A sufficient condition is given for meanrth-power error in quantizing an unbounded random variable to vary inversely with therth power of the number of quantizing levels. An example is given of a distribution for which this behavior is unattainable.

Estimates are given for the least common denominator of rational-coefficient realizations of certain classes of multithreshold logic circuits. The result is useful in computer simulation of these functions.

It is shown, under weak assumptions on the density function of a random variable and under weak assumptions on the error criterion, that uniform quantizing yields an output entropy which asymptotically is smaller than that for any other quantizer, independent of the density function or the error criterion. The asymptotic behavior of the rate distortion function is determined for the class of\nuth law loss functions, and the entropy of the uniform quantizer is compared with the rate distortion function for this class of loss functions. The extension of these results to the quantizing of sequences is also given. It is shown that the discrepancy between the entropy of the uniform quantizer and the rate distortion function apparently lies with the inability of the optimal quantizing shapes to cover large dimensional spaces without overlap. A comparison of the entropies of the uniform quantizer and of the minimum-alphabet quantizer is also given.

The distribution of the ratio of minimum distance to code length of a random linear code approaches a step distribution as the code length becomes arbitrarily large at fixed code rate. The location of the step is at the smaller value ofpsatisfying1 + p \log_{2}p + (1 - p) \log_{2} (1 - p) = k/n.