825 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
New Korkin-Zoloratev inequalities : implementation and numerical data
This technical report discusses the mathematical details and the implementation of the methods discussed in the accompanying paper [PZ06]. In particular a method to find a finite list of inequalities that certify Korkin–Zolotarev reducedness of a quadratic form is presented. Moreover a semidefinite programming relaxation of the space of KZ-reduced quadratic forms is described in detail, together with a branching strategy to optimize over this space. Finally the implementation of these methods is discussed, together with some hints on how to compile and use the programs. The two digital appendices, which can be obtained from the SPOR reports website†, contain an implementation of the methods discussed and numerical data that prove the theorems in [PZ06]
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
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
New Korkin-Zolotarev inequalities
Korkin and Zolotarev showed that if \sum_i A_i\Big(x_i-\sum_{j>i} \alpha_{ij}x_j\Big)^2 is the Lagrange expansion of a Korkin–Zolotarev (KZ-) reduced positive definite quadratic form, then and . They showed that the implied bound is not attained by any KZ-reduced form. We propose a method to optimize numerically over the set of Lagrange expansions of KZ-reduced quadratic forms using a semidefinite relaxation combined with a branch and bound process. We use a rounding technique to derive exact results from the numerical data. Applying these methods, we prove several new linear inequalities on the of any KZ-reduced form, one of them being . We also give a form with . These new inequalities are then used to study the cone of outer coefficients of KZ-reduced forms, to find bounds on Hermite's constant, and to give better estimates on the quality of -block KZ-reduced lattice bases
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