Code Covering Radius

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. 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 . Equivalently, every Boolean function on nine variables has Boolean function nonlinearity at most 242, and this value is attained.

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.

Cite this as:

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

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