36 research outputs found
On perturbations of highly connected dyadic matroids
Geelen, Gerards, and Whittle [3] announced the following result: let be a prime power, and let be a proper minor-closed class of
-representable matroids, which does not contain
for sufficiently high . There exist integers
such that every vertically -connected matroid in is a
rank- perturbation of a frame matroid or the dual of a frame matroid
over . They further announced a characterization of the
perturbations through the introduction of subfield templates and frame
templates.
We show a family of dyadic matroids that form a counterexample to this
result. We offer several weaker conjectures to replace the ones in [3], discuss
consequences for some published papers, and discuss the impact of these new
conjectures on the structure of frame templates.Comment: Version 3 has a new title and a few other minor corrections; 38
pages, including a 6-page Jupyter notebook that contains SageMath code and
that is also available in the ancillary file
The highly connected even-cycle and even-cut matroids
The classes of even-cycle matroids, even-cycle matroids with a blocking pair,
and even-cut matroids each have hundreds of excluded minors. We show that the
number of excluded minors for these classes can be drastically reduced if we
consider in each class only the highly connected matroids of sufficient size.Comment: Version 2 is a major revision, including a correction of an error in
the statement of one of the main results and improved exposition. It is 89
pages, including a 33-page Jupyter notebook that contains SageMath code and
that is also available in the ancillary file
On the existence of asymptotically good linear codes in minor-closed classes
Let be a sequence of codes such that each
is a linear -code over some fixed finite field
, where is the length of the codewords, is the
dimension, and is the minimum distance. We say that is
asymptotically good if, for some and for all , , , and . Sequences of
asymptotically good codes exist. We prove that if is a class of
GF-linear codes (where is prime and ), closed under
puncturing and shortening, and if contains an asymptotically good
sequence, then must contain all GF-linear codes. Our proof
relies on a powerful new result from matroid structure theory
An obstacle to a decomposition theorem for near-regular matroids
Seymour's Decomposition Theorem for regular matroids states that any matroid
representable over both GF(2) and GF(3) can be obtained from matroids that are
graphic, cographic, or isomorphic to R10 by 1-, 2-, and 3-sums. It is hoped
that similar characterizations hold for other classes of matroids, notably for
the class of near-regular matroids. Suppose that all near-regular matroids can
be obtained from matroids that belong to a few basic classes through k-sums.
Also suppose that these basic classes are such that, whenever a class contains
all graphic matroids, it does not contain all cographic matroids. We show that
in that case 3-sums will not suffice.Comment: 11 pages, 1 figur
Matroid 3-connectivity and branch width
We prove that, for each nonnegative integer k and each matroid N, if M is a
3-connected matroid containing N as a minor, and the the branch width of M is
sufficiently large, then there is a k-element subset X of E(M) such that one of
M\X and M/X is 3-connected and contains N as a minor.Comment: 21 page
Fan-extensions in fragile matroids
If S is a set of matroids, then the matroid M is S-fragile if, for every
element e in E(M), either M\e or M/e has no minor isomorphic to a member of S.
Excluded-minor characterizations often depend, implicitly or explicitly, on
understanding classes of fragile matroids. In certain cases, when F is a
minor-closed class of S-fragile matroids, and N is in F, the only members of F
that contain N as a minor are obtained from N by increasing the length of fans.
We prove that if this is the case, then we can certify it with a finite
case-analysis. The analysis involves examining matroids that are at most two
elements larger than N.Comment: Small revisions and correction
On the Highly Connected Dyadic, Near-Regular, and Sixth-Root-of-Unity Matroids
Subject to announced results by Geelen, Gerards, and Whittle, we completely
characterize the highly connected members of the classes of dyadic,
near-regular, and sixth-root-of-unity matroids.Comment: 23 pages, SageMath worksheet in ancillary files. arXiv admin note:
text overlap with arXiv:1902.0713