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.