91 research outputs found
Orbits of Antichains in Ranked Posets
AbstractWe consider the permutation f of antichains of a ranked poset P , moving the set of lower units of any monotone boolean function on P to the set of its upper zeros. A duality relation on orbits of this permutation is found, which is used for proving a conjecture by M. Deza and K. Fukuda. For P a direct product of two chains, possible lengths of orbits are completely determined
Codes, Graphs and Schemes from Nonlinear Functions
AMS classifications: 05E30; 05B20; 94B0
Codes, graphs and schemes from nonlinear functions
AbstractWe consider functions on binary vector spaces which are far from linear functions in different senses. We compare three existing notions: almost perfect nonlinear functions, almost bent (AB) functions, and crooked (CR) functions. Such functions are of importance in cryptography because of their resistance to linear and differential attacks on certain cryptosystems. We give a new combinatorial characterization of AB functions in terms of the number of solutions to a certain system of equations, and a characterization of CR functions in terms of the Fourier transform. We also show how these functions can be used to construct several combinatorial structures; such as semi-biplanes, difference sets, distance regular graphs, symmetric association schemes, and uniformly packed (BCH and Preparata) codes
On diameter perfect constant-weight ternary codes
From cosets of binary Hamming codes we construct diameter perfect
constant-weight ternary codes with weight (where is the code length)
and distances 3 and 5. The class of distance 5 codes has parameters unknown
before. Keywords: constant-weight codes, ternary codes, perfect codes, diameter
perfect codes, perfect matchings, Preparata codesComment: 15 pages, 2 figures; presented at 2004 Com2MaC Conference on
Association Schemes, Codes and Designs; submitted to Discrete Mathematic
Promotion and Rowmotion
We present an equivariant bijection between two actions--promotion and
rowmotion--on order ideals in certain posets. This bijection simultaneously
generalizes a result of R. Stanley concerning promotion on the linear
extensions of two disjoint chains and recent work of D. Armstrong, C. Stump,
and H. Thomas on root posets and noncrossing partitions. We apply this
bijection to several classes of posets, obtaining equivariant bijections to
various known objects under rotation. We extend the same idea to give an
equivariant bijection between alternating sign matrices under rowmotion and
under B. Wieland's gyration. Finally, we define two actions with related orders
on alternating sign matrices and totally symmetric self-complementary plane
partitions.Comment: 25 pages, 22 figures; final versio
Coloring translates and homothets of a convex body
We obtain improved upper bounds and new lower bounds on the chromatic number
as a linear function of the clique number, for the intersection graphs (and
their complements) of finite families of translates and homothets of a convex
body in \RR^n.Comment: 11 pages, 2 figure
Permutation complexity of the fixed points of some uniform binary morphisms
An infinite permutation is a linear order on the set N. We study the
properties of infinite permutations generated by fixed points of some uniform
binary morphisms, and find the formula for their complexity.Comment: In Proceedings WORDS 2011, arXiv:1108.341
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