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MDS Conjecture

The MDS conjecture asserts that the maximum length of a linear code of dimension k over the finite field F_q attaining the Singleton bound is q+1 for 2<=k<=q, except that it is q+2 when q is even and k=3 or k=q-1. Such a code is a maximum distance separable code (Fan 2026).

For q=p^h with p an odd prime, define

B(p,q)=|_((p-2)q+6p-10)/(2p-3)_|,

where |_x_| is the floor function. Fan (2026) proved the conjecture throughout the ranges 2<=k<=B(p,q) and q+2-B(p,q)<=k<=q. The odd-characteristic conjecture remains open outside known ranges.

Chen and Huang (2026) reported a proof of the full linear conjecture in even characteristic. For even prime power q, a nontrivial linear [n,k] maximum distance separable code over F_q has n<=q+1, except that n<=q+2 when k=3 or k=q-1. Here nontrivial means 2<=k<=n-2. For q>=4, the bounds are attained throughout the admissible dimensions 2<=k<=q-1.

See also

Finite Field, Linear Code, Maximum Distance Separable Code, Singleton Bound

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References

Chen, B. and Huang, J. "On the MDS Conjecture in Even Characteristic." 20 Oct 2026. https://arxiv.org/abs/2610.28208.Fan, X. "A Linear-in-q Range of Dimensions for the MDS Conjecture over F_q in Odd Characteristic." 22 Sep 2026. https://arxiv.org/abs/2610.03740.

Cite this as:

Weisstein, Eric W. "MDS Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MDSConjecture.html

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