1
Zero-Free and Redundant Quaternion
Orthogonal Designs
Sarah Spence Adams, Nathaniel Karst, Jonathan Pollack, Jennifer Seberry
Abstract
The success of applying generalized complex orthogonal designs as space-time block codes
recently motivated the definition of quaternion orthogonal designs as potential building blocks
for space-time-polarization block codes. Since complex orthogonal designs with no zero entries
have proven important in applications, this paper offers a construction for an infinite family of
restricted quaternion orthogonal designs with no zero entries. The construction is particularly
useful because it requires only readily available complex orthogonal designs. This paper also
offers a construction for an infinite family of unrestricted quaternion orthogonal designs that
have a regular redundancy incorporated into their structure. This construction is useful because
it provides the first simple construction for an infinite family of quaternion designs with any
number of columns.
Key words and phrases: Orthogonal designs, complex orthogonal designs, quaternions, quaternion
orthogonal designs, space-time block codes.
AMS Subject Classification: Primary 05B20, Secondary 94A05, 62K05, 62K10
I. I NTRODUCTION AND D EFINITIONS
In recent work, Seberry, Finlayson, Adams, Wysocki, and Xia have motivated the study of
orthogonal designs over the quaternion domain for future use in signals processing as spacetime-polarization block codes [8], [13], [14]. This idea was sparked by previous work in spacetime coding and in polarization diversity [3], [15], [10], [5], [6], [7], [11], [16], [9]. Alamouti’s
scheme [3] for wireless transmissions utilizing two transmit antennas motivated Tarokh et al. [15]
to define generalized complex orthogonal designs for wireless transmissions utilizing multiple
transmit antennas. The resulting space-time block codes effectively combine space and time
diversity and provide a major step in moving the capacity of wireless communication systems
towards the theoretical limits [15], [10]. The success of space-time block codes led to the idea of
developing new codes that could effectively combine space and time diversity with an additional
form of diversity. Polarization diversity has been shown to offer performance improvements [5],
[6], [7], [11], [16], and the polarization states have been shown to be presentable via quaternion
representations [9]. Therefore, we focus our research on developing a theory for orthogonal
designs over the quaternion domain and anticipate the application of these designs as space-timepolarization codes.
S. Spence Adams is an Assistant Professor of Mathematics at the Franklin W. Olin College of Engineering,
Olin Way, Needham, MA 02492-1245. Supported in part by an AWM-NSF Mentoring Travel Grant. Email:
sarah.adams@olin.edu
N. Karst is an undergraduate at the Franklin W. Olin College of Engineering, Olin Way, Needham, MA 02492-1245.
Email: nathaniel.karst@students.olin.edu
J. Pollack is an undergraduate at the Franklin W. Olin College of Engineering, Olin Way, Needham, MA 02492-1245.
Email: jonathan.pollack@students.olin.edu
J. Seberry is a Professor in the Centre for Computer Security Research, School of IT and Computer Science,
University of Wollongong, NSW 2522, Australia
2
In this paper, we provide an infinite family of restricted quaternion orthogonal designs that
have the desirable property of containing no zero entries. This has been shown to be important
in earlier work on space-time block codes [12]. We also provide an infinite family of quaternion
orthogonal designs that incorporate a regular redundancy in their structure. It may be possible
to exploit the regular redundancy for error-control in future applications. At the least, this latter
construction provides the first simple construction for an infinite family of unrestricted quaternion
orthogonal designs for any number of columns.
Here, we provide a brief treatment of the definitions necessary to build orthogonal designs over
the quaternion domain, while Seberry et al. provide a more comprehensive treatment including
several examples [13].
Definition 1: A generalized complex orthogonal design A of type (s1 , s2 , . . . , su ) on commuting complex variables z1 , . . . ,P
zu , is an r × n matrix with entries from {0, ±z1 , . . . , ±zk , ±z1∗ ,
. . . , ±zu∗ } satisfying AH A = uh=1 sh |zh |2 In , where In is the n × n identity matrix, H is the
Hermitian transform, and so the columns of A are formally orthogonal.
In this paper, we restrict our attention to generalized complex orthogonal designs (GCODs)
where sh = 1 for all h and each variable appears exactly once per column and at most once per
row.
Definition 2: A quaternion variable a is defined as
a = a1 + a2 i + a3 j + a4 k
= (a1 + a2 i) + (a3 + a4 i)j,
where ah , h = 1, . . . , 4 are real variables and i, j, k are elements of the quaternions, a noncommutative extension of the complex numbers defined by Q = {±1, ±i, ±j, ±k}, where i2 =
j2 = k2 = ijk = −1, ij = k, jk = i, ki = j, and 1 is the unit [4].
Definition 3: If a is a quaternion variable, then its quaternion conjugate, denoted aQ , is defined
as aQ = a1 − a2 i − a3 j − a4 k.
The definition of the quaternion conjugate implies thatPthe product aaQ , equivalently aQ a, is
equal to (a1 + a2 i + a3 j + a4 k)(a1 − a2 i − a3 j − a4 k) = 4h=1 a2h . So, aaQ is real, and we write
aaQ = |a|2 .
Definition 4: The inner product of two quaternion variables a = a1 + a2 i + a3 j + a4 k, and
b = b1 + b2 i + b3 j + b4 k, is said to be a · b = aQ b.
Definition 5: Given a matrix A = (a`,m ), where au are quaternion variables, we define its
quaternion transform by AQ = (aQ
m,` ).
Definition 6: A rectangular restricted quaternion orthogonal design of type (s1 , s2 , . . . , su ),
denoted RQOD(n; s1 , s2 , . . . , su ), on the complex variables z1 , z2 , . . ., zu is an r × n matrix A with entries from {0, q1 z1 , q2 z2 , . . . , qu zu }, where each qh is an element of Q =
{±1, ±i, ±j, ± k} such that
u
X
A A=(
sh |zh |2 )In ,
Q
h=1
so again the columns of A are formally orthogonal.
Definition 7: A rectangular quaternion orthogonal design of type (s1 , s2 , . . . , su ), denoted
QOD(n; s1 , s2 , . . ., su ), on the quaternion variables a1 ,a2 , . . ., au is an r × n matrix A
with entries from {0, q1 a1 , q2 a2 , . . . , qu au }, where each qh is a linear combination of Q =
{±1, ±i, ±j, ± k} such that
u
X
A A=(
sh |ah |2 )In .
Q
h=1
3
It is clear that restricted quaternion orthogonal designs are special cases of quaternion orthogonal designs.
We say that an orthogonal design is zero-free if it contains no zeros as entries.
A. Zero-Free Restricted Quaternion Orthogonal Designs
In this Section, we show that given a maximum rate generalized complex orthogonal design A
with an even number of columns 2m, it is possible to form a companion matrix B so that A + jB
is a zero-free RQOD with 2m columns. Since the deletion of any column within an RQOD is
also an RQOD, this construction produces zero-free RQODs for any number of columns.
Definition 8: Let A be a generalized complex orthogonal design with size r × n, where r is
even. We define a companion matrix B by permuting the rows of A according to a permutation
π with order 2, so that π 2 = 1 and π −1 = π .
Given any generalized complex orthogonal design A of size r × n, where r is even, it is easy
to show that an order 2 permutation always exists so that we may always build a companion
matrix B . For example, if r1 , r2 , . . . , rr denote the r rows of A in order, then the rows of B are
r2 , r1 , r4 , r3 , . . . , rr , rr−1 in order. It is clear that this permutation consisting of pairs of mutually
exclusive row swaps has order 2, since applying the permutation to the companion matrix B will
simply re-swap the pairs, resulting the original matrix A. This permutation has cycle notation
(12)(34) · · · (r − 1, r).
In previous work, we showed that given A and a companion matrix B , A + jB is an RQOD:
Theorem 1: [13] Let A be a generalized complex orthogonal design of order r × n, and let B
be a companion matrix formed by applying a permutation π of order 2 to the rows of A. Then
A + jB is a restricted quaternion orthogonal design RQOD(n; 2, 2, . . . , 2).
Here, we will use Theorem 1 and structural properties of maximum rate generalized CODs to
produce zero-free RQODs.
Definition 9: Let A be a generalized complex orthogonal design with size r × n, where r is
an even number. We say that A is zero-maskable if the rows of A can be paired together in
mutually exclusive pairs such that the two rows in a given pair are never both zero in the same
column. If we form a companion matrix B whose rows are, in order, the zero-masking partners
of the rows of A, in order, then B is called a zero-masking companion for A.
We note that in previous work, we applied the concept of zero-masking to the columns of a
design [14]. We now believe that applying this concept to the rows of a design is more productive.
Corollary 1: Let A be a generalized complex orthogonal design of order r × n. If A is zero
maskable, then let B be a zero-masking companion for A. Then A + jB is a restricted quaternion
orthogonal design RQOD(n; 2, 2, . . . , 2) with no zero entries.
Example 1: The following simple, pure generalized complex orthogonal design A achieves the
maximum rate of 3/4 [10], but has twice the minimum decoding delay [1].












z1
0
−z2∗
−z3∗
z6∗
0
z5
−z4
0
z1
−z4∗
−z5∗
0
z6∗
−z3
z2
z2
z4
z1∗
0
−z5∗
z3∗
z6
0
z3
z5
0
z1∗
z4∗
−z2∗
0
z6






.





4
One example of a zero-masking companion for A is the following matrix B :


0
z1
z4
z5
 z1
0
z2
z3 


 −z3∗ −z5∗
0
z1∗ 


 −z2∗ −z4∗
z1∗
0 


∗
∗
∗ ,
 0
z
z
−z
6
3
2


 z∗
0
−z5∗
z4∗ 
 6

 −z4
z2
0
z6 
z5
−z3
z6
0
which is formed by applying the permutation π = (12)(34)(56)(78) to the rows of A. Clearly
the RQOD A + jB will have no zero entries.
We now review some results concerning maximum rate and minimum decoding delay generalized complex orthogonal designs (GCOD). Recall that the rate of a GCOD is defined to be
the ratio of the number of complex variables to the number of rows, and the minimum decoding
delay for a given number of columns is defined as the minimum number of rows necessary to
achieve a maximum rate design [15], [10], [2].
Result 1: [10] The maximum rate of a generalized complex orthogonal design with 2m − 1
or 2m columns is m+1
2m .
Result 2: [1] The theoretical minimum decoding delay r̃ for a¡ maximum
rate generalized
¢
2m
complex orthogonal design A with 2m − 1 or 2m columns is r̃ = m−1
.
Result 3: [1] If the number of columns 2m − 1 or 2m in a maximum rate generalized complex
orthogonal design A is congruent
¡ 2m ¢to 0, 1, or 3 modulo 4, then A can achieve the theoretical
minimum decoding delay of m−1
. If the number of columns 2m is congruent to 2 modulo 4,
¡ 2m ¢
then A cannot achieve the theoretical minimum delay, but A can achieve 2 m−1
in the best case.
Remark 1: For a maximum rate GCOD A with the number
of
columns
2m−1
or
2m congruent
¡ 2m ¢
to 0, 1, or 3 modulo 4 and the number of rows equal to m−1
, we say that A is of minimum
¡ 2m ¢
delay. If A has 2m columns where 2m is congruent to 2 modulo 4 and A has 2 m−1
rows, we
say that A is of minimum delay.
Result 4: [2] Given a maximum rate generalized complex orthogonal design A with 2m
columns, every possible pattern of m − 1 zeros must occur a positive integer number of times
within the rows of A.
We are now ready to provide our first major result:
Theorem 2: Let A be a maximum rate, minimum delay generalized complex orthogonal design
with 2m columns. Then it is possible to form a zero masking companion matrix B , so that A+jB
is an RQOD with no zero entries.
Proof: Let A be a maximum rate, minimum delay generalized complex orthogonal design
(GCOD)
4, then A has size
¡ 2m ¢ with 2m columns. By Remark 1, if 2m is congruent to¡ 02mmodulo
¢
m−1 × 2m; if 2m is congruent to 2 modulo 4, then A has size 2 m−1 × 2m. In both cases, the
number of rows is even, which can be confirmed via straight-forward algebraic manipulations.
We note that the case where 2m is congruent to 0 modulo 4 is more attractive for applications,
since its decoding delay is half that for the case where 2m is congruent to 2 modulo 4.
Theorem 1 then shows that given any companion matrix B for A, the matrix A + jB is an
RQOD. Corollary 1 shows that given any zero-masking companion matrix B for A, the matrix
A+jB is a zero-free RQOD. So, it remains to show that we can always construct a zero-masking
companion for any maximum rate minimum delay GCOD with 2m columns. In other words, we
must show that the rows of A can be partitioned into mutually exclusive pairs so that no two
rows in any given pair are zero in the same column.
5
Liang showed that for any variable zi in a maximum rate GCOD A, it is possible to rearrange
the matrix through suitable row and column permutations and multiplications of rows or columns
by −1 to obtain a submatrix:


zi
0
... 0
 0

zi
... 0
Mi


 ..

..
..
..
 .

. .
.


 0

0
. . . zi

,
Bi = 
∗

z
0
.
.
.
0
i


∗

0 zi . . . 0 



..
.. . .
.. 
H

. . 
.
.
−M
i
0
0
. . . zi∗
where Bi is 2m × 2m when A has 2m columns [10]. Liang showed that for maximum rate
GCODs, the Mi have no zero entries, and for m > 2, the variables in Mi are distinct (without
considering conjugation or sign as distinctions) [10].
We say that a maximum rate GCOD A is in “Bi form” if (1) the submatrix Bi can be created
in A through only row swaps and possible multiplications by -1; and (2) for 2m columns, Mi
is m × m. We have previously shown that any arrangement of a maximum rate A is in Bi form
for some i [1, Corollary II.12]. So, we may suppose that given A, it is in Bi form for some i.
We let Xi denote the first m rows of Bi and let Yi denote the last m rows of Bi .
By the structure of the submatrix Bi , clearly any row within Xi is zero-masked by any row
within Yi . In any manner, form mutually exclusive pairs between rows in Xi and rows in Yi . As
the number of rows in Bi is even, namely 2m, and the number of rows in Xi is equal to the
number of rows in Yi , any such pairing will always work. We call this “handling of Bi ” the first
iteration of row pairing.
Now, keeping these pairings on record, swap any column from the first m columns of A
with any of the last m columns of A. Such a column swap affects all rows in A by the same
permutation of coordinates. We have previously shown that such a swap destroys the Bi form
but moves A into Bj form for some j 6= i [1, Lemma II.8]. For convenience of discussion,
perform any necessary row swaps and multiplications by -1 until Bj appears at the top of A.
Now, consider the rows in Bj that did not appear (up to permutation) in Bi . These rows are
currently unpaired, as they were not considered in the first iteration of row pairing. We must
show that the number of unpaired rows in Xj is exactly equal to the number of unpaired rows
in Yj , since this would allow us to form zero-masked pairs between rows in Xj and rows in Yj .
We must consider the situation where a row (up to permutation) is contained in both Bi and
Bj . If certain permutations of a given row r appear in both Bi and Bj , i 6= j , we say that Bi
and Bj “share” the row r. We now show that if Bi and Bj share a row r, then they must share
exactly two rows. For, suppose r is shared by Bi and Bj . For convenience, we follow Liang’s
convention to say that an entry of ±zi or ±zi∗ has color i [10]. Then, since each row in Bi
contains an element of color i and since each row in Bj contains an element of color j , this
row r contains one variable of color i and one of color j . (Recall that no variable of any color
appears more than once in any row, by the definition of a GCOD.)
Now, consider the permutation of r that appears in Bi . Due to the structure of Bi , the element
of color i must appear along the main diagonal of Bi and the variable of color j must appear
within Mi or within −MiH . Suppose without loss of generality that it appears within Mi . Then,
this row r must appear within Xi , the top half of Bi . Then, since Bi contains both submatrices
Mi and −MiH , there must be a second row r0 in Bi that contains another variable of color j in
−MiH . Clearly, this second element of color j will have the opposite sign and conjugation of the
6
originally considered element of color j , and this second row r0 must be within Yi . Since every
row in Bi contains an element of color i, this row r0 must also contain an element of color i.
We now claim that, under the same permutation that puts r in Bj , r0 is also put in Bj : This is
true since it has been shown that every instance of zj (up to conjugation and sign) must appear
in Bj [1, Proposition II.4]. For the same reasoning that placed (say) r in Xi and r0 in Yi , we
know that within Bj , the permutation of one of these rows appears in Xj and the permutation
of the other appears in Yj . Specifically, we know that (up to signs) one of these rows contains
zj (and thus appears in Xj ) and the other contains zj∗ (and thus appears in Yj ).
So, we have shown that if Bi and Bj share one row, then they share two rows. We claim that
Bi and Bj can not share more than two rows: For maximum rate, minimum delay codes with
2 or 4 columns, we can see this from inspection [10], [1]. For more than 4 columns, if Bi and
Bj shared more than two rows, then this would require, for example, more than one variable
of color j inside Mi . This would contradict Liang’s result that the variables in Mi have distinct
colors for any maximum rate design with more than 4 columns [10].
So, we can conclude that if Bi and Bj , i 6= j , share one row, then they share exactly two
rows. Furthermore, we showed that one of the shared rows appears in Xi (or Xj ) and the other
appears in Yi (or Yj ).
Now, suppose r is a row in Bi , and without loss of generality assume it is in Xi . During
the handling of Bi , r is paired together with some row in Yi . Then, suppose a permutation of
r appears again in Bj . Our arguments above show that there is also some second row r0 that is
shared by Bi and Bj . Furthermore, as shown above, since r appears in Xi , r0 must appear in Yi .
So, during the handling of Bi , r in Xi was paired with some row in Yi and r0 in Yi was paired
with some row in Xi . (It is even possible that r and r0 were paired together during the handling
of Bi .) So, during the handling of Bj , the permutations of r and r0 that appear in Bj do not need
to be paired with any other rows; these rows already have been assigned partners. As explained
above, we know that the appropriate permutation of one of r and r0 must appear in Xj , while the
appropriate permutation of the other must appear in Yj . So, the number of unpaired rows in Xj
and the number of unpaired rows in Yj both decrease by 1. Thus, the number of unpaired rows
in Xj is equal to the number of unpaired rows in Yj . Therefore, since any row in Xj can be
paired with any row in Yj , it is still possible to produce zero masking pairs between the unpaired
rows in Xj and the unpaired rows in Yj .
We now repeat this process by performing another column swap to move A into a new Bk
form, where k 6= i, k 6= j . Again, any row in Xk that has already been paired during the handling
of a previous Bs form will imply there is another row in Yk that has also already been paired.
Again it is implied that there will always be an equal number of unpaired rows in Xk as unpaired
rows in Yk .
We continue to perform column swaps as necessary until we have handled all Bi forms, for
i = 1, 2, . . . , v , where v is the number of variables. (It may be possible to terminate this algorithm
without examining every Bi form, due to the overlap between Bi ’s, but here we give the general
algorithm.) Since every row is contained in some Bi form (in fact, each row appears in m + 1
Bi forms [10]), we are guaranteed that each row will be paired.
Since we have shown it is possible to partition the rows of A into mutually exclusive zeromasked pairs, it follows directly that we can produce a zero-masking companion matrix for A.
Thus, we have shown that given any maximum rate minimum delay GCOD A with 2m columns,
we can find a zero-masking companion matrix B such that A + jB is a zero-free RQOD with
2m columns. Since deleting any column of A + jB produces an RQOD with 2m − 1 columns,
we can use this construction to produce a zero-free RQOD with any number of columns.
7
Theorem 2 is important because it allows us to use a maximum rate minimum delay GCOD
with 2m columns to generate a zero-free RQOD with 2m columns, or with 2m − 1 columns after
a column deletion. Therefore, we have a general algorithm for producing zero-free RQODs for
any number of columns. The proof remains true when applied to maximum rate non-minimum
delay designs. However, given the choice, applications prefer minimum delay designs. Liang
[10] provided arguably the best-known algorithm for maximum rate designs with any number of
columns (but these all achieve twice the theoretical minimum delay), while other algorithms do
exist for constructing maximum rate, minimum delay designs [17].
II. F ULL Q UATERNION O RTHOGONAL D ESIGNS
In this Section, we provide a construction for quaternion orthogonal designs (QODs), as
opposed to restricted quaternion orthogonal designs (RQODs). We recall the difference between
RQODs and QODs. RQODs have entries of the form qz , where q is an element of Q =
{±1, ±i, ±j, ±k} and z is a complex variable. So, we may generally write an entry of an RQOD
as a1 q + a2 qi =a1 q + a2 q0 , where q,q0 ∈ Q and a1 , a2 are real variables. QODs have entries of
the form a, where a is a full quaternion variable of the form a = a1 + a2 i + a3 j + a4 k, where the
ai are real variables. It has thus far proven generally more difficult to obtain QODs, since it is
more difficult to handle matrix entries of full quaternion variables [13]. The following theorem
provides the first simple construction to yield QODs for any number of columns.
Theorem 3: Let A be a maximum rate complex
¡ 2m ¢ orthogonal design with 2m columns and twice
the theoretical minimum delay, hence size 2 m−1 × 2m. Then it is possible to form a companion
matrix B , so that A + jB is an unrestricted quaternion orthogonal design.
Proof: We have previously shown that a maximum rate GCOD with 2m columns that
achieves twice the theoretical minimum decoding delay has each pattern of m−1 zeros appearing
exactly twice in the rows of A [1]. Therefore, we can form mutually exclusive pairs by matching
together rows whose zero patterns are identical. The matrix formed via the combination A + jB
will keep zero entries in each place that A had a zero entries. The importance is that given the
(s, t) entry of A as the complex variable z = a+bi, and given the (s, t) entry of B as the complex
variable z 0 = c + di, where a, b, c, d are real variables, we have the (s, t) entry of A + jB as
a + bi + cj − dk, which is a full quaternion variable. Hence, A + jB is an unrestricted quaternion
orthogonal design.
Through possible column deletions in A+jB , Theorem 3 provides the first simple construction
to produce orthogonal designs whose entries are full quaternion variables for any number of
columns. Liang’s algorithm provides the necessary maximum rate GCODs that achieve twice the
theoretical minimum decoding delay [10]. Hence, this construction can be readily implemented.
We note that the ratio of nonzero entries per column to total number of rows in A + jB is
equal to this ratio in the maximum rate A.
The QODs formed via the construction described in the proof of Theorem 3 have a property
that may be exploitable in future applications of these designs as space-time-polarization codes:
A regular pattern of redundancy is built into the entries of A + jB . This redundancy is present
due to the construction technique based on orthogonal, maximum rate GCODs A and B . In
particular, for each entry of A + jB of the form zi − jzs∗ , there are entries of A + jB of the form
zi∗ + jzs , −zs∗ + jzi , and zs + jz1∗ . We conjecture that it may be possible to exploit this redundancy
for error-control coding. Our future work utilizes this redundancy in our determination of the
maximum rate for quaternion orthogonal designs.
III. C ONCLUSIONS
The first contribution of this article was to provide a technique to produce zero-free RQODs. In
particular, we showed that given a maximum rate, minimum delay generalized complex orthogonal
8
design A with 2m columns, it is possible to find a zero-masking companion matrix B such that
A + jB is a zero-free RQOD with 2m columns. Since deleting a column from any orthogonal
design produces an orthogonal design, it is possible to use the proposed construction to build
zero-free RQODs with any number of columns. We expect this zero-free construction to be
desirable in future applications of these designs as space-time-polarization codes.
The second contribution of this article was to provide a related construction to build unrestricted
QODs. In particular, we showed that given a maximum rate, twice the theoretical minimum delay,
generalized complex orthogonal design A with 2m columns, it is possible to find a companion
matrix B such that the ratio of the number of nonzero entries per column to total number of
rows in the resulting QOD A + jB is equal to the ratio of the number of nonzero entries per
column to total number of rows in the original maximum rate A. This construction produces
designs A + jB whose entries are full, rather than restricted, quaternion variables. Moreover, a
regular redundancy in the entries of A + jB is induced. It may be possible in future applications
to exploit this redundancy for error-control coding.
Both constructions in this article require only readily available maximum rate designs, either
with the minimum achievable decoding delay [17] or twice the theoretical minimum delay [10].
Therefore, these infinite families of RQODs and QODs are easily obtainable.
In future work, we build upon the ideas presented in this paper to prove the maximum rate
(maximum ratio of quaternion variables to number of rows) achievable for any number of columns
and the minimum number of rows required to achieve maximum rate.
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