94 research outputs found
The quaternary complex Hadamard matrices of orders 10, 12, and 14
A complete classification of quaternary complex Hadamard matrices of orders
10, 12 and 14 is given, and a new parametrization scheme for obtaining new
examples of affine parametric families of complex Hadamard matrices is
provided. On the one hand, it is proven that all 10x10 and 12x12 quaternary
complex Hadamard matrices belong to some parametric family, but on the other
hand, it is shown by exhibiting an isolated 14x14 matrix that there cannot be a
general method for introducing parameters into these types of matrices.Comment: 14+8 pages, preprin
Classification of generalized Hadamard matrices H(6,3) and quaternary Hermitian self-dual codes of length 18
All generalized Hadamard matrices of order 18 over a group of order 3,
H(6,3), are enumerated in two different ways: once, as class regular symmetric
(6,3)-nets, or symmetric transversal designs on 54 points and 54 blocks with a
group of order 3 acting semi-regularly on points and blocks, and secondly, as
collections of full weight vectors in quaternary Hermitian self-dual codes of
length 18. The second enumeration is based on the classification of Hermitian
self-dual [18,9] codes over GF(4), completed in this paper. It is shown that up
to monomial equivalence, there are 85 generalized Hadamard matrices H(6,3), and
245 inequivalent Hermitian self-dual codes of length 18 over GF(4).Comment: 17 pages. Minor revisio
On Generalized Hadamard Matrices and Difference Matrices:
We give some very interesting matrices which are orthogonal over groups and,
as far as we know, referenced, but in fact undocumented. This note is not
intended to be published but available for archival reasons.Comment: 15 page
Hadamard matrices modulo 5
In this paper we introduce modular symmetric designs and use them to study
the existence of Hadamard matrices modulo 5. We prove that there exist
5-modular Hadamard matrices of order n if and only if n != 3, 7 (mod 10) or n
!= 6, 11. In particular, this solves the 5-modular version of the Hadamard
conjecture.Comment: 7 pages, submitted to JC
Morphisms of Butson classes
We introduce the concept of a morphism from the set of Butson Hadamard
matrices over kth roots of unity to the set of Butson matrices over th
roots of unity. As concrete examples of such morphisms, we describe
tensor-product-like maps which reduce the order of the roots of unity appearing
in a Butson matrix at the cost of increasing the dimension. Such maps can be
constructed from Butson matrices with eigenvalues satisfying certain natural
conditions. Our work unifies and generalises Turyn's construction of real
Hadamard matrices from Butson matrices over the 4th roots and the work of
Compton, Craigen and de Launey on `unreal' Butson matrices over the 6th roots.
As a case study, we classify all morphisms from the set of Butson
matrices over kth roots of unity to the set of Butson matrices
over th roots of unity where
Complex conference matrices, Complex Hadamard matrices and equiangular tight frames
We construct new, previously unknown parametric families of complex
conference matrices and of complex Hadamard matrices of square orders and
related them to complex equiangular tight frames.Comment: 9 page
Constructions of complementary sequence sets and complete complementary codes by 2-level autocorrelation sequences and permutation polynomials
In this paper, a recent method to construct complementary sequence sets and
complete complementary codes by Hadamard matrices is deeply studied. By taking
the algebraic structure of Hadamard matrices into consideration, our main
result determine the so-called -linear terms and -quadratic
terms. As a first consequence, a powerful theory linking Golay complementary
sets of -ary ( prime) sequences and the generalized Reed-Muller codes by
Kasami et al. is developed. These codes enjoy good error-correcting capability,
tightly controlled PMEPR, and significantly extend the range of coding options
for applications of OFDM using subcarriers. As another consequence, we
make a previously unrecognized connection between the sequences in CSSs and
CCCs and the sequence with 2-level autocorrelation, trace function and
permutation polynomial (PP) over the finite fields
The genetic code, 8-dimensional hypercomplex numbers and dyadic shifts
Matrix forms of the representation of the multi-level system of
molecular-genetic alphabets have revealed algebraic properties of this system.
Families of genetic (4*4)- and (8*8)-matrices show unexpected connections of
the genetic system with Walsh functions and Hadamard matrices, which are known
in theory of noise-immunity coding, digital communication and digital
holography. Dyadic-shift decompositions of such genetic matrices lead to sets
of sparse matrices. Each of these sets is closed in relation to multiplication
and defines relevant algebra of hypercomplex numbers. It is shown that genetic
Hadamard matrices are identical to matrix representations of Hamilton
quaternions and its complexification in the case of unit coordinates. The
diversity of known dialects of the genetic code is analyzed from the viewpoint
of the genetic algebras. An algebraic analogy with Punnett squares for
inherited traits is shown. Our results are used in analyzing genetic phenomena.
The statement about existence of the geno-logic code in DNA and epigenetics on
the base of the spectral logic of systems of Boolean functions is put forward.
Our results show promising ways to develop algebraic-logical biology, in
particular, in connection with the logic holography on Walsh functions.Comment: 108 pages, 73 figures, added text, added reference
Identifying Complex Hadamard Submatrices of the Fourier Matrices via Primitive Sets
For a given selection of rows and columns from a Fourier matrix, we give a
number of tests for whether the resulting submatrix is Hadamard based on the
primitive sets of those rows and columns. In particular, we demonstrate that
whether a given selection of rows and columns of a Fourier matrix forms a
Hadamard submatrix is exactly determined by whether the primitive sets of those
rows and columns are compatible with respect to the size of the Fourier matrix.
This allows the partitioning of all submatrices into equivalence classes that
will consist entirely of Hadamard or entirely of non-Hadamard submatrices and
motivates the creation of compatibility graphs that represent this structure.
We conclude with some results that facilitate the construction of these graphs
for submatrix sizes 2 and 3.Comment: 22 pages, 6 figure
New Sequence Sets with Zero-Correlation Zone
A method for constructing sets of sequences with zero-correlation zone (ZCZ
sequences) and sequence sets with low cross correlation is proposed. The method
is to use families of short sequences and complete orthogonal sequence sets to
derive families of long sequences with desired correlation properties. It is a
unification of works of Matsufuji and Torii \emph{et al.}, and there are more
choices of parameters of sets for our method. In particular, ZCZ sequence sets
generated by the method can achieve a related ZCZ bound. Furthermore, the
proposed method can be utilized to derive new ZCZ sets with both longer ZCZ and
larger set size from known ZCZ sets. These sequence sets are applicable in
broadband satellite IP networks.Comment: 28 pages, submitted to IEEE-IT on May 18, 200
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