for p>p/sub max/ is important because for many codes, when p is fixed the optimal rule for any coset corresponds to either the optimal rule for small p, or the optimal rule for p near 1/2. The optimal decoding strategy varies with the generator matrix used in general, but it is demonstrated that the optimal generator matrix is separable when d/sup /spl perp//>2. The results are shown to extend in a natural way to the case where the goal is to minimize the probability that arbitrary collections of information bits are in error. We examine some specific codes, and compare the performance under optimal decoding to the performance of previously suggested strategies.>