An algebraic-coding equivalence to the maximum distance separable conjecture

In this paper, we provide Algebraic-Coding necessary and sufficient conditions for the Maximum Distance Separable Conjecture to hold. Introduction: The MDS ConjectureLet k be an integer such that 2 ≤ k ≤ q = p r , where p is prime and r is a positive integer.A k × n maximum distance separable (k × n MDS) code M is a k × n matrix with entries in F q such that every set of k columns of M is linearly independent.The Maximum Distance Separable (MDS) conjecture is a well-known problem in coding theory and algebraic geometry with important consequences for example to the study of arcs in finite projective spaces [7] and to coding theory [6,3].The conjecture, first posed by Singleton in 1964 [11] gives a possible upper-bound on the size of a k × n MDS code.More precisely, the MDS conjecture says the following:Conjecture 1.1.The maximum width, n, of a k × n MDS code with entries in F q is q + 1, unless q is even and k ∈ {3, q -1}, in which case the maximum width is q + 2.We remark that there exist k × n MDS codes that attain the maximum possible width as given by the MDS conjecture.These are the Reed-Solomon codes defined in Definition 1.3 (e).See [3].Henceforth all matrices will have entries in F q .When we speak to linear combinations, we mean nontrivial F q -linear combinations.