6,408 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
Counting Steiner triple systems with classical parameters and prescribed rank
By a famous result of Doyen, Hubaut and Vandensavel \cite{DHV}, the 2-rank of
a Steiner triple system on points is at least , and equality
holds only for the classical point-line design in the projective geometry
. It follows from results of Assmus \cite{A} that, given any integer
with , there is a code containing
representatives of all isomorphism classes of STS with 2-rank at most
. Using a mixture of coding theoretic, geometric, design
theoretic and combinatorial arguments, we prove a general formula for the
number of distinct STS with 2-rank at most contained
in this code. This generalizes the only previously known cases, , proved
by Tonchev \cite{T01} in 2001, , proved by V. Zinoviev and D. Zinoviev
\cite{ZZ12} in 2012, and (V. Zinoviev and D. Zinoviev \cite{ZZ13},
\cite{ZZ13a} (2013), D. Zinoviev \cite{Z16} (2016)), while also unifying and
simplifying the proofs. This enumeration result allows us to prove lower and
upper bounds for the number of isomorphism classes of STS with 2-rank
exactly (or at most) . Finally, using our recent systematic
study of the ternary block codes of Steiner triple systems \cite{JT}, we obtain
analogous results for the ternary case, that is, for STS with 3-rank at
most (or exactly) . We note that this work provides the first
two infinite families of 2-designs for which one has non-trivial lower and
upper bounds for the number of non-isomorphic examples with a prescribed
-rank in almost the entire range of possible ranks.Comment: 27 page
Self-Dual Codes
Self-dual codes are important because many of the best codes known are of
this type and they have a rich mathematical theory. Topics covered in this
survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight
enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems,
bounds, mass formulae, enumeration, extremal codes, open problems. There is a
comprehensive bibliography.Comment: 136 page
Breakup Conditions of Projectile Spectators from Dynamical Observables
Momenta and masses of heavy projectile fragments (Z >= 8), produced in
collisions of 197Au with C, Al, Cu and Pb targets at E/A = 600 MeV, were
determined with the ALADIN magnetic spectrometer at SIS. An analysis of
kinematic correlations between the two and three heaviest projectile fragments
in their rest frame was performed. The sensitivity of these correlations to the
conditions at breakup was verified within the schematic SOS-model. The data
were compared to calculations with statistical multifragmentation models and to
classical three-body calculations. Classical trajectory calculations reproduce
the dynamical observables. The deduced breakup parameters, however, differ
considerably from those assumed in the statistical multifragmentation models
which describe the charge correlations. If, on the other hand, the analysis of
kinematic and charge correlations is performed for events with two and three
heavy fragments produced by statistical multifragmentation codes, a good
agreement with the data is found with the exception that the fluctuation widths
of the intrinsic fragment energies are significantly underestimated. A new
version of the multifragmentation code MCFRAG was therefore used to investigate
the potential role of angular momentum at the breakup stage. If a mean angular
momentum of 0.75/nucleon is added to the system, the energy fluctuations
can be reproduced, but at the same time the charge partitions are modified and
deviate from the data.
PACS numbers: 25.70.Mn, 25.70.Pq, 25.75.Ld, 25.75.-qComment: 38 pages, RevTeX with 21 included figures; Also available from
http://www-kp3.gsi.de/www/kp3/aladin_publications.htm
Syntactic Structures and Code Parameters
We assign binary and ternary error-correcting codes to the data of syntactic
structures of world languages and we study the distribution of code points in
the space of code parameters. We show that, while most codes populate the lower
region approximating a superposition of Thomae functions, there is a
substantial presence of codes above the Gilbert-Varshamov bound and even above
the asymptotic bound and the Plotkin bound. We investigate the dynamics induced
on the space of code parameters by spin glass models of language change, and
show that, in the presence of entailment relations between syntactic parameters
the dynamics can sometimes improve the code. For large sets of languages and
syntactic data, one can gain information on the spin glass dynamics from the
induced dynamics in the space of code parameters.Comment: 14 pages, LaTeX, 12 png figure
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