On a condition equivalent to the Maximal Distance Separable conjecture

We prove that the following are equivalent. We denote by Pq the vector space of functions from a finite field Fq to itself, which can be represented as the space Pq := Fq[x]/(x q − x) of polynomial functions. We denote by On ⊂ Pq the set of polynomials that are either the zero polynomial, or have at most n distinct roots in Fq. Given two subspaces Y,Z of Pq, we denote by 〈Y,Z〉 their span. • Let k, q integers, with q a prime power and 2 ≤ k ≤ q. Suppose that either: – q is odd – q is even and k 6∈ {3, q − 1}. Then there do not exist distinct subspaces Y and Z of Pq such that: – dim(〈Y,Z〉) = k – dim(Y ) = dim(Z) = k − 1. – 〈Y,Z〉 ⊂ Ok−1 – Y,Z ⊂ Ok−2 – Y ∩ Z ⊂ Ok−3. • The MDS conjecture is true.