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 $⟨Y,Z⟩$ their span. We prove that the following are equivalent. A) Let $k,q$ integers, with $q$ a prime power and $2\le k\le q$. Suppose that either: 1) $q$ is odd 2) $q$ is even and $k\notin \{3,q-1\}$. Then there do not exist distinct subspaces $Y$ and $Z$ of ${\mathcal{P}}_q$ such that: 1') $dim(⟨Y,Z⟩)=k$ 2') $dim(Y)=dim(Z)=k-1$. 3') $⟨Y,Z⟩\subset {\mathcal{O}}_{k-1}$ 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 $(q,k)$.

arXiv commentThe paper 1705.06136 is the correct version of this paper. This paper is included in 1705.06136