Nearly Orthogonal (+1,-1) Matrices in
Higher Dimensions: An Application to
Signal Processing
Jennifer Seberry and Tianbing Xia
University of Wollongong
JSPS-DST Asian Academic Seminar 2013,
Discrete Mathematics and its Applications
November 3-10, 2013
University of Tokyo
1
Why Study Higher
Dimensional?
Shlichta, 1979, said: “proper higher dimensional
matrices may find application in error-correcting codes,
where their hierarchy of orthogonalities permit a variety
of checking procedures. Other types of Hadamard
matrices may be of use in security codes on the basis of
their resemblance to random binary matrices”.
Communication encoding uses frequency, phase, time,
amplitude, polarity (light) to increase bandwidth;
believe we can use higher dimensions to augment these
Applications in spectroscopy (see Horadam)
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Examples of propriety
Shlichta’s , de Launey’s, Yang’s and Hammer&Seberry’s
higher dimensional Hadamard matrices can be proper:
that is all the 2-dimensional subspaces are orthogonal
i.e. Hadamard matrices
Higher n-dimensional Hadamard matrices made from
Walsh functions are not proper in any dimension smaller
than n
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Examples of 2^3 Hadamard
matrices
Shlichta gave a construction for 2^n
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Shlichta 1979
•In his beautiful paper on “Higher dimensional matrices”
Shlichta constructed higher dimensional matrices of
side 2 which are orthogonal…but what does orthogonal
mean?
•Proper: every two dimensional subspace is an
Hadamard matrix…this works for complex, jacket,
Butson, weighing matrices, and orthogonal designs
•Propriety: when the k dimensional subspace in
direction m is orthogonal that is the k-1 dimensional
subspaces are super imposed
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Example from Hammer and Seberry
From IEEE
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Statement of de Launey and
Yang’s construction
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Example of de Launey’s and
Yang’s construction in higher
dimensions
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Higher Dimensional Version
Example
9
Higher Dimensional Version
Example
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Example of Hammer & Seberry’s
difference set developed construction
This only applies to Menon difference sets including
Ming-yuan Xia’s splendid work
Kathy Horadam & C Lin extended this construction to
any group developed Hadamard matrices and all
co-cyclic Hadamard matrices: they extended perfect
binary arrays
11
Example of Seberry & Xia’s difference
set developed construction in higher
dimensions
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Hammer & Seberry: Other
Examples
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Comparison of efficiency of
construction
Higher dimensional matrices via group development
will be faster to construct than the de Launey-Yang
construction but
The group developed higher
r dimensional Hadamard
matrices depend on difference sets and Menon
difference sets have been hard to find: but we can use
them for near orthogonal higher dimensional matrices
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What if we change orthogonal
to near orthogonal?
 de Launey and Levin were looking at this problem in
2009
Windpassinger, Pertsch, Cioppa have more recently
looked at near orthogonal higher dimensional matrices
for codes, diffraction and experimental design
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Near orthogonal and higher
dimensional: I
Take any proper higher dimensional Hadamard matrix
and remove one coordinate from each direction. The
resulting matrix will have had either (1,1) or (1, -1) or
(-1,1) or (-1,-1) removed from the inner product of any
two rows in a 2-dimensional plane. Thus we will have
near orthogonality for any such pair
•However this may be very time consuming or even
difficult to construct, repair or analyse…
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Near orthogonal and higher
dimensional: II
The group developed construction from difference sets
other than Menon difference sets
Use Paley, Stanton-Sprott and Singer difference sets
with parameters 2-(4t-1,2t-1,t-1). These can be group
developed for 4t-1 any odd prime power to get near
orthogonal higher dimensional matrices
Using group developed 2 x 2 matrices we can use the
Hammer-Seberry construction to extend to any number
of dimensions; we note that in some pairs of dimensions
there will be n^2 correlation but other pairs are fine
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Example using difference sets
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Near orthogonal and higher
dimensional: III
•Use the Hammer and Seberry construction to make
Williamson Hadamard into high dimensions.
•With appropriate definitions this can also be extended
to Williamson type, Ito’s, Goethal-Seidel type,
•Wallis-Whiteman type, Xia type and others to form
higher dimensional matrices…then use deletion of any
coordinate as above
•It is easy to see how this can be used for Hadamard
error codes with 8t codewords of length 4t, distance 2t
and dimension 8t (we have yet to relate this to Reed19
Solomon codes)
Example from Hammer &
Seberry
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Near orthogonal and higher
dimensional: IV
Use the ideas of the Hammer & Seberry for the
Williamson construction extended to high dimensions
but with
 sequences with small correlation
 two supplementary difference sets extended to a group
generated form
Complex sequences
Higher dimensional quaternion based codes
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Applications to signal
processing
The high dimensional orthogonal matrix can be used to
encode the whole of a large file. For example, a database
or a video with error-correction for single errors and
burst errors by choice of the encoding matrix
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More uses and applications
Any thing that previously used zero autocorrelation can
be adapted to near zero autocorrelation
Applications in spectroscopy, acoustics,
communication, security(?)....
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References and
Acknowledgements
We acknowledge Kathy Horadam’s magnificent,
pioneering and insightful research and book:
K.J. Horadam, Hadamard Matrices and their
Applications, Princeton University Press, Princeton, 2006
Y.X. Yang, Theory and Applications of Higher-
Dimensional Hadamard Matrices, Kluwer, Dordrecht,
2001.
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References
Warwick de Launey and Daniel M Gordon: On the density of the set of
known Hadamard orders. Cryptography and Communications 2(2): 233246 (2010)
Warwick de Launey and David A Levin, : (1, -1)-Matrices with NearExtremal Properties. SIAM J. Discrete Math. 23(3): 1422-1440 (2009)
Warwick de Launey: On the asymptotic existence of Hadamard
matrices. J. Comb. Theory, Ser. A 116(4): 1002-1008 (2009)
Warwick de Launey, Kathy J. Horadam: A Weak Difference Set
Construction for Higher Dimensional Designs. Des. Codes Cryptography
3(1): 75-87 (1993)
Warwick de Launey: A note on N -dimensional Hadamard matrices of
order 2t and Reed-Muller codes. IEEE Transactions on Information
Theory 37(3): 664- (1991)
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References
Kathy J. Horadam: Hadamard matrices and their applications: Progress
2007-2010. Cryptography and Communications 2(2): 129-154 (2010)
ChristophWindpassinger and Robert F. H. Fischer: Low-Complexity
Near-Maximum-Likelihood Detection and Precoding for MIMO Systems
using Lattice Reduction , ITW2003, Paris, France, March 31 – April 4,
2003
Thomas Pertsch, Ulf Peschel, Falk Lederer, Jonas Burghoff, Matthias
Will, Stefan Nolte, and Andreas Tünnermann: Discrete diffraction in twodimensional arrays of coupled waveguides in silica, Optics Letters, Vol.
29, Issue 5, pp. 468-470 (2004)
Thomas M Cioppa, Efficient near orthogonal and space-filling
experimental designs for high dimensional complex designs, Dissertation,
Naval Postgraduate School, Monterey, CA, (2002)
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References +
Paul J Shlichta, Higher dimensional Hadamard
matrices, IEEE Trans Inf Th 25, 5 (1979) 566-572
J Hammer and Jennifer Seberry, Higher dimensional
orthogonal designs and applications, IEEE Trans Inf Th
27 (1981) 772-779
Yixian Yang, The proofs of some conjectures on higher
dimensional Hadamard matrices, Kexue Tongbao
(translated from Chinese) 31, 24 (1986) 1662-1667
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