19 research outputs found
An Analytic and Probabilistic Approach to the Problem of Matroid Representibility
We introduce various quantities that can be defined for an arbitrary matroid,
and show that certain conditions on these quantities imply that a matroid is
not representable over . Mostly, for a matroid of rank , we
examine the proportion of size- subsets that are dependent, and give
bounds, in terms of the cardinality of the matroid and a prime power, for
this proportion, below which the matroid is not representable over
. We also explore connections between the defined quantities and
demonstrate that they can be used to prove that random matrices have high
proportions of subsets of columns independent
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
Counting generalized Reed-Solomon codes
In this article we count the number of generalized Reed-Solomon (GRS) codes
of dimension k and length n, including the codes coming from a non-degenerate
conic plus nucleus. We compare our results with known formulae for the number
of 3-dimensional MDS codes of length n=6,7,8,9
A note on full weight spectrum codes
A linear code is said to be a full weight spectrum (FWS)
code if there exist codewords of each nonzero weight less than or equal to . In this brief communication we determine necessary and sufficient conditions
for the existence of linear full weight spectrum (FWS) codes.
Central to our approach is the geometric view of linear codes, whereby columns
of a generator matrix correspond to points in
Extending small arcs to large arcs
This is a post-peer-review, pre-copyedit version of an article published in European Journal of Mathematics. The final authenticated version is available online at: https://doi.org/10.1007/s40879-017-0193-xAn arc is a set of vectors of the k-dimensional vector space over the finite field with q elements Fq , in which every subset of size k is a basis of the space, i.e. every k-subset is a set of linearly independent vectors. Given an arc G in a space of odd characteristic, we prove that there is an upper bound on the largest arc containing G. The bound is not an explicit bound but is obtained by computing properties of a matrix constructed from G. In some cases we can also determine the largest arc containing G, or at least determine the hyperplanes which contain exactly k-2 vectors of the large arc. The theorems contained in this article may provide new tools in the computational classification and construction of large arcs.Postprint (author's final draft