Singleton Bound

The Singleton bound states that a code containing at least two codewords and having length over an alphabet of size , with minimum Hamming distance , satisfies

Deleting any coordinates maps distinct codewords to distinct words, which proves the bound. For a linear code of dimension over a finite field with elements, it becomes . A code attaining equality is a maximum distance separable code (Singleton 1964, p. 116; Fan 2026).

See also

Code, Coding Theory, Hamming Distance, Linear Code, Maximum Distance Separable Code, MDS Conjecture

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References

Fan, X. "A Linear-in-

Range of Dimensions for the MDS Conjecture over

in Odd Characteristic." 22 Sep 2026. https://arxiv.org/abs/2610.03740.Singleton, R. C. "Maximum Distance

-Nary Codes." IEEE Trans. Inform. Th. 10, 116-118, 1964. https://doi.org/10.1109/TIT.1964.1053661.

Cite this as:

Weisstein, Eric W. "Singleton Bound." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SingletonBound.html

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