Improved asymptotic bounds for error-correcting codes
Selmer Martin Johnson · IEEE Transactions on Information Theory · 1963
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.