3,274 research outputs found
Codes, graphs and designs from maximal subgroups of alternating groups
Philosophiae Doctor - PhD (Mathematics)The main theme of this thesis is the construction of linear codes from adjacency matrices or sub-matrices of adjacency matrices of regular graphs. We first examine the binary codes from the row span of biadjacency matrices and their transposes for some classes of bipartite graphs. In this case we consider a sub-matrix of an adjacency matrix of a graph as the generator of the code. We then shift our attention to uniform subset graphs by exploring the automorphism groups of graph covers and some classes of uniform subset graphs. In the sequel, we explore equal codes from adjacency matrices of non-isomorphic uniform subset graphs and finally consider codes generated by an adjacency matrix formed by adding adjacency matrices of two classes of uniform subset graphs
Codes from adjacency matrices of uniform subset graphs
Studies of the p-ary codes from the adjacency matrices of uniform subset graphs Γ(n,k,r)Γ(n,k,r) and their reflexive associates have shown that a particular family of codes defined on the subsets are intimately related to the codes from these graphs. We describe these codes here and examine their relation to some particular classes of uniform subset graphs. In particular we include a complete analysis of the p-ary codes from Γ(n,3,r)Γ(n,3,r) for p≥5p≥5 , thus extending earlier results for p=2,3p=2,3
Graphs, designs and codes related to the n-cube
For integers n 1; k 0, and k n, the graph k
n has vertices the 2n vectors of
Fn
2 and adjacency defined by two vectors being adjacent if they differ in k coordinate
positions. In particular 1
n is the n-cube, usually denoted by Qn. We examine the binary
codes obtained from the adjacency matrices of these graphs when k D 1; 2; 3, following
the results obtained for the binary codes of the n-cube in Fish [Washiela Fish, Codes
from uniform subset graphs and cyclic products, Ph.D. Thesis, University of the Western
Cape, 2007] and Key and Seneviratne [J.D. Key, P. Seneviratne, Permutation decoding for
binary self-dual codes from the graph Qn where n is even, in: T. Shaska, W. C Huffman,
D. Joyner, V. Ustimenko (Eds.), Advances in Coding Theory and Cryptology, in: Series on
Coding Theory and Cryptology, vol. 2, World Scientific Publishing Co. Pte. Ltd., Hackensack,
NJ, 2007, pp. 152 159 ]. We find the automorphism groups of the graphs and of their
associated neighbourhood designs for k D 1; 2; 3, and the dimensions of the ternary
codes for k D 1; 2. We also obtain 3-PD-sets for the self-dual binary codes from 2
n when
n 0 .mod 4/, n 8
Graphs, designs and codes related to the n-cube
For integers n 1; k 0, and k n, the graph k
n has vertices the 2n vectors of
Fn
2 and adjacency defined by two vectors being adjacent if they differ in k coordinate
positions. In particular 1
n is the n-cube, usually denoted by Qn. We examine the binary
codes obtained from the adjacency matrices of these graphs when k D 1; 2; 3, following
the results obtained for the binary codes of the n-cube in Fish [Washiela Fish, Codes
from uniform subset graphs and cyclic products, Ph.D. Thesis, University of the Western
Cape, 2007] and Key and Seneviratne [J.D. Key, P. Seneviratne, Permutation decoding for
binary self-dual codes from the graph Qn where n is even, in: T. Shaska, W. C Huffman,
D. Joyner, V. Ustimenko (Eds.), Advances in Coding Theory and Cryptology, in: Series on
Coding Theory and Cryptology, vol. 2, World Scientific Publishing Co. Pte. Ltd., Hackensack,
NJ, 2007, pp. 152 159 ]. We find the automorphism groups of the graphs and of their
associated neighbourhood designs for k D 1; 2; 3, and the dimensions of the ternary
codes for k D 1; 2. We also obtain 3-PD-sets for the self-dual binary codes from 2
n when
n 0 .mod 4/, n 8
Frames, Graphs and Erasures
Two-uniform frames and their use for the coding of vectors are the main
subject of this paper. These frames are known to be optimal for handling up to
two erasures, in the sense that they minimize the largest possible error when
up to two frame coefficients are set to zero. Here, we consider various
numerical measures for the reconstruction error associated with a frame when an
arbitrary number of the frame coefficients of a vector are lost. We derive
general error bounds for two-uniform frames when more than two erasures occur
and apply these to concrete examples. We show that among the 227 known
equivalence classes of two-uniform (36,15)-frames arising from Hadamard
matrices, there are 5 that give smallest error bounds for up to 8 erasures.Comment: 28 pages LaTeX, with AMS macros; v.3: fixed Thm 3.6, added comment,
Lemma 3.7 and Proposition 3.8, to appear in Lin. Alg. App
High Dimensional Random Walks and Colorful Expansion
Random walks on bounded degree expander graphs have numerous applications,
both in theoretical and practical computational problems. A key property of
these walks is that they converge rapidly to their stationary distribution.
In this work we {\em define high order random walks}: These are
generalizations of random walks on graphs to high dimensional simplicial
complexes, which are the high dimensional analogues of graphs. A simplicial
complex of dimension has vertices, edges, triangles, pyramids, up to
-dimensional cells. For any , a high order random walk on
dimension moves between neighboring -faces (e.g., edges) of the complex,
where two -faces are considered neighbors if they share a common
-face (e.g., a triangle). The case of recovers the well studied
random walk on graphs.
We provide a {\em local-to-global criterion} on a complex which implies {\em
rapid convergence of all high order random walks} on it. Specifically, we prove
that if the -dimensional skeletons of all the links of a complex are
spectral expanders, then for {\em all} the high order random walk
on dimension converges rapidly to its stationary distribution.
We derive our result through a new notion of high dimensional combinatorial
expansion of complexes which we term {\em colorful expansion}. This notion is a
natural generalization of combinatorial expansion of graphs and is strongly
related to the convergence rate of the high order random walks.
We further show an explicit family of {\em bounded degree} complexes which
satisfy this criterion. Specifically, we show that Ramanujan complexes meet
this criterion, and thus form an explicit family of bounded degree high
dimensional simplicial complexes in which all of the high order random walks
converge rapidly to their stationary distribution.Comment: 27 page
CONTEST : a Controllable Test Matrix Toolbox for MATLAB
Large, sparse networks that describe complex interactions are a common feature across a number of disciplines, giving rise to many challenging matrix computational tasks. Several random graph models have been proposed that capture key properties of real-life networks. These models provide realistic, parametrized matrices for testing linear system and eigenvalue solvers. CONTEST (CONtrollable TEST matrices) is a random network toolbox for MATLAB that implements nine models. The models produce unweighted directed or undirected graphs; that is, symmetric or unsymmetric matrices with elements equal to zero or one. They have one or more parameters that affect features such as sparsity and characteristic pathlength and all can be of arbitrary dimension. Utility functions are supplied for rewiring, adding extra shortcuts and subsampling in order to create further classes of networks. Other utilities convert the adjacency matrices into real-valued coefficient matrices for naturally arising computational tasks that reduce to sparse linear system and eigenvalue problems
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