142 research outputs found
Hadamard Matrices of Orders 60 and 64 with Automorphisms of Orders 29 and 31
A classification of Hadamard matrices of order 2p + 2 with an automorphism of order p is given for p = 29 and 31. The ternary self-dual codes spanned by the newly found Hadamard matrices of order 60 with an automorphism of order 29 are computed, as well as the binary doubly even self-dual codes of length 120 with generator matrices defined by related Hadamard designs. Several new ternary near-extremal self-dual codes, as well as binary near-extremal doubly even self-dual codes with previously unknown weight enumerators are found
Design Lines
The two basic equations satisfied by the parameters of a block design define
a three-dimensional affine variety in . A point
of that is not in some sense trivial lies on four lines lying in
. These lines provide a degree of organization for certain general
classes of designs, and the paper is devoted to exploring properties of the
lines. Several examples of families of designs that seem naturally to follow
the lines are presented.Comment: 16 page
Low-Coherence Frames from Group Fourier Matrices
Many problems in areas such as compressive sensing and coding theory seek to design a set of equal-norm vectors with large angular separation. This idea is essentially equivalent to constructing a frame with low coherence. The elements of such frames can in turn be used to build high-performance spherical codes, quantum measurement operators, and compressive sensing measurement matrices, to name a few applications.
In this work, we allude to the group-frame construction first described by Slepian and further explored in the works of Vale and Waldron. We present a method for selecting representations of a nite group to construct a group frame that achieves low coherence. Our technique produces a tight frame with a small number of distinct inner product values between the frame elements, in a sense approximating a Grassmannian frame. We identify special cases in which our construction yields some previously-known frames with optimal coherence meeting the Welch lower bound, and other cases in which the entries of our frame vectors come from small alphabets. In particular, we apply our technique to the problem choosing a subset of rows of a Hadamard matrix so that the resulting columns form a low-coherence frame. Finally, we give an explicit calculation of the average coherence of our frames, and nd regimes in which they satisfy the Strong Coherence Property described by Mixon, Bajwa, and Calderbank
Self-Dual Codes
Self-dual codes are important because many of the best codes known are of
this type and they have a rich mathematical theory. Topics covered in this
survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight
enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems,
bounds, mass formulae, enumeration, extremal codes, open problems. There is a
comprehensive bibliography.Comment: 136 page
Symplectic spreads, planar functions and mutually unbiased bases
In this paper we give explicit descriptions of complete sets of mutually
unbiased bases (MUBs) and orthogonal decompositions of special Lie algebras
obtained from commutative and symplectic semifields, and
from some other non-semifield symplectic spreads. Relations between various
constructions are also studied. We show that the automorphism group of a
complete set of MUBs is isomorphic to the automorphism group of the
corresponding orthogonal decomposition of the Lie algebra .
In the case of symplectic spreads this automorphism group is determined by the
automorphism group of the spread. By using the new notion of pseudo-planar
functions over fields of characteristic two we give new explicit constructions
of complete sets of MUBs.Comment: 20 page
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