17,894 research outputs found
Symmetric Bush-type generalized Hadamard matrices and association schemes
We define Bush-type generalized Hadamard matrices over abelian groups and
construct symmetric Bush-type generalized Hadamard matrices over the additive
group of finite field , a prime power. We then show and study
an association scheme obtained from such generalized Hadamard matrices.Comment: 10 page
Generalized Hadamard Product and the Derivatives of Spectral Functions
In this work we propose a generalization of the Hadamard product between two
matrices to a tensor-valued, multi-linear product between k matrices for any . A multi-linear dual operator to the generalized Hadamard product is
presented. It is a natural generalization of the Diag x operator, that maps a
vector into the diagonal matrix with x on its main diagonal.
Defining an action of the orthogonal matrices on the space of
k-dimensional tensors, we investigate its interactions with the generalized
Hadamard product and its dual. The research is motivated, as illustrated
throughout the paper, by the apparent suitability of this language to describe
the higher-order derivatives of spectral functions and the tools needed to
compute them. For more on the later we refer the reader to [14] and [15], where
we use the language and properties developed here to study the higher-order
derivatives of spectral functions.Comment: 24 page
Error-Correcting Codes Associated With Generalized Hadamard Matrices Over Groups
Classical Hadamard matrices are orthogonal matrices whose elements are ±1. It is well-known that error correcting codes having large minimum distance between codewords can be associated with these Hadamard matrices. Indeed, the success of early Mars deep-space probes was strongly dependent upon this communication technology.
The concept of Hadamard matrices with elements drawn from an Abelian group is a natural generalization of the concept. For the case in which the dimension of the matrix is q and the group consists of the p-th roots of unity, these generalized Hadamard matrices are called “Butson Hadamard Matrices BH(p, q)”, first discovered by A. T. Butson [6].
In this dissertation it is shown that an error correcting code whose codewords consist of real numbers in finite Galois field Gf( p) can be associated in a simple way with each Butson Hadamard matrix BH(p, q), where p \u3e 0 is a prime number. Distance properties of such codes are studied, as well as conditions for the existence of linear codes, for which standard decoding techniques are available.
In the search for cyclic linear generalized Hadamard codes, the concept of an M-invariant infinite sequence whose elements are integers in a finite field is introduced. Such sequences are periodic of least period, T, and have the interesting property, that arbitrary identical rearrangements of the elements in each period yields a periodic sequence with the same least period. A theorem characterizing such M-invariant sequences leads to discovery of a simple and efficient polynomial method for constructing generalized Hadamard matrices whose core is a linear cyclic matrix and whose row vectors constitute a linear cyclic error correcting code.
In addition, the problem is considered of determining parameter sequences {tn} for which the corresponding potential generalized Hadamard matrices BH(p, ptn) do not exist. By analyzing quadratic Diophantine equations, new methods for constructing such parameter sequences are obtained. These results show the rich number theoretic complexity of the existence question for generalized Hadamard matrices
Temperley-Lieb R-matrices from generalized Hadamard matrices
New sets of rank n-representations of Temperley-Lieb algebra TL_N(q) are
constructed. They are characterized by two matrices obeying a generalization of
the complex Hadamard property. Partial classifications for the two matrices are
given, in particular when they reduce to Fourier or Butson matrices.Comment: 17 page
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