Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Group Symmetries of
Complementary Code Matrices
Brooke Logan
Mathematics Department
Rowan University, New Jersey
Combinatorics and Computer Algebra- CoCoA 2015
Colorado State University
Fort Collins, Colorado
July 21, 2015
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
This work was done in collaboration with Professor
Hieu D. Nguyen at Rowan University.
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
And the topic was inspired by Greg Coxson
Figure: JMM 2014 Undergraduate Poster Session
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Barker Codes and Golay Pairs
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Barker Codes and Golay Pairs
• When a radar signal is sent out the returned signal is
compared to it by computing the aperiodic auto correlation
function, A, as follows:
8
N j
>
< P a ā , if 0  j  N 1;
i i+j
Ax (j) = i=1
(1)
>
:A ( j),
if
N + 1  j < 0.
x
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Barker Codes
Definition
A Barker Code, x, is a code of length N consisting of ±1 such that
Ax (0) = N and |Ax (j)|  1 for all others.
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Barker Codes
Definition
A Barker Code, x, is a code of length N consisting of ±1 such that
Ax (0) = N and |Ax (j)|  1 for all others.
Definition
The Barker Conjecture states that there exists no Barker
Sequences of length N > 13 and has been proven in the case of all
odd N values.
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Barker Codes
Definition
A Barker Code, x, is a code of length N consisting of ±1 such that
Ax (0) = N and |Ax (j)|  1 for all others.
Definition
The Barker Conjecture states that there exists no Barker
Sequences of length N > 13 and has been proven in the case of all
odd N values.
Note: The even case has been proven up to
N = 4 ⇥ 198048300122642987380412
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example of a Binary Code
The autocorrelation function for a single code, x = {1, 1, 1, 1} of
length N = 4, can be computed as follows:
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example of a Binary Code
The autocorrelation function for a single code, x = {1, 1, 1, 1} of
length N = 4, can be computed as follows:
• Ax (0) =
Brooke Logan
N
P
i=1
xi xic = (1 · 1) + (1 · 1) + (1 · 1) + ( 1 ·
July 21, 2015
1) = 4
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example of a Binary Code
The autocorrelation function for a single code, x = {1, 1, 1, 1} of
length N = 4, can be computed as follows:
• Ax (0) =
N
P
i=1
xi xic = (1 · 1) + (1 · 1) + (1 · 1) + ( 1 ·
• Ax (1) = Āx ( 1) =
Brooke Logan
NP1
i=1
1) = 4
c
xi xi+1
= (1 · 1) + (1 · 1) + (1 ·
July 21, 2015
1) = 1
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example of a Binary Code
The autocorrelation function for a single code, x = {1, 1, 1, 1} of
length N = 4, can be computed as follows:
• Ax (0) =
N
P
i=1
xi xic = (1 · 1) + (1 · 1) + (1 · 1) + ( 1 ·
• Ax (1) = Āx ( 1) =
• Ax (2) = Āx ( 2) =
• Ax (3) = Āx ( 3) =
Brooke Logan
NP1
i=1
NP2
i=1
NP3
i=1
1) = 4
c
xi xi+1
= (1 · 1) + (1 · 1) + (1 ·
c
xi xi+2
= (1 · 1) + (1 ·
c
xi xi+3
= (1 ·
July 21, 2015
1) =
1) = 1
1) = 0
1
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example of a Binary Code
x = {1, 1, 1, 1} with Ax = { 1, 0, 1, 4, 1, 0, 1}
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example of a Binary Code
y = {1, 1, 1, 1} with Ay = {1, 0, 1, 4, 1, 0, 1}
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example of a Binary Code
y = {1, 1, 1, 1} with Ay = {1, 0, 1, 4, 1, 0, 1}
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example of a Pair of Codes
Composite autocorrelation for pair of codes x = {1, 1, 1, 1} and
y = {1, 1, 1, 1}:
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example of a Pair of Codes
Composite autocorrelation for pair of codes x = {1, 1, 1, 1} and
y = {1, 1, 1, 1}:
• Ax + Ay
• { 1, 0, 1, 4, 1, 0,
Brooke Logan
1} + {1, 0, 1, 4, 1, 0, 1}
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example of a Pair of Codes
Composite autocorrelation for pair of codes x = {1, 1, 1, 1} and
y = {1, 1, 1, 1}:
• Ax + Ay
• { 1, 0, 1, 4, 1, 0,
1} + {1, 0, 1, 4, 1, 0, 1}
{0, 0, 0, 8, 0, 0, 0}
This is called a Golay Pair.
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Graph of Ax , Ay and Ax + Ay
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Complementary Code Matrices
Definition
A complementary code matrix (or CCM) is a N ⇥ K matrix M
whose row Gramian, B = M · M ⇤ , is diagonally regular with
diagonal entries equal to K . Here, * represents the conjugate
transpose.
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Complementary Code Matrices
Definition
A complementary code matrix (or CCM) is a N ⇥ K matrix M
whose row Gramian, B = M · M ⇤ , is diagonally regular with
diagonal entries equal to K . Here, * represents the conjugate
transpose.
p-phase: exp
⇣
2⇡i
p
+
2k⇡i
p
⌘
with k = 0, . . . , p
1
We define the set of N ⇥ K p-phase CCMs as CN,K (p).
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example
0
1
B 1
B
@ 1
1
Brooke Logan
1
1
1
1
1
i
i
1
1
1
i C
C
i A
1
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example
0
1
B 1
B
@ 1
1
Brooke Logan
1
1
1
1
1
i
i
1
1 0
1
1
C
B
i C B 1
·
i A @ 1
1
1
July 21, 2015
1
1
i
i
1
1
i
i
1
1
1 C
C=
1 A
1
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example
0
1
B 1
B
@ 1
1
Brooke Logan
1
1
1
1
1
i
i
1
0
1 0
1
1
C
B
i C B 1
·
i A @ 1
1
1
4
B 2
B
@ 2
0
July 21, 2015
2
4
0
2
2
0
4
2
1
1
1
1
i
i
i
i
1
0
2 C
C
2 A
4
1
1
1 C
C=
1 A
1
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Our Research
• Described relations between symmetries of Complmentary
Code Matrices
• Adapted an existing algorithm to speed up search for CCMs
• Classified the matrices into di↵erent equivalence classes
• Created a new construction method
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Symmetries
Coxson-Haloupek [1]
Suppose that M is an N ⇥ K CCM. Then we can obtain an
equivalent N ⇥ K CCM using the following transformations.
(i) Column multiplication by a unimodular complex number.
(ii) Column conjugate reversal.
(iii) Matrix conjugation.
(iv) Progressive multiplication by consecutive powers of a
unimodular complex number.
(v) Column permutation.
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example
0
1
B 1
B
@ i
i
1
1
i
i
Brooke Logan
1
1
i
i
1
1
1 C
C
i A
i
x1 ⇥i
!
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example
0
1
B 1
B
@ i
i
1
1
i
i
Brooke Logan
1
1
i
i
1
1
1 C
C
i A
i
0
i
B
x1 ⇥i B i
!@
1
1
July 21, 2015
1
1
i
i
1
1
i
i
1
1
1 C
C
i A
i
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example
0
i
B i
B
@ 1
1
Brooke Logan
1
1
i
i
1
1
i
i
1
1
1 C
C
i A
i
rev (x̄2 )
July 21, 2015
!
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example
0
i
B i
B
@ 1
1
Brooke Logan
1
1
i
i
1
1
i
i
1
1
1 C
C
i A
i
0
i
B
rev (x̄2 )
i
!B
@ 1
1
July 21, 2015
i
i
1
1
1
1
i
i
1
1
1 C
C
i A
i
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example
0
i
B i
B
@ 1
1
Brooke Logan
i
i
1
1
1
1
i
i
1
1
1 C
C
i A
i
conjugate
July 21, 2015
!
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example
0
i
B i
B
@ 1
1
Brooke Logan
i
i
1
1
1
1
i
i
1
1
1 C
C
i A
i
B
!B
@
conjugate
July 21, 2015
0
i
i
1
1
i
i
1
1
1
1
i
i
1
1
1 C
C
i A
i
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example
0
B
B
@
i
i
1
1
Brooke Logan
i
i
1
1
1
1
i
i
1
1
1 C
C
i A
i
Q(i)
July 21, 2015
!
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example
0
B
B
@
i
i
1
1
Brooke Logan
i
i
1
1
1
1
i
i
1
1
1 C
C
i A
i
0
1
B
Q(i)
i
!B
@ i
1
July 21, 2015
1
i
i
1
i
1
1
i
1
i
1 C
C
1 A
i
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example
0
1
B i
B
@ i
1
Brooke Logan
1
i
i
1
i
1
1
i
1
i
1 C
C
1 A
i
(x1 ,x3 )(x2 ,x4 )
July 21, 2015
!
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example
0
1
B i
B
@ i
1
Brooke Logan
1
i
i
1
i
1
1
i
1
i
1 C
C
1 A
i
0
i
B
(x1 ,x3 )(x2 ,x4 )
1
!B
@ 1
i
July 21, 2015
i
1
1
i
1
i
i
1
1
1
i C
C
i A
1
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Example
0
1
B 1
B
@ i
i
0
B
B
@
i
i
1
1
Brooke Logan
1
1
i
i
1
1
i
i
i
i
1
1
1
1
i
i
10
1
B
1 C
CB
A
@
i
i
i
i
1
1
10
1
1
B i
1 C
CB
i A@ i
i
1
July 21, 2015
1
1
i
i
1
1
i
i
1
i
i
1
i
1
1
i
10
1
B
1 C
CB
A
@
i
i
i
i
1
1
10
i
i
B 1
1 C
CB
1 A@ 1
i
i
i
i
1
1
i
1
1
i
1
1
i
i
1
i
i
1
1
1
1 C
C
i A
i
1
1
i C
C
i A
1
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Quick Proof
U = {↵1 , ↵2 , . . . , ↵K }
T = {t1 , t2 , . . . , tK } where t = 0, 1
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Quick Proof
U = {↵1 , ↵2 , . . . , ↵K }
T = {t1 , t2 , . . . , tK } where t = 0, 1
CU ⇢T M = CU ⇢T [x1 , x2 , . . . , xK ]
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Quick Proof
U = {↵1 , ↵2 , . . . , ↵K }
T = {t1 , t2 , . . . , tK } where t = 0, 1
CU ⇢T M = CU ⇢T [x1 , x2 , . . . , xK ]
= CU [! 2
Brooke Logan
t1
(x1 ), ! 2
July 21, 2015
t2
(x2 ), . . . , ! 2
tK
(xK )]
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Quick Proof
U = {↵1 , ↵2 , . . . , ↵K }
T = {t1 , t2 , . . . , tK } where t = 0, 1
CU ⇢T M = CU ⇢T [x1 , x2 , . . . , xK ]
Brooke Logan
= CU [! 2
t1
= [↵1 ! 2
t1
(x1 ), ! 2
t2
(x1 ), ↵2 ! 2
July 21, 2015
(x2 ), . . . , ! 2
t2
tK
(xK )]
(x2 ), . . . , ↵K ! 2
tK
(xK )]
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Quick Proof
U = {↵1 , ↵2 , . . . , ↵K }
T = {t1 , t2 , . . . , tK } where t = 0, 1
CU ⇢T M = CU ⇢T [x1 , x2 , . . . , xK ]
= CU [! 2
t1
= [↵1 ! 2
t1
= ⇢T [!
Brooke Logan
(x1 ), ! 2
t2
(x1 ), ↵2 ! 2
2 t1
(↵1 )x1 , !
July 21, 2015
(x2 ), . . . , ! 2
t2
tK
(xK )]
(x2 ), . . . , ↵K ! 2
2 t2
(↵2 )x2 , . . . , !
tK
2 tK
(xK )]
(↵K )xK ]
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Quick Proof
U = {↵1 , ↵2 , . . . , ↵K }
T = {t1 , t2 , . . . , tK } where t = 0, 1
CU ⇢T M = CU ⇢T [x1 , x2 , . . . , xK ]
= CU [! 2
t1
= [↵1 ! 2
t1
= ⇢T [!
(x1 ), ! 2
t2
(x1 ), ↵2 ! 2
2 t1
(↵1 )x1 , !
(x2 ), . . . , ! 2
t2
tK
(xK )]
(x2 ), . . . , ↵K ! 2
2 t2
(↵2 )x2 , . . . , !
tK
2 tK
(xK )]
(↵K )xK ]
= ⇢T CUT [x1 , x2 , x3 , . . . , xK ]
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Quick Proof
U = {↵1 , ↵2 , . . . , ↵K }
T = {t1 , t2 , . . . , tK } where t = 0, 1
CU ⇢T M = CU ⇢T [x1 , x2 , . . . , xK ]
= CU [! 2
t1
= [↵1 ! 2
t1
= ⇢T [!
(x1 ), ! 2
t2
(x1 ), ↵2 ! 2
2 t1
(↵1 )x1 , !
(x2 ), . . . , ! 2
t2
tK
(xK )]
(x2 ), . . . , ↵K ! 2
2 t2
(↵2 )x2 , . . . , !
tK
2 tK
(xK )]
(↵K )xK ]
= ⇢T CUT [x1 , x2 , x3 , . . . , xK ]
= ⇢T CUT M
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Definition
The complementary group, G , of the set of all N ⇥ K p-phase
CCMs is defined to be the group generated by the symmetries
S, P, CU , ⇢T , Q( ) and their relations given in the following lemma.
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Lemma
Let M be a N ⇥ K p-phase CCM, M = [x1 , x2 , x3 , . . . , xK ] and
xk = [m1,k , m2,k , m3,k , . . . , mN,k ]T . Then
(i) CU ⇢T M = ⇢T CUT M
(ii) CU SM = SCŪ M
(iii) CU Q( )M = Q( )CU M
(iv) CU PM = PCUP M
(v) ⇢T SM = S⇢T M
(vi) ⇢T PM = P⇢TP 1 M
(vii) SQ( )M = Q( ¯)SM
(viii) SPM = PSM
(ix) Q( )⇢T M = CUT , ⇢T Q M
(x) Q( )PM = PQ( )M
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Theorem
The cardinality of the complementary group G of CN,K (p) is
bounded by
|G |  2K +1 p K +1 K !
Brooke Logan
July 21, 2015
(2)
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Proof
Each CCM preserving operation on the matrix M can be
represented in the following form.
SPCu ⇢T Q( )
|S| = 2
|P| = K !
|CU | = p K
|⇢T | = 2K
|Q( )| = p
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Proof
Each CCM preserving operation on the matrix M can be
represented in the following form.
SPCu ⇢T Q( )
|S| = 2
|P| = K !
|CU | = p K
|⇢T | = 2K
|Q( )| = p
So this will will produce a max of
|G |  |S||P||CU ||⇢T ||Q( )| = 2K !p K 2K p = 2K +1 p K +1 K !
CCMs from M.
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Applications of the Symmetries
0
m1,1
B m2,1
M=B
@ m3,1
m4,1
Brooke Logan
July 21, 2015
m1,2
m2,2
m3,2
m4,2
m1,3
m2,3
m3,3
m4,3
1
m1,4
m2,4 C
C
m3,4 A
m4,4
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Applications of the Symmetries
0
m1,1
B m2,1
M=B
@ m3,1
m4,1
m1,2
m2,2
m3,2
m4,2
m1,3
m2,3
m3,3
m4,3
1
m1,4
m2,4 C
C
m3,4 A
m4,4
U1 = (m̄1,1 , m̄1,2 , m̄1,3 , m̄1,4 )
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Applications of the Symmetries
0
m1,1
B m2,1
M=B
@ m3,1
m4,1
m1,2
m2,2
m3,2
m4,2
m1,3
m2,3
m3,3
m4,3
1
m1,4
m2,4 C
C
m3,4 A
m4,4
U1 = (m̄1,1 , m̄1,2 , m̄1,3 , m̄1,4 )
U2 = (m2,1 m̄1,1 , m2,1 m̄1,1 , m2,1 m̄1,1 , m2,1 m̄1,1 )
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Applications of the Symmetries
0
m1,1
B m2,1
M=B
@ m3,1
m4,1
m1,2
m2,2
m3,2
m4,2
m1,3
m2,3
m3,3
m4,3
1
m1,4
m2,4 C
C
m3,4 A
m4,4
U1 = (m̄1,1 , m̄1,2 , m̄1,3 , m̄1,4 )
U2 = (m2,1 m̄1,1 , m2,1 m̄1,1 , m2,1 m̄1,1 , m2,1 m̄1,1 )
= m̄2,1 m1,1
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Applications of the Symmetries
0
m1,1
B m2,1
M=B
@ m3,1
m4,1
m1,2
m2,2
m3,2
m4,2
m1,3
m2,3
m3,3
m4,3
1
m1,4
m2,4 C
C
m3,4 A
m4,4
U1 = (m̄1,1 , m̄1,2 , m̄1,3 , m̄1,4 )
U2 = (m2,1 m̄1,1 , m2,1 m̄1,1 , m2,1 m̄1,1 , m2,1 m̄1,1 )
= m̄2,1 m1,1
Q( )CU2 CU1 M
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Applications of the Symmetries
0
1
1
1
1
1
B 1
m2,2 m2,3 m2,4 C
B
C
@ m3,1 m3,2 m3,3 m3,4 A
m4,1 m4,2 m4,3 m4,4
416 ! 411
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Applications of the Symmetries
0
1
r1
B r2 C
B
C
@ r3 A
r4
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Applications of the Symmetries
0
1
r1
B r2 C
B
C
@ r3 A ·
r4
Brooke Logan
r¯1 r¯2 r¯3 r¯4
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Applications of the Symmetries
0
1
r1
B r2 C
B
C
@ r3 A ·
r4
Brooke Logan
r¯1 r¯2 r¯3 r¯4
July 21, 2015
0
r1 r¯1
B r2 r¯1
=B
@ r3 r¯1
r4 r¯1
r1 r¯2
r2 r¯1
r3 r¯2
r4 r¯2
r1 r¯3
r2 r¯1
r3 r¯3
r4 r¯3
1
r1 r¯4
r2 r¯1 C
C
r3 r¯4 A
r4 r¯4
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Applications of the Symmetries
0
1
r1
B r2 C
B
C
@ r3 A ·
r4
Brooke Logan
r¯1 r¯2 r¯3 r¯4
0
r1 r¯1
B r2 r¯1
=B
@ r3 r¯1
r4 r¯1
r4 r¯1 = 0
July 21, 2015
r1 r¯2
r2 r¯1
r3 r¯2
r4 r¯2
r1 r¯3
r2 r¯1
r3 r¯3
r4 r¯3
1
r1 r¯4
r2 r¯1 C
C
r3 r¯4 A
r4 r¯4
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Equivalence Classes
N ⇥K
4-phase CCMs
2x4
3x4
4x4
5x4
6x4
Brooke Logan
Coxson-Russo
Algorithm
36
95
231
5246
23448
July 21, 2015
Total Number of
Equivalence Classes
2
5
24
133
1448
Hadamard
Representations
2
5
17
0
0
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Equivalence Classes
N ⇥K
4-phase CCMs
2x4
3x4
4x4
5x4
6x4
Coxson-Russo
Algorithm
36
95
231
5246
23448
Total Number of
Equivalence Classes
2
5
24
133
1448
Hadamard
Representations
2
5
17
0
0
Exhaustive 4 ⇥ 4 4-phase CCM ⇡ 4.3 million
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
4 ⇥ 4 Equivalence Class Representations
Brooke Logan
1.
[[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1], [ 1, 1, 1, 1]]
2.
[[1, 1, 1, 1], [1, 1, 1, i], [ 1, 1, i, 1], [1, 1, 1, 1]]
3.
[[1, 1, 1, 1], [1, 1, i, i], [1, 1, i, i], [ 1, 1, 1, 1]]
4.
[[1, 1, 1, 1], [1, 1, i, i], [i, i, 1, 1], [1, 1, 1, 1]]
5.
[[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1]]
6.
[[1, 1, 1, 1], [1, i, i, 1], [ 1, i, i, 1], [ 1, 1, 1, 1]]
7.
[[1, 1, 1, 1], [1, 1, 1, 1], [1, i, 1, i], [ 1, i, 1, i]]
8.
[[1, 1, 1, 1], [1, 1, i, i], [1, 1, 1, 1], [ 1, 1, i, i]]
9.
[[1, 1, 1, 1], [1, 1, i, i], [i, 1, 1, i], [1, 1, i, i]]
10.
[[1, 1, 1, 1], [1, 1, i, i], [ 1, 1, 1, 1], [1, 1, i, i]]
11.
[[1, 1, 1, 1], [1, 1, i, i], [i, i, i, i], [1, 1, i, i]]
12.
[[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, i, i], [1, 1, i, i]]
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
4 ⇥ 4 Equivalence Class Representations Continued
Brooke Logan
13.
[[1, 1, 1, 1], [1, 1, i, 1], [1, i, 1, 1], [i, 1, 1, i]]
14.
[[1, 1, 1, 1], [1, 1, i, 1], [1, i, i, i], [i, 1, 1, i]]
15.
[[1, 1, 1, 1], [1, 1, 1, i], [i, 1, i, i], [ i, i, 1, 1]]
16.
[[1, 1, 1, 1], [1, 1, 1, i], [ 1, 1, i, 1], [i, i, 1, 1]]
17.
[[1, 1, 1, 1], [1, 1, i, 1], [i, 1, 1, 1], [1, i, 1, i]]
18.
[[1, 1, 1, 1], [1, 1, i, i], [i, i, i, i], [ i, i, 1, 1]]
19.
[[1, 1, 1, 1], [1, 1, 1, 1], [i, i, i, i], [ i, i, i, i]]
20.
[[1, 1, 1, 1], [1, 1, 1, 1], [i, i, i, i], [i, i, i, i]]
21.
[[1, 1, 1, 1], [1, i, i, 1], [ i, 1, 1, i], [ i, i, i, i]]
22.
[[1, 1, 1, 1], [1, i, 1, i], [ i, i, i, i], [ i, 1, i, 1]]
23.
[[1, 1, 1, 1], [1, i, 1, i], [ 1, 1, 1, 1], [ 1, i, 1, i]]
24.
[[1, 1, 1, 1], [1, i, 1, i], [1, 1, 1, 1], [1, i, 1, i]]
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Kronecker Product
Let A and B be 2 ⇥ 2 p-phase CCMs.
✓
◆ ✓
◆
a1,1 a1,2
b1,1 b1,2
=
a2,1 a2,2
b2,1 b2,2
0
a1,1 b1,1
B a1,1 b2,1
B
@ a2,1 b1,1
a2,1 b2,1
Brooke Logan
a1,1 b1,2
a1,1 b2,2
a2,1 b1,2
a2,1 b2,2
July 21, 2015
1
a1,2 b1,1 a1,2 b1, 2
a1,2 b2,1 a1,2 b2,2 C
C
a2,2 b1,1 a2,2 b1 1, 2 A
a2,2 b2,1 a2,2 b2,2
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Concatentation Theorem
Let A and B be 4 ⇥ 2 p-phase CCMs.
20
1 0
13 0
a1,1 a1,2
b1,1 b1,2
6B a2,1 a2,2 C B b2,1 b2,2 C7 B
6B
C B
C7 B
4@ a3,1 a3,2 A , @ b3,1 b3,2 A5 = @
a4,1 a4,2
b4,1 b4,2
Brooke Logan
July 21, 2015
a1,1
a2,1
a3,1
a4,1
a1,2
a2,2
a3,2
a4,2
b1,1
b2,1
b3,1
b4,1
1
b1,2
b2,2 C
C
b3,2 A
b4,2
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Dual Pair Theorem
0
B
M=B
@ i
i
Brooke Logan
July 21, 2015
1
i
1
i
i
i
i
i
1
i
i
1
i C
C
1 A
i
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Dual Pair Theorem
0
B
M=B
@ i
i
1
i
1
i
i
i
i
i
1
i
M = A + iB
Brooke Logan
July 21, 2015
i
1
i C
C
1 A
i
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Dual Pair Theorem
0
B
M=B
@ i
i
1
i
1
i
i
i
i
i
1
i
M = A + iB
0
1
B 0
A=B
@ 0
0
Brooke Logan
1
0
0
0
0
0
1
0
1
i C
C
1 A
i
1
0
0
0
C
B
0 C
1
,B = B
@ 1
1 A
0
1
July 21, 2015
i
0
1
1
1
1
1
0
1
1
1
1 C
C
0 A
1
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Dual Pair Theorem
Assume that A and B are N ⇥ K ternary, { 1, 0, 1} “CCMs”.
Then Z = A + iB is a N ⇥ K quad-phase CCM if
(i) | An,k | + | Bn,k |= 1 {8n, k|1  n  N and 1  k  K }
(ii) BA⇤ B ⇤ A is diagonally regular
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Proof
Prove Z is quad-phase.
Prove ZZ ⇤ is Diagonally Regular.
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Proof
Prove Z is quad-phase.
Assumed | An,k | + | Bn,k |= 1 which implies
1, 1, i, i.
Prove ZZ ⇤ is Diagonally Regular.
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Proof
Prove Z is quad-phase.
Assumed | An,k | + | Bn,k |= 1 which implies
1, 1, i, i.
Prove ZZ ⇤ is Diagonally Regular.
Z = A+iB so the following holds true.
ZZ ⇤ = (A + iB)(A + iB)⇤
= (A + iB)(A⇤
⇤
= AA + i(BA
Brooke Logan
July 21, 2015
⇤
iB ⇤ )
B ⇤ A) + BB ⇤
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Proof
Prove Z is quad-phase.
Assumed | An,k | + | Bn,k |= 1 which implies
1, 1, i, i.
Prove ZZ ⇤ is Diagonally Regular.
Z = A+iB so the following holds true.
ZZ ⇤ = (A + iB)(A + iB)⇤
= (A + iB)(A⇤
⇤
= AA + i(BA
⇤
iB ⇤ )
B ⇤ A) + BB ⇤
QED!
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Constructing the Equivalency Classes
4-phase CCMs
2x4
3x4
4x4
5x4
6x4
Brooke Logan
Equivalence
Classes
2
5
24
133
1448
July 21, 2015
Kronecker
Product
n/a
n/a
2
n/a
2
Concatentation
Theorem
2
1
6
3
27
CCM
Dual Pair
2
5
22
94
471
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
For More Information...
• CCM Website: elvis.rowan.edu/datamining/ccm/
• Equivlency Class Links:
elvis.rowan.edu/datamining/ccm/equivalence/
• Our Paper: arxiv.org/abs/1506.00011
• My email: brookelogan974@gmail.com
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
Thank You!
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
References I
G. Coxson and W. Haloupek
Construction of Complementary Code Matrices for Waveform
Design.
Aerospace and Electronic Systems, IEEE Transactions on
(Volume:49 , Issue: 3 ) November 25, 2012.
G. Coxson and J. Russo
Efficient Exhaustive Search for Binary Complementary Code
Sets.
Information Sciences and Systems (CISS), 2013 47th Annual
Conference on 20-22 March 2013
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices
Introduction
Known Symmetries
Searching for N ⇥ 4 CCMs
Using the symmetries
Construction Methods
Conclusion
References II
R. Gibson
Quaternary Golay Sequence Pairs Master’s Thesis.
Simon Fraser University (Fall 2008).
G. Coxson, B. Logan, H. Nguyen
Row-Correlation Function: A New Approach to
Complementary Code Matrices,
Proceedings of 52nd Annual Allerton Conference on
Communication, Control, and Computing (2014), 1358-1361.
B. Logan and H. Nguyen
Group Symmetries of Complementary Code Matrices,
CoRR 2015
Brooke Logan
July 21, 2015
Group Symmetries of Complementary Code Matrices