399 research outputs found
Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms
Mathematical programming is a branch of applied mathematics and has recently
been used to derive new decoding approaches, challenging established but often
heuristic algorithms based on iterative message passing. Concepts from
mathematical programming used in the context of decoding include linear,
integer, and nonlinear programming, network flows, notions of duality as well
as matroid and polyhedral theory. This survey article reviews and categorizes
decoding methods based on mathematical programming approaches for binary linear
codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory.
Published July 201
The Bergman complex of a matroid and phylogenetic trees
We study the Bergman complex B(M) of a matroid M: a polyhedral complex which
arises in algebraic geometry, but which we describe purely combinatorially. We
prove that a natural subdivision of the Bergman complex of M is a geometric
realization of the order complex of its lattice of flats. In addition, we show
that the Bergman fan B'(K_n) of the graphical matroid of the complete graph K_n
is homeomorphic to the space of phylogenetic trees T_n.Comment: 15 pages, 6 figures. Reorganized paper and updated references. To
appear in J. Combin. Theory Ser.
Schottky Algorithms: Classical meets Tropical
We present a new perspective on the Schottky problem that links numerical
computing with tropical geometry. The task is to decide whether a symmetric
matrix defines a Jacobian, and, if so, to compute the curve and its canonical
embedding. We offer solutions and their implementations in genus four, both
classically and tropically. The locus of cographic matroids arises from
tropicalizing the Schottky-Igusa modular form.Comment: 17 page
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