1,824 research outputs found
Spectral Bounds for Quasi-Twisted Codes
New lower bounds on the minimum distance of quasi-twisted codes over finite
fields are proposed. They are based on spectral analysis and eigenvalues of
polynomial matrices. They generalize the Semenov-Trifonov and Zeh-Ling bounds
in a manner similar to how the Roos and shift bounds extend the BCH and HT
bounds for cyclic codes.Comment: Accepted ISIT 201
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
Dynamical Systems on Spectral Metric Spaces
Let (A,H,D) be a spectral triple, namely: A is a C*-algebra, H is a Hilbert
space on which A acts and D is a selfadjoint operator with compact resolvent
such that the set of elements of A having a bounded commutator with D is dense.
A spectral metric space, the noncommutative analog of a complete metric space,
is a spectral triple (A,H,D) with additional properties which guaranty that the
Connes metric induces the weak*-topology on the state space of A. A
*-automorphism respecting the metric defined a dynamical system. This article
gives various answers to the question: is there a canonical spectral triple
based upon the crossed product algebra AxZ, characterizing the metric
properties of the dynamical system ? If is the noncommutative analog
of an isometry the answer is yes. Otherwise, the metric bundle construction of
Connes and Moscovici is used to replace (A,) by an equivalent dynamical
system acting isometrically. The difficulties relating to the non compactness
of this new system are discussed. Applications, in number theory, in coding
theory are given at the end
Symplectic self-orthogonal quasi-cyclic codes
In this paper, we obtain sufficient and necessary conditions for quasi-cyclic
codes with index even to be symplectic self-orthogonal. Then, we propose a
method for constructing symplectic self-orthogonal quasi-cyclic codes, which
allows arbitrary polynomials that coprime to construct symplectic
self-orthogonal codes. Moreover, by decomposing the space of quasi-cyclic
codes, we provide lower and upper bounds on the minimum symplectic distances of
a class of 1-generator quasi-cyclic codes and their symplectic dual codes.
Finally, we construct many binary symplectic self-orthogonal codes with
excellent parameters, corresponding to 117 record-breaking quantum codes,
improving Grassl's table (Bounds on the Minimum Distance of Quantum Codes.
http://www.codetables.de)
Improved Spectral Bound for Quasi-Cyclic Codes
Spectral bounds form a powerful tool to estimate the minimum distances of
quasi-cyclic codes. They generalize the defining set bounds of cyclic codes to
those of quasi-cyclic codes. Based on the eigenvalues of quasi-cyclic codes and
the corresponding eigenspaces, we provide an improved spectral bound for
quasi-cyclic codes. Numerical results verify that the improved bound
outperforms the Jensen bound in almost all cases. Based on the improved bound,
we propose a general construction of quasi-cyclic codes with excellent designed
minimum distances. For the quasi-cyclic codes produced by this general
construction, the improved spectral bound is always sharper than the Jensen
bound
Quasi-Cyclic Codes
Quasi-cyclic codes form an important class of algebraic codes that includes
cyclic codes as a special subclass. This chapter focuses on the algebraic
structure of quasi-cyclic codes, first. Based on these structural properties,
some asymptotic results, a few minimum distance bounds and further applications
such as the trace representation and characterization of certain subfamilies of
quasi-cyclic codes are elaborated. This survey will appear as a chapter in "A
Concise Encyclopedia of Coding Theory" to be published by CRC Press.Comment: arXiv admin note: text overlap with arXiv:1906.0496
Orthogonal transmultiplexers : extensions to digital subscriber line (DSL) communications
An orthogonal transmultiplexer which unifies multirate filter bank theory and communications theory is investigated in this dissertation. Various extensions of the orthogonal transmultiplexer techniques have been made for digital subscriber line communication applications.
It is shown that the theoretical performance bounds of single carrier modulation based transceivers and multicarrier modulation based transceivers are the same under the same operational conditions. Single carrier based transceiver systems such as Quadrature Amplitude Modulation (QAM) and Carrierless Amplitude and Phase (CAP) modulation scheme, multicarrier based transceiver systems such as Orthogonal Frequency Division Multiplexing (OFDM) or Discrete Multi Tone (DMT) and Discrete Subband (Wavelet) Multicarrier based transceiver (DSBMT) techniques are considered in this investigation.
The performance of DMT and DSBMT based transceiver systems for a narrow band interference and their robustness are also investigated. It is shown that the performance of a DMT based transceiver system is quite sensitive to the location and strength of a single tone (narrow band) interference. The performance sensitivity is highlighted in this work. It is shown that an adaptive interference exciser can alleviate the sensitivity problem of a DMT based system. The improved spectral properties of DSBMT technique reduces the performance sensitivity for variations of a narrow band interference. It is shown that DSBMT technique outperforms DMT and has a more robust performance than the latter. The superior performance robustness is shown in this work.
Optimal orthogonal basis design using cosine modulated multirate filter bank is discussed. An adaptive linear combiner at the output of analysis filter bank is implemented to eliminate the intersymbol and interchannel interferences. It is shown that DSBMT is the most suitable technique for a narrow band interference environment.
A blind channel identification and optimal MMSE based equalizer employing a nonmaximally decimated filter bank precoder / postequalizer structure is proposed. The performance of blind channel identification scheme is shown not to be sensitive to the characteristics of unknown channel. The performance of the proposed optimal MMSE based equalizer is shown to be superior to the zero-forcing equalizer
Spectral pseudorandomness and the road to improved clique number bounds for Paley graphs
We study subgraphs of Paley graphs of prime order induced on the sets of
vertices extending a given independent set of size to a larger independent
set. Using a sufficient condition proved in the author's recent companion work,
we show that a family of character sum estimates would imply that, as , the empirical spectral distributions of the adjacency matrices of any
sequence of such subgraphs have the same weak limit (after rescaling) as those
of subgraphs induced on a random set including each vertex independently with
probability , namely, a Kesten-McKay law with parameter . We prove
the necessary estimates for , obtaining in the process an alternate
proof of a character sum equidistribution result of Xi (2022), and provide
numerical evidence for this weak convergence for . We also conjecture
that the minimum eigenvalue of any such sequence converges (after rescaling) to
the left edge of the corresponding Kesten-McKay law, and provide numerical
evidence for this convergence. Finally, we show that, once , this
(conjectural) convergence of the minimum eigenvalue would imply bounds on the
clique number of the Paley graph improving on the current state of the art due
to Hanson and Petridis (2021), and that this convergence for all
would imply that the clique number is .Comment: 43 pages, 1 table, 6 figure
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