16,231 research outputs found
Coding Theory and Algebraic Combinatorics
This chapter introduces and elaborates on the fruitful interplay of coding
theory and algebraic combinatorics, with most of the focus on the interaction
of codes with combinatorial designs, finite geometries, simple groups, sphere
packings, kissing numbers, lattices, and association schemes. In particular,
special interest is devoted to the relationship between codes and combinatorial
designs. We describe and recapitulate important results in the development of
the state of the art. In addition, we give illustrative examples and
constructions, and highlight recent advances. Finally, we provide a collection
of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in
Information and Coding Theory", ed. by I. Woungang et al., World Scientific,
Singapore, 201
Quasi-Linear Cellular Automata
Simulating a cellular automaton (CA) for t time-steps into the future
requires t^2 serial computation steps or t parallel ones. However, certain CAs
based on an Abelian group, such as addition mod 2, are termed ``linear''
because they obey a principle of superposition. This allows them to be
predicted efficiently, in serial time O(t) or O(log t) in parallel.
In this paper, we generalize this by looking at CAs with a variety of
algebraic structures, including quasigroups, non-Abelian groups, Steiner
systems, and others. We show that in many cases, an efficient algorithm exists
even though these CAs are not linear in the previous sense; we term them
``quasilinear.'' We find examples which can be predicted in serial time
proportional to t, t log t, t log^2 t, and t^a for a < 2, and parallel time log
t, log t log log t and log^2 t.
We also discuss what algebraic properties are required or implied by the
existence of scaling relations and principles of superposition, and exhibit
several novel ``vector-valued'' CAs.Comment: 41 pages with figures, To appear in Physica
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