16,572 research outputs found
Super-regular Steiner 2-designs
A design is additive under an abelian group G (briefly, G-additive) if, up to isomorphism, its point set is contained in G and the elements of each block sum up to zero. The only known Steiner 2-designs that are G-additive for some G have block size which is either a prime power or a prime power plus one. Indeed they are the point-line designs of the affine spaces AG(n,q), the point-line designs of the projective planes PG(2,q), the point-line designs of the projective spaces PG(n,2) and a sporadic example of a 2-(8191,7,1) design. In the attempt to find new examples, possibly with a block size which is neither a prime power nor a prime power plus one, we look for Steiner 2-designs which are strictly G-additive (the point set is exactly G) and G-regular (any translate of any block is a block as well) at the same time. These designs will be called “G-super-regular”. Our main result is that there are infinitely many values of v for which there exists a super-regular, and therefore additive, 2-(v,k,1) design whenever k is neither singly even nor of the form 2n3≥12. The case k≡2 (mod 4) is a genuine exception whereas k=2n3≥12 is at the moment a possible exception. We also find super-regular 2-(pn,p,1) designs with p∈{5,7} and n≥3 which are not isomorphic to the point-line design of AG(n,p)
Super-regular Steiner 2-designs
A design is additive under an abelian group (briefly, -additive) if,
up to isomorphism, its point set is contained in and the elements of each
block sum up to zero. The only known Steiner 2-designs that are -additive
for some have block size which is either a prime power or a prime power
plus one. Indeed they are the point-line designs of the affine spaces
, the point-line designs of the projective planes , and the
point-line designs of the projective spaces . In the attempt to find
new examples, possibly with a block size which is neither a prime power nor a
prime power plus one, we look for Steiner 2-designs which are strictly
-additive (the point set is exactly ) and -regular (any translate of
any block is a block as well) at the same time. These designs will be
called\break "-super-regular". Our main result is that there are infinitely
many values of for which there exists a super-regular, and therefore
additive, - design whenever is neither singly even nor of the
form . The case (mod 4) is a definite exception whereas
is at the moment a possible exception. We also find
super-regular - designs with and which are
not isomorphic to the point-line design of .Comment: 31 page
Computational complexity of reconstruction and isomorphism testing for designs and line graphs
Graphs with high symmetry or regularity are the main source for
experimentally hard instances of the notoriously difficult graph isomorphism
problem. In this paper, we study the computational complexity of isomorphism
testing for line graphs of - designs. For this class of
highly regular graphs, we obtain a worst-case running time of for bounded parameters . In a first step, our approach
makes use of the Babai--Luks algorithm to compute canonical forms of
-designs. In a second step, we show that -designs can be reconstructed
from their line graphs in polynomial-time. The first is algebraic in nature,
the second purely combinatorial. For both, profound structural knowledge in
design theory is required. Our results extend earlier complexity results about
isomorphism testing of graphs generated from Steiner triple systems and block
designs.Comment: 12 pages; to appear in: "Journal of Combinatorial Theory, Series A
Entanglement-assisted quantum low-density parity-check codes
This paper develops a general method for constructing entanglement-assisted
quantum low-density parity-check (LDPC) codes, which is based on combinatorial
design theory. Explicit constructions are given for entanglement-assisted
quantum error-correcting codes (EAQECCs) with many desirable properties. These
properties include the requirement of only one initial entanglement bit, high
error correction performance, high rates, and low decoding complexity. The
proposed method produces infinitely many new codes with a wide variety of
parameters and entanglement requirements. Our framework encompasses various
codes including the previously known entanglement-assisted quantum LDPC codes
having the best error correction performance and many new codes with better
block error rates in simulations over the depolarizing channel. We also
determine important parameters of several well-known classes of quantum and
classical LDPC codes for previously unsettled cases.Comment: 20 pages, 5 figures. Final version appearing in Physical Review
Disconnecting strongly regular graphs
In this paper, we show that the minimum number of vertices whose removal
disconnects a connected strongly regular graph into non-singleton components,
equals the size of the neighborhood of an edge for many graphs. These include
blocks graphs of Steiner -designs, many Latin square graphs and strongly
regular graphs whose intersection parameters are at most a quarter of their
valency
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
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