The MDS conjecture asserts that the maximum length of a linear code of dimension 
over the finite field 
attaining the Singleton bound
is 
for 
, except that it is 
when 
is even and 
or 
. Such a code is a maximum
distance separable code (Fan 2026).
For 
with 
an odd prime, define
 |
where 
is the floor function. Fan (2026) proved the conjecture
throughout the ranges 
and 
.
The conjecture remains open outside known ranges.
Fan (2026) reports that GPT-5.6 Sol and GPT-6 Astra assisted with exploratory proof development and computations and accepts responsibility for the results.
Range of Dimensions for the MDS Conjecture over 
in Odd Characteristic." 22 Sep 2026.