Cycle Bases for Lattices of Binary Matroids with No Fano Dual Minor and Their One-Element Extensions

@article{Fleiner1999CycleBF,
  title={Cycle Bases for Lattices of Binary Matroids with No Fano Dual Minor and Their One-Element Extensions},
  author={Tam{\'a}s Fleiner and Winfried Hochst{\"a}ttler and Monique Laurent and Martin Loebl},
  journal={J. Comb. Theory, Ser. B},
  year={1999},
  volume={77},
  pages={25-38}
}
In this paper we study the question of existence of a basis consisting only of cycles for the lattice Z(M) generated by the cycles of a binary matroid M. We show that if M has no Fano dual minor, then any set of fundamental circuits can be completed to a cycle basis of Z(M); moreover, for any one-element extension M? of such a matroid M, any cycle basis for Z(M) can be completed to a cycle basis for Z(M?). 
Bases of cocycle lattices and submatrices of a Hadamard matrix
TLDR
It is shown that the cocycle lattice of a recursively deened class of matroids, including all binaryMatroids of rank four, always has a basis consisting of cocycles.

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