Nearly Orthogonal (+1,-1) Matrices in Higher Dimensions: An Application to Signal Processing

Jennifer Seberry and Tianbing XiaUniversity of Wollongong

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JSPS-DST Asian Academic Seminar 2013, Discrete Mathematics and its Applications

November 3-10, 2013University of Tokyo

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 theseApplications in spectroscopy (see Horadam)

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Examples of proprietyShlichta’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 matricesHigher 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 matricesShlichta 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

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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 workKathy Horadam & C Lin extended this construction to any group developed Hadamard matrices and all co-cyclic Hadamard matrices: they extended perfect binary arrays

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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 butThe group developed higher 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

rr

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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: ITake 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: IIThe group developed construction from difference sets other than Menon difference setsUse 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 matricesUsing 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 Reed-Solomon codes)

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Example from Hammer & Seberry

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Near orthogonal and higher dimensional: IVUse 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 formComplex 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 applicationsAny thing that previously used zero autocorrelation can be adapted to near zero autocorrelationApplications in spectroscopy, acoustics, communication, security(?)....

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References and AcknowledgementsWe 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): 233-246 (2010)Warwick de Launey and David A Levin, : (1, -1)-Matrices with Near-Extremal 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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ReferencesKathy 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, 2003Thomas Pertsch, Ulf Peschel, Falk Lederer, Jonas Burghoff, Matthias Will, Stefan Nolte, and Andreas Tünnermann: Discrete diffraction in two-dimensional 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) 566-572J Hammer and Jennifer Seberry, Higher dimensional orthogonal designs and applications, IEEE Trans Inf Th 27 (1981) 772-779Yixian Yang, The proofs of some conjectures on higher dimensional Hadamard matrices, Kexue Tongbao (translated from Chinese) 31, 24 (1986) 1662-1667 27