On a condition equivalent to the Maximum Distance Separable conjecture

Abstract

We denote by $\mathcal{P}_q$ the vector space of functions from a finite field $\mathbb{F}_q$ to itself, which can be represented as the space $\mathcal{P}_q := \mathbb{F}_q[x]/(x^q-x)$ of polynomial functions. We denote by $\mathcal{O}_n \subset \mathcal{P}_q$ the set of polynomials that are either the zero polynomial, or have at most $n$ distinct roots in $\mathbb{F}_q$. Given two subspaces $Y,Z$ of $\mathcal{P}_q$, we denote by $\langle Y,Z \rangle$ their span. We prove that the following are equivalent. A) Let $k, q$ integers, with $q$ a prime power and $2 \leq k \leq q$. Suppose that either: 1) $q$ is odd 2) $q$ is even and $k \not\in \{3, q-1\}$. Then there do not exist distinct subspaces $Y$ and $Z$ of $\mathcal{P}_q$ such that: 1') $dim(\langle Y, Z \rangle) = k$ 2') $dim(Y) = dim(Z) = k-1$. 3') $\langle Y, Z \rangle \subset \mathcal{O}_{k-1}$ 4') $Y, Z \subset \mathcal{O}_{k-2}$ 5') $Y\cap Z \subset \mathcal{O}_{k-3}$. B) The MDS conjecture is true for the given $(q,k)$.