16,816 research outputs found
Quarter-fraction factorial designs constructed via quaternary codes
The research of developing a general methodology for the construction of good
nonregular designs has been very active in the last decade. Recent research by
Xu and Wong [Statist. Sinica 17 (2007) 1191--1213] suggested a new class of
nonregular designs constructed from quaternary codes. This paper explores the
properties and uses of quaternary codes toward the construction of
quarter-fraction nonregular designs. Some theoretical results are obtained
regarding the aliasing structure of such designs. Optimal designs are
constructed under the maximum resolution, minimum aberration and maximum
projectivity criteria. These designs often have larger generalized resolution
and larger projectivity than regular designs of the same size. It is further
shown that some of these designs have generalized minimum aberration and
maximum projectivity among all possible designs.Comment: Published in at http://dx.doi.org/10.1214/08-AOS656 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A linear construction for certain Kerdock and Preparata codes
The Nordstrom-Robinson, Kerdock, and (slightly modified) Pre\- parata codes
are shown to be linear over \ZZ_4, the integers . The Kerdock and
Preparata codes are duals over \ZZ_4, and the Nordstrom-Robinson code is
self-dual. All these codes are just extended cyclic codes over \ZZ_4. This
provides a simple definition for these codes and explains why their Hamming
weight distributions are dual to each other. First- and second-order
Reed-Muller codes are also linear codes over \ZZ_4, but Hamming codes in
general are not, nor is the Golay code.Comment: 5 page
On the number of nonequivalent propelinear extended perfect codes
The paper proves that there exist an exponential number of nonequivalent
propelinear extended perfect binary codes of length growing to infinity.
Specifically, it is proved that all transitive extended perfect binary codes
found by Potapov are propelinear. All such codes have small rank, which is one
more than the rank of the extended Hamming code of the same length. We
investigate the properties of these codes and show that any of them has a
normalized propelinear representation
Catalytic quantum error correction
We develop the theory of entanglement-assisted quantum error correcting
(EAQEC) codes, a generalization of the stabilizer formalism to the setting in
which the sender and receiver have access to pre-shared entanglement.
Conventional stabilizer codes are equivalent to dual-containing symplectic
codes. In contrast, EAQEC codes do not require the dual-containing condition,
which greatly simplifies their construction. We show how any quaternary
classical code can be made into a EAQEC code. In particular, efficient modern
codes, like LDPC codes, which attain the Shannon capacity, can be made into
EAQEC codes attaining the hashing bound. In a quantum computation setting,
EAQEC codes give rise to catalytic quantum codes which maintain a region of
inherited noiseless qubits.
We also give an alternative construction of EAQEC codes by making classical
entanglement assisted codes coherent.Comment: 30 pages, 10 figures. Notation change: [[n,k;c]] instead of
[[n,k-c;c]
New cubic self-dual codes of length 54, 60 and 66
We study the construction of quasi-cyclic self-dual codes, especially of
binary cubic ones. We consider the binary quasi-cyclic codes of length 3\ell
with the algebraic approach of [9]. In particular, we improve the previous
results by constructing 1 new binary [54, 27, 10], 6 new [60, 30, 12] and 50
new [66, 33, 12] cubic self-dual codes. We conjecture that there exist no more
binary cubic self-dual codes with length 54, 60 and 66.Comment: 8 page
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