On the undetected error probability of nonlinear binary constant weight codes

The undetected error probability (UEP) of binary (n, 2/spl delta/, m) nonlinear constant weight codes over the binary symmetric channel (BSC) is investigated, where n is the blocklength, m is the weight of codeword and 2/spl delta/ is the minimum distance of the codes. The distance distribution of the (n, 2, m) nonlinear constant weight codes is evaluated. It is proven in this paper that the (5, 2, 2) code, (5, 2, 3) code, (6, 2, 3) code, (7, 2, 4) code, (7, 2, 3) code and (8, 2, 4) code are the only proper error-detecting codes in the (n, 2, m) nonlinear constant weight codes for n/spl ges/5, in the sense that their UEP is increased monotonically with the channel error rate p, of course all these proper codes are m-out-of-n codes. Furthermore, it is conjectured that except for the cases of n/spl les/4/spl delta/, there are no proper error-detecting binary (n, 2/spl delta/, m) nonlinear constant weight codes, for n>8 and /spl deltaspl ges/1.<
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