The code covering radius of an error-correcting code is the greatest Hamming distance of a word to its nearest codeword:

Here is the alphabet and is the Hamming distance. A word attaining this greatest distance is a deep hole. Thus Hamming distance balls of radius about the codewords cover .

For the first-order Reed-Muller code on nine variables, Wang et al. (2026) proved that .

For the third-order Reed-Muller code on eight variables, Zhao and Wu (2026) proved that .

See also

Boolean Function Nonlinearity, Error-Correcting Code, Hamming Distance, Reed-Muller Code

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References

Wang, Q.; Lou, Y.; and Wang, Z. "The Covering Radius of the Reed-Muller Code Is 242." 13 Oct 2026. https://arxiv.org/abs/2610.17871.Zhao, Z. and Wu, Y. "The Covering Radius of the Reed-Muller Code Is 56." 2 Nov 2026. https://arxiv.org/abs/2611.03186v1.

Cite this as:

Weisstein, Eric W. "Code Covering Radius." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CodeCoveringRadius.html

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