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Branch-Width and Well-Quasi-Ordering in Matroids and Graphs
We prove that a class of matroids representable over a fixed finite field and with bounded branch-width is well-quasi-ordered under taking minors. With some extra work, the result implies Robertson
Partial Fields and Matroid Representation
A partial fieldPis an algebraic structure that behaves very much like a field except that addition is a partial binary operation, that is, for somea,b?P,a+bmay not be defined. We develop a theory of
Stabilizers of Classes of Representable Matroids
  • G. Whittle
  • Mathematics
    J. Comb. Theory, Ser. B
  • 1 September 1999
One of the main theorems of this paper proves that if M is minor-closed and closed under duals, and N is 3- connected, then to show that N is a stabilizer it suffices to check 3-connected matroids in M that are single-element extensions or coextensions of N, or are obtained by a single- element extension followed by asingle-element coextension.
The Highly Connected Matroids in Minor-Closed Classes
For any minor-closed class of matroids over a fixed finite field, we state an exact structural characterization for the sufficiently connected matroids in the class. We also state a number of
The Parametrized Complexity of Some Fundamental Problems in Coding Theory
It is proved here for the first time that THETA SERIES is NP-complete and it is shown that the NEAREST VECTOR problem for integer lattices is hard for W[1], which is the counterparts of WEIGHT DISTRUBUTION and MAXIMUM-LIKELIHOOD DECODING for lattices.
Totally Free Expansions of Matroids
It is proved that, within a class of matroids that is closed under minors and duality, the totally free matroIDS can be found by an inductive search and it is employed to show that, for all r?4, there are unique and easily described rank-r quaternary and quinternaryMatroids, the first being the free spike.
On Inequivalent Representations of Matroids over Finite Fields
This paper proves the conjecture that, for each prime powerq, there is an integern(q) such that 6 is a sharp value forn(5), and shows that the conjecture is false for all larger values ofq.