In this paper improved upper bounds are obtained forA(n,d), the maximum number of binary code vectors in a code of block lengthnand minimum distanced. Known bounds are presented in a unified way and then refined, giving improvement over best previously published results in almost all cases. Finally, tabulations of the improved results are given. Asymptotically, the new bounds agree with those given by Elias.

The author has previously developed a new upper bound on nonsystematic binary error-correcting codes, using a sphere-packing approach and combinatorial analysis. A significant refinement is now added; together with a detailed study of the asymptotic behavior of the upper bound, this enables one to show that any large code must {\em correct} almost all sequences with a larger number of errors than the code was designed for. This excess is expressed numerically as a fraction of the designed error-correcting capability of the code. The fraction is a function of the ratio of the sequence length and the designed error-correcting capability. A possible application might be in the use of a larger code giving almost certain error correction rather than a smaller one with certain correction capability.

By refining Hamming's geometric sphere-packing model a new upper bound for nonsystematic binary error-correcting codes is found. Only combinatorial arguments are used. Whereas Hamming's upper bound estimate fore-error-correcting codes involved a count of all points\leq eHamming distance from the set of code points, the model is extended here to include consideration of points which are>edistance away from the code set. The percentage improvement from Hamming's bounds is sometimes quite sizable for cases of two or more errors to be corrected. The new bound improves on Wax's bounds in all but four of the cases he lists.