283 research outputs found
On quaternary complex Hadamard matrices of small orders
One of the main goals of design theory is to classify, characterize and count
various combinatorial objects with some prescribed properties. In most cases,
however, one quickly encounters a combinatorial explosion and even if the
complete enumeration of the objects is possible, there is no apparent way how
to study them in details, store them efficiently, or generate a particular one
rapidly. In this paper we propose a novel method to deal with these
difficulties, and illustrate it by presenting the classification of quaternary
complex Hadamard matrices up to order 8. The obtained matrices are members of
only a handful of parametric families, and each inequivalent matrix, up to
transposition, can be identified through its fingerprint.Comment: 7 page
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
A Generalised Hadamard Transform
A Generalised Hadamard Transform for multi-phase or multilevel signals is
introduced, which includes the Fourier, Generalised, Discrete Fourier,
Walsh-Hadamard and Reverse Jacket Transforms. The jacket construction is
formalised and shown to admit a tensor product decomposition. Primary matrices
under this decomposition are identified. New examples of primary jacket
matrices of orders 8 and 12 are presented.Comment: To appear in the proceedings of the 2005 IEEE International Symposium
on Information Theory, Adelaide, Australia, September 4-9, 200
Kerdock Codes Determine Unitary 2-Designs
The non-linear binary Kerdock codes are known to be Gray images of certain
extended cyclic codes of length over . We show that
exponentiating these -valued codewords by produces stabilizer states, that are quantum states obtained using
only Clifford unitaries. These states are also the common eigenvectors of
commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the
Pauli group. We use this quantum description to simplify the derivation of the
classical weight distribution of Kerdock codes. Next, we organize the
stabilizer states to form mutually unbiased bases and prove that
automorphisms of the Kerdock code permute their corresponding MCS, thereby
forming a subgroup of the Clifford group. When represented as symplectic
matrices, this subgroup is isomorphic to the projective special linear group
PSL(). We show that this automorphism group acts transitively on the Pauli
matrices, which implies that the ensemble is Pauli mixing and hence forms a
unitary -design. The Kerdock design described here was originally discovered
by Cleve et al. (arXiv:1501.04592), but the connection to classical codes is
new which simplifies its description and translation to circuits significantly.
Sampling from the design is straightforward, the translation to circuits uses
only Clifford gates, and the process does not require ancillary qubits.
Finally, we also develop algorithms for optimizing the synthesis of unitary
-designs on encoded qubits, i.e., to construct logical unitary -designs.
Software implementations are available at
https://github.com/nrenga/symplectic-arxiv18a, which we use to provide
empirical gate complexities for up to qubits.Comment: 16 pages double-column, 4 figures, and some circuits. Accepted to
2019 Intl. Symp. Inf. Theory (ISIT), and PDF of the 5-page ISIT version is
included in the arXiv packag
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
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