195 research outputs found
A lower bound for the size of the smallest critical set in the back circulant latin square
The back circulant latin square of order n is the latin square based on the addition table for the integers modulo n. A critical set is a partial latin square that has a unique completion to a latin square, and is minimal with respect to this property. In this note we show that the size of a critical set in the back circulant latin square of order n is at least n ā“/Ā³/2 - n - nĀ²/Ā³/2 + 2nĀ¹/Ā³ - 1
A uniqueness result for -homogeneous latin trades
summary:A latin trade is a subset of a latin square which may be replaced with a disjoint mate to obtain a new latin square. A -homogeneous latin trade is one which intersects each row, each column and each entry of the latin square either or times. In this paper, we show that a construction given by Cavenagh, Donovan and DrƔpal for -homogeneous latin trades in fact classifies every minimal -homogeneous latin trade. We in turn classify all -homogeneous latin trades. A corollary is that any -homogeneous latin trade may be partitioned into three, disjoint, partial transversals
Scaled Projected-Directions Methods with Application to Transmission Tomography
Statistical image reconstruction in X-Ray computed tomography yields
large-scale regularized linear least-squares problems with nonnegativity
bounds, where the memory footprint of the operator is a concern. Discretizing
images in cylindrical coordinates results in significant memory savings, and
allows parallel operator-vector products without on-the-fly computation of the
operator, without necessarily decreasing image quality. However, it
deteriorates the conditioning of the operator. We improve the Hessian
conditioning by way of a block-circulant scaling operator and we propose a
strategy to handle nondiagonal scaling in the context of projected-directions
methods for bound-constrained problems. We describe our implementation of the
scaling strategy using two algorithms: TRON, a trust-region method with exact
second derivatives, and L-BFGS-B, a linesearch method with a limited-memory
quasi-Newton Hessian approximation. We compare our approach with one where a
change of variable is made in the problem. On two reconstruction problems, our
approach converges faster than the change of variable approach, and achieves
much tighter accuracy in terms of optimality residual than a first-order
method.Comment: 19 pages, 7 figure
COMPLEX HADAMARD MATRICES AND APPLICATIONS
A complex Hadamard matrix is a square matrix H ā M N (C) whose entries are on the unit circle, |H ij | = 1, and whose rows and pairwise orthogonal. The main example is the Fourier matrix, F N = (w ij) with w = e 2Ļi/N. We discuss here the basic theory of such matrices, with emphasis on geometric and analytic aspects. CONTENT
Critical sets of full Latin squares
This thesis explores the properties of critical sets of the full n-Latin square and related combinatorial structures including full designs, (m,n,2)-balanced Latin rectangles and n-Latin cubes.
In Chapter 3 we study known results on designs and the analogies between critical sets of the full n-Latin square and minimal defining sets of the full designs.
Next in Chapter 4 we fully classify the critical sets of the full (m,n,2)-balanced Latin square, by describing the precise structures of these critical sets from the smallest to the largest.
Properties of different types of critical sets of the full n-Latin square are investigated in Chapter 5. We fully classify the structure of any saturated critical set of the full n-Latin square. We show in Theorem 5.8 that such a critical set has size exactly equal to nĀ³ - 2nĀ² - n. In Section 5.2 we give a construction which provides an upper bound for the size of the smallest critical set of the full n-Latin square. Similarly in Section 5.4, another construction gives a lower bound for the size of the largest non-saturated critical set. We conjecture that these bounds are best possible.
Using the results from Chapter 5, we obtain spectrum results on critical sets of the full n-Latin square in Chapter 6. In particular, we show that a critical set of each size between (n - 1)Ā³ + 1 and n(n - 1)Ā² + n - 2 exists.
In Chapter 7, we turn our focus to the completability of partial k-Latin squares. The relationship between partial k-Latin squares and semi-k-Latin squares is used to show that any partial k-Latin square of order n with at most (n - 1) non-empty cells is completable.
As Latin cubes generalize Latin squares, we attempt to generalize some of the results we have established on k-Latin squares so that they apply to k-Latin cubes. These results are presented in Chapter 8
Novel Code-Construction for (3, k) Regular Low Density Parity Check Codes
Communication system links that do not have the ability to retransmit generally rely
on forward error correction (FEC) techniques that make use of error correcting codes
(ECC) to detect and correct errors caused by the noise in the channel. There are
several ECCās in the literature that are used for the purpose. Among them, the low
density parity check (LDPC) codes have become quite popular owing to the fact that
they exhibit performance that is closest to the Shannonās limit.
This thesis proposes a novel code-construction method for constructing not only (3, k)
regular but also irregular LDPC codes. The choice of designing (3, k) regular LDPC
codes is made because it has low decoding complexity and has a Hamming distance,
at least, 4. In this work, the proposed code-construction consists of information submatrix
(Hinf) and an almost lower triangular parity sub-matrix (Hpar). The core design
of the proposed code-construction utilizes expanded deterministic base matrices in
three stages. Deterministic base matrix of parity part starts with triple diagonal matrix
while deterministic base matrix of information part utilizes matrix having all elements
of ones. The proposed matrix H is designed to generate various code rates (R) by
maintaining the number of rows in matrix H while only changing the number of
columns in matrix Hinf.
All the codes designed and presented in this thesis are having no rank-deficiency, no
pre-processing step of encoding, no singular nature in parity part (Hpar), no girth of
4-cycles and low encoding complexity of the order of (N + g2) where g2Ā«N. The
proposed (3, k) regular codes are shown to achieve code performance below 1.44 dB
from Shannon limit at bit error rate (BER) of 10
ā6
when the code rate greater than
R = 0.875. They have comparable BER and block error rate (BLER) performance
with other techniques such as (3, k) regular quasi-cyclic (QC) and (3, k) regular
random LDPC codes when code rates are at least R = 0.7. In addition, it is also shown
that the proposed (3, 42) regular LDPC code performs as close as 0.97 dB from
Shannon limit at BER 10
ā6
with encoding complexity (1.0225 N), for R = 0.928 and
N = 14364 ā a result that no other published techniques can reach
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