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An Algebraic-Coding Equivalence to the Maximum Distance Separable Conjecture
We formulate an Algebraic-Coding Equivalence to the Maximum Distance
Separable Conjecture. Specifically, we present novel proofs of the following
equivalent statements. Let be a fixed pair of integers satisfying
is a prime power and . We denote by the vector
space of functions from a finite field to itself, which can be
represented as the space of
polynomial functions. We denote by the
set of polynomials that are either the zero polynomial, or have at most
distinct roots in . Given two subspaces of ,
we denote by their span. We prove that the following are
equivalent.
[A] Suppose that either: 1. is odd 2. is even and .
Then there do not exist distinct subspaces and of
such that:
3. 4. . 5. 6. 7.
.
[B] Suppose is odd, or, if is even, . There is
no integer with such that the Reed-Solomon code
over of dimension can have columns
added to it, such that:
8. Any submatrix of containing
the first columns of is independent. 9. is independent.
[C] The MDS conjecture is true for the given .Comment: This is version: 5.6.18. arXiv admin note: substantial text overlap
with arXiv:1611.0235