9 research outputs found
Directed Graph Representation of Half-Rate Additive Codes over GF(4)
We show that (n,2^n) additive codes over GF(4) can be represented as directed
graphs. This generalizes earlier results on self-dual additive codes over
GF(4), which correspond to undirected graphs. Graph representation reduces the
complexity of code classification, and enables us to classify additive (n,2^n)
codes over GF(4) of length up to 7. From this we also derive classifications of
isodual and formally self-dual codes. We introduce new constructions of
circulant and bordered circulant directed graph codes, and show that these
codes will always be isodual. A computer search of all such codes of length up
to 26 reveals that these constructions produce many codes of high minimum
distance. In particular, we find new near-extremal formally self-dual codes of
length 11 and 13, and isodual codes of length 24, 25, and 26 with better
minimum distance than the best known self-dual codes.Comment: Presented at International Workshop on Coding and Cryptography (WCC
2009), 10-15 May 2009, Ullensvang, Norway. (14 pages, 2 figures
Classification of real Bott manifolds and acyclic digraphs
We completely characterize real Bott manifolds up to affine diffeomorphism in
terms of three simple matrix operations on square binary matrices obtained from
strictly upper triangular matrices by permuting rows and columns
simultaneously. We also prove that any graded ring isomorphism between the
cohomology rings of real Bott manifolds with coefficients is
induced by an affine diffeomorphism between the real Bott manifolds.
Our characterization can also be described in terms of graph operations on
directed acyclic graphs. Using this combinatorial interpretation, we prove that
the decomposition of a real Bott manifold into a product of indecomposable real
Bott manifolds is unique up to permutations of the indecomposable factors.
Finally, we produce some numerical invariants of real Bott manifolds from the
viewpoint of graph theory and discuss their topological meaning. As a
by-product, we prove that the toral rank conjecture holds for real Bott
manifolds.Comment: 27 pages, 5 figures. It is a combination of arXiv:0809.2178 and
arXiv:1002.4704, including some new result