20 research outputs found
Around Pelikan's conjecture on very odd sequences
Very odd sequences were introduced in 1973 by J. Pelikan who conjectured that
there were none of length >=5. This conjecture was disproved by MacWilliams and
Odlyzko in 1977 who proved there are in fact many very odd sequences. We give
connections of these sequences with duadic codes, cyclic difference sets,
levels (Stufen) of cyclotomic fields, and derive some new asymptotic results on
their lengths and on S(n), which denotes the number of very odd sequences of
length n.Comment: 21 pages, two tables. Revised version with improved presentation and
correction of some typos and minor errors that will appear in Manuscripta
Mathematic
Cyclic Codes from Cyclotomic Sequences of Order Four
Cyclic codes are an interesting subclass of linear codes and have been used
in consumer electronics, data transmission technologies, broadcast systems, and
computer applications due to their efficient encoding and decoding algorithms.
In this paper, three cyclotomic sequences of order four are employed to
construct a number of classes of cyclic codes over \gf(q) with prime length.
Under certain conditions lower bounds on the minimum weight are developed. Some
of the codes obtained are optimal or almost optimal. In general, the cyclic
codes constructed in this paper are very good. Some of the cyclic codes
obtained in this paper are closely related to almost difference sets and
difference sets. As a byproduct, the -rank of these (almost) difference sets
are computed
Cyclotomic Constructions of Cyclic Codes with Length Being the Product of Two Primes
Cyclic codes are an interesting type of linear codes and have applications in
communication and storage systems due to their efficient encoding and decoding
algorithms. They have been studied for decades and a lot of progress has been
made. In this paper, three types of generalized cyclotomy of order two and
three classes of cyclic codes of length and dimension
are presented and analysed, where and are two distinct primes.
Bounds on their minimum odd-like weight are also proved. The three
constructions produce the best cyclic codes in certain cases.Comment: 19 page
Quantum error control codes
It is conjectured that quantum computers are able to solve certain problems more
quickly than any deterministic or probabilistic computer. For instance, Shor's algorithm
is able to factor large integers in polynomial time on a quantum computer.
A quantum computer exploits the rules of quantum mechanics to speed up computations.
However, it is a formidable task to build a quantum computer, since the
quantum mechanical systems storing the information unavoidably interact with their
environment. Therefore, one has to mitigate the resulting noise and decoherence
effects to avoid computational errors.
In this dissertation, I study various aspects of quantum error control codes - the key component of fault-tolerant quantum information processing. I present the
fundamental theory and necessary background of quantum codes and construct many
families of quantum block and convolutional codes over finite fields, in addition to
families of subsystem codes. This dissertation is organized into three parts:
Quantum Block Codes. After introducing the theory of quantum block codes, I
establish conditions when BCH codes are self-orthogonal (or dual-containing)
with respect to Euclidean and Hermitian inner products. In particular, I derive
two families of nonbinary quantum BCH codes using the stabilizer formalism. I study duadic codes and establish the existence of families of degenerate quantum
codes, as well as families of quantum codes derived from projective geometries.
Subsystem Codes. Subsystem codes form a new class of quantum codes in which
the underlying classical codes do not need to be self-orthogonal. I give an
introduction to subsystem codes and present several methods for subsystem
code constructions. I derive families of subsystem codes from classical BCH and
RS codes and establish a family of optimal MDS subsystem codes. I establish
propagation rules of subsystem codes and construct tables of upper and lower
bounds on subsystem code parameters.
Quantum Convolutional Codes. Quantum convolutional codes are particularly
well-suited for communication applications. I develop the theory of quantum
convolutional codes and give families of quantum convolutional codes based
on RS codes. Furthermore, I establish a bound on the code parameters of
quantum convolutional codes - the generalized Singleton bound. I develop a
general framework for deriving convolutional codes from block codes and use it
to derive families of non-catastrophic quantum convolutional codes from BCH
codes.
The dissertation concludes with a discussion of some open problems