A Family of Quasi Orthogonal Matrices
with Two Levels Constructed via Twin Prime Difference Sets
N. A. Balonin* and Jennifer Seberry†
October 4, 2014
We show that if B is the incidence matrix of a (υ, k, λ) difference set, then there
exists a two-level quasi-orthogonal matrix, S. This result gives new infinite
families.
We apply this result to the twin prime family of difference sets obtaining a new
infinite family.
Keywords: Hadamard matrices; quasi-orthogonal matrices; difference sets; Hadamard
difference sets; 05B20.
1. Introduction
Difference sets are of considerable use and interest to image processing (compression, masking)
to statisticians undertaking medical or agricultural research, to position smaller telescopes to make
very large deep space telescopes and to spacing the tread on rubber tyres for vehicles.
In this and further papers we use some names, definitions, notation differently than we have in
the past [2]. This, we hope, will cause less confusion, bring our nomenclature closer to common usage
and conform for mathematical purists. We have chosen the use of the word level, instead of value for
the entries of a matrix, to conform to earlier writings. We note that the strict definition of an
orthogonal matrix, X, of order n, is that X⊺X =XX⊺=In where In is the identity matrix of order n. In
this paper we consider X⊺X=XX⊺=ω In where ω is a constant. We call these quasi-orthogonal
matrices [4, 6].
*
Saint Petersburg State University of Aerospace Instrumentation, 67, B. Morskaia St., 190000, St. Petersburg,
Russian Federation. Email: korbendfs@mail.ru
† Centre for Computer and Information Security Research, School of Computer Science and Software
Engineering, EIS, University of Wollongong, NSW 2522,
Australia. Email: jennifer seberry@uow.edu.au
1
2. Definitions
This paper studies the construction of some quasi-orthogonal matrices. We examine the Singer
difference sets.
Definition 1. Let D = {d1, d2, …, dk} be a subset of the integers 0, 1, 2, … , v − 1. If the
collection Δ = {di – dj: i, j, 1 …. k, i not equal to j} contains each element 1, 2, … , v−1 exactly λ
times D will be called a (v, k, λ) difference set.
This is said to be additive notation. Equivalently, (ν, k, λ) difference set in a multiplicative group
G of order ν is k–subset D of G such, that every element g≠1 of G has exactly λ
representations g=d1d2–1 with d1, d2 from G.
Parameter ν is called the order of the difference set.
Difference sets due to James Singer (1938, [13]) appeared first. Marshall Hall [9] wrote an
extensive survey in 1956. Difference sets of regular Hadamard matrices were discussed in [14, 15]
We note that for every (υ, k, λ) difference set there is a complementary (υ, υ − k, υ − 2k + λ)
difference set made by choosing the subset of 1, 2, … , v not in D.
Definition 2. Let D = {d1, d2, …, dk} be a difference set. Then the v×v, matrix B = (bij) is said to
be the incidence matrix of D if bij = 1 for j − i in D and 0 is j − i is not in D.
Example 1. Let D be the subset {1, 3, 4, 5, 9} of the integers 0, 1, 2, … , 10. Hence we take all
the differences modulo 11. Then Δ contains 1 – 3 = −2 = 9 ; 1 − 4 = −3 = 8; 1 – 5 = −4 = 7; 1 – 9 =
−8 = 3; 3 – 1 = 2; 3 – 4 = −1 = 10; 3 – 5 = −2 = 9; 3 – 9 = −6 = 5; 4 – 1 = 3; 4 – 3 =1; 4 – 5 = −1 = 10;
4 – 9 = −5 = 6; 5 – 1 = 4; 5 − 3 = 2; 5 − 4 = 1; 9 – 1 = 8; 9 – 3 = 6 = 5; 9 – 4 = 5; 9 – 5 = 4; which is
each non-zero integer 0, 1, 2, …10 exactly twice.
The incidence matrix of D is B = circ(0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0).
Definition 3. Let p and q = p + 2 be primes. Twin prime difference sets have parameters
(pq,
pq  1 pq  1
,
)
2
4
Examples of matrices with these parameters are placed in the survey of Marshall Hall [9], they
were generalized in [16].
Definition 4. The values of the entries of a matrix are called levels.
There is a trivial one-level matrix for every order n; it is the zero matrix of order n. Hadamard
matrices are two-level matrices and symmetric conference matrices and weighing matrices are threelevel matrices. Quasi-orthogonal matrices with maximal determinant of odd orders have been
discovered with a larger number of levels [2].
2
We will denote the level/values of two-level matrices as a, –b; for positive 0 ≤ b ≤ a = 1.
Definition 5. A real square matrix S of order n is called quasi-orthogonal if it satisfies
S TS = SST = ω In, where In is the n×n identity matrix, and ω is a constant real number.
In this and future work we will only use quasi-orthogonal to refer to matrices with moduli of real
elements ≤ 1 [4], where at least one entry in each row and column must be 1. Hadamard matrices [7],
symmetric conference matrices [3], and weighing matrices [12] are the best known of these matrices
with entries from the unit disk [11].
The matrix orthogonality equation S TS = SST = ω In, is a set of n2 scalar equations, giving two
kinds of formulae: g(a, b) = ω, there are n such equations, and f(a, b)=0, there are n2–n such equations.
We concentrate on two of them: g(a, b)=ω, f(a, b)=0.
The entries in ωIn which are on the diagonal, are given by the radius equation ω=g(a, b), they
depend on the choice of a, b. If a=1, then ω ≤ n.
The maximal weight ω = n arises from Hadamard matrices, symmetric conference matrices
have ω = n − 1. Quasi-orthogonal matrices can have also irrational values for the weight. The second
equation f(a, b)=0 we name the characteristic equation, as it allows us to find a formulae for level
b ≤ a. Level a = 1 is pre-determined for all quasi-orthogonal matrices.
3. Construction for Two-Level Quasi-Orthogonal Matrices
We now use difference sets to construct two-level quasi-orthogonal matrices.
Construction 1 Consider twin prime difference sets ( pq,
pq  1 pq  1
,
) [9, 16], where the
2
4
parameters p and q = p + 2 are primes. Then there exists an twin prime difference set. Then, when its
incidence matrix has its ones replaced by a and zeros with –b , we obtain a two-level quasi-orthogonal
matrix S satisfying the orthogonality equation (1)
S TS = SST = ω In
(1)
Giving the radius equation (2)
(pq+1) a2 +(pq −1) b2 = 2ω
(2)
and the characteristic equation (3) for levels a and b
(pq+1) a2 − 2(pq+1) ab + (pq − 3) b2 = 0
(3)
If b > a we have to choose the second level to be 1/b, to ensure entries are from the unit disk for the
3
complementary difference set for b 
pq  1
pq  1  2 pq  1
pq
a, det(S)  ω 2 .
For the “+” sign, choosing a = 1, we have the first solution
b
pq  1
pq  1  2 pq  1
, ω
pq  1 pq  1 2

b .
2
2
For the “–” sign, choosing a = 1, and inverted level 1/b we have
b
pq  1  2 pq  1
pq  1 pq  1 2

b .
, ω
2
2
pq  1
for the complementary difference set (pq,
pq  1 pq  3
,
) with smaller b and smaller determinant.
2
4
Let us take as a principal solution the matrix with higher determinant, the complementary
solution has the same structure: it is an analogue of the equivalent version for an Hadamard matrix.
The first (extremal) solution has been called a Mersenne matrix [6]. The conditions for existence of
Mersenne matrices for some orders 4t − 1 has been observed in [1], [5].
Example 2. For the p=3, q=5 difference set {15, 8, 4} leads to the quasi-orthogonal matrix
circ(−b , a, −b, −b, a, a, −b, a, −b, a, a, a, a, −b, −b) with othogonality equation
S TS = SST = (8a2 + 7b2) In = ω In,
The radius equation is 8a2 + 7b2 = ω and the characteristic equation is 16a2 − 32ab + 12b2 = 0,
15
det S   ω 2 . The principal solution, see Fig. 1a (entries a and −b given as white and black squares),
has
a  1, b 
16
16  2 16

2
28
 11.1111.
, ω  8a 2  7b 2  8 
9
3
The second solution, circ(a, −b, a, a, −b, a, −b, a, −b, a, a, a, a, −b, −b), from the
complementary difference set, see Fig. 1b (which has a smaller b than the first value of b, here and
later, given as red squares), has
a  1, b 
16  2 16 1
 , ω  7a 2  8b 2  7  2  9.
16
2
The two determinants are 6.9691×107 and 1.43×107 respectively.
4
(a) The principal solution
(b) The complementary solution
Figure 2: Quasi-orthogonal matrices for order 15: CM(15) part of the Mersenne Family
4. Acknowledgements
The authors would like to acknowledge Professor Mikhael Sergeev for his advice regarding the
content of this paper. The authors also wish to sincerely thank Tamara Balonina for converting this
paper into printing format. We acknowledge the use of the http://www.mathscinet.ru and
http://www.wolframalpha.com sites for the number and symbol calculations in this paper.
5. Conclusion
We note that there exist (υ, k, λ) difference sets for υ = 4t + 1, 4t, 4t − 1, 4t – 2.
The La Jolla Difference Set Repository [8] gives many parameter sets which can make circulant
incidence matrices from difference sets. It opens new possibilities for image processing (compression,
masking) and other areas we touched in our introduction.
References
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Balonin N. A., and Seberry, Jennifer. A review and new symmetric conference matrices.
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