25,343 research outputs found
Lattice polytopes in coding theory
In this paper we discuss combinatorial questions about lattice polytopes
motivated by recent results on minimum distance estimation for toric codes. We
also prove a new inductive bound for the minimum distance of generalized toric
codes. As an application, we give new formulas for the minimum distance of
generalized toric codes for special lattice point configurations.Comment: 11 pages, 3 figure
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
Remarks on Bounds for Quantum Codes
We present some results that show that bounds from classical coding theory
still work in many cases of quantum coding theory
Kneser-Hecke-operators in coding theory
The Kneser-Hecke-operator is a linear operator defined on the complex vector
space spanned by the equivalence classes of a family of self-dual codes of
fixed length. It maps a linear self-dual code over a finite field to the
formal sum of the equivalence classes of those self-dual codes that intersect
in a codimension 1 subspace. The eigenspaces of this self-adjoint linear
operator may be described in terms of a coding-theory analogue of the Siegel
-operator
Applications of finite geometry in coding theory and cryptography
We present in this article the basic properties of projective geometry, coding theory, and cryptography, and show how
finite geometry can contribute to coding theory and cryptography. In this way, we show links between three research areas, and in particular, show that finite geometry is not only interesting from a pure mathematical point of view, but also of interest for applications. We concentrate on introducing the basic concepts of these three research areas and give standard references for all these three research areas. We also mention particular results involving ideas from finite geometry, and particular results in cryptography involving ideas from coding theory
Principles and Parameters: a coding theory perspective
We propose an approach to Longobardi's parametric comparison method (PCM) via
the theory of error-correcting codes. One associates to a collection of
languages to be analyzed with the PCM a binary (or ternary) code with one code
words for each language in the family and each word consisting of the binary
values of the syntactic parameters of the language, with the ternary case
allowing for an additional parameter state that takes into account phenomena of
entailment of parameters. The code parameters of the resulting code can be
compared with some classical bounds in coding theory: the asymptotic bound, the
Gilbert-Varshamov bound, etc. The position of the code parameters with respect
to some of these bounds provides quantitative information on the variability of
syntactic parameters within and across historical-linguistic families. While
computations carried out for languages belonging to the same family yield codes
below the GV curve, comparisons across different historical families can give
examples of isolated codes lying above the asymptotic bound.Comment: 11 pages, LaTe
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