On a condition equivalent to the Maximum Distance Separable conjecture

TL;DR

This paper establishes an equivalence between a specific polynomial subspace configuration condition over finite fields and the truth of the Maximum Distance Separable (MDS) conjecture for certain parameters, linking algebraic geometry and coding theory.

Contribution

It proves that a particular geometric condition on polynomial subspaces is equivalent to the MDS conjecture holding for given parameters, providing a new perspective on this longstanding problem.

Findings

The polynomial subspace condition is impossible under certain field and dimension constraints.

The equivalence offers a new approach to verify the MDS conjecture for specific cases.

The results connect algebraic properties of polynomial functions with coding theory conjectures.

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 $(q,k)$.