173 research outputs found
Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets
H. Cohn et. al. proposed an association scheme of 64 points in R^{14} which
is conjectured to be a universally optimal code. We show that this scheme has a
generalization in terms of Kerdock codes, as well as in terms of maximal real
mutually unbiased bases. These schemes also related to extremal line-sets in
Euclidean spaces and Barnes-Wall lattices. D. de Caen and E. R. van Dam
constructed two infinite series of formally dual 3-class association schemes.
We explain this formal duality by constructing two dual abelian schemes related
to quaternary linear Kerdock and Preparata codes.Comment: 16 page
Efficient Two-Stage Group Testing Algorithms for Genetic Screening
Efficient two-stage group testing algorithms that are particularly suited for
rapid and less-expensive DNA library screening and other large scale biological
group testing efforts are investigated in this paper. The main focus is on
novel combinatorial constructions in order to minimize the number of individual
tests at the second stage of a two-stage disjunctive testing procedure.
Building on recent work by Levenshtein (2003) and Tonchev (2008), several new
infinite classes of such combinatorial designs are presented.Comment: 14 pages; to appear in "Algorithmica". Part of this work has been
presented at the ICALP 2011 Group Testing Workshop; arXiv:1106.368
Problems on q-Analogs in Coding Theory
The interest in -analogs of codes and designs has been increased in the
last few years as a consequence of their new application in error-correction
for random network coding. There are many interesting theoretical, algebraic,
and combinatorial coding problems concerning these q-analogs which remained
unsolved. The first goal of this paper is to make a short summary of the large
amount of research which was done in the area mainly in the last few years and
to provide most of the relevant references. The second goal of this paper is to
present one hundred open questions and problems for future research, whose
solution will advance the knowledge in this area. The third goal of this paper
is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author
New characterisations of the Nordstrom–Robinson codes
In his doctoral thesis, Snover proved that any binary code
is equivalent to the Nordstrom-Robinson code or the punctured
Nordstrom-Robinson code for or respectively. We
prove that these codes are also characterised as \emph{completely regular}
binary codes with or , and moreover, that they are
\emph{completely transitive}. Also, it is known that completely transitive
codes are necessarily completely regular, but whether the converse holds has up
to now been an open question. We answer this by proving that certain completely
regular codes are not completely transitive, namely, the (Punctured) Preparata
codes other than the (Punctured) Nordstrom-Robinson code
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Z2Z4-additive cyclic codes, generator polynomials and dual codes
A -additive code is called cyclic if
the set of coordinates can be partitioned into two subsets, the set of
and the set of coordinates, such that any
cyclic shift of the coordinates of both subsets leaves the code invariant.
These codes can be identified as submodules of the -module
. The parameters
of a -additive cyclic code are stated in terms of
the degrees of the generator polynomials of the code. The generator polynomials
of the dual code of a -additive cyclic code are
determined in terms of the generator polynomials of the code
- …