An Algebraic-Coding Equivalence to the Maximum Distance Separable Conjecture

Abstract

We formulate an Algebraic-Coding Equivalence to the Maximum Distance Separable Conjecture. Specifically, we present novel proofs of the following equivalent statements. Let (q, k) be a fixed pair of integers satisfying q is a prime power and 2 ≤ k ≤ q. We denote by P q the vector space of functions from a finite field F q to itself, which can be represented as the space P q := F q [x]/(x q − x) of polynomial functions. We denote by O n ⊂ P q the set of polynomials that are either the zero polynomial, or have at most n distinct roots in F q . Given two subspaces Y, Z of P q , we denote by Y, Z their span. We prove that the following are equivalent.A Suppose that either:(a) q is odd (b) q is even and k ∈ {3, q − 1}.Then there do not exist distinct subspaces Y and Z of P q such that:B Suppose q is odd, or, if q is even, k ∈ {3, q − 1}. There is no integer s with q ≥ s > k such that the Reed-Solomon code R over F q of dimension s can have s − k + 2 columns B = {b 1 , . . . , b s−k+2 } added to it, such that:(a) Any s × s submatrix of R ∪ B containing the first s − k columns of B is independent. (b) B ∪ {[0, 0, . . . , 0, 1]} is independent.C The MDS conjecture is true for the given (q, k).