Embedding cocyclic D-optimal designs in cocyclic Hadamard matrices

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by  Víctor Álvarez

EMBEDDING COCYCLIC D-OPTIMAL DESIGNS IN COCYCLIC
2 HADAMARD MATRICES
1
0
2 V´ICTOR A´LVAREZ∗, JOSE´ ANDRE´S ARMARIO∗, MAR´IA DOLORES FRAU∗, AND
n FE´LIX GUDIEL∗
a
J
Abstract. Inthispaperamethodforembeddingcocyclicsubmatriceswith“large”determinants
0
oforders2tincertaincocyclicHadamardmatricesoforders4tisdescribed(tanoddinteger). Ifthese
2
determinants attain the largest possible value, we are embedding D-optimal designs. Applications

] tothepivotvaluesthatappear whenGaussianEliminationwithcompletepivotingisperformedon
O thesecocyclicHadamardmatricesarestudied.
C
. Key words. D-optimal Designs, Cocyclic Hadamard matrices, Embedded matrices, Gaussian
h
eliminationpivots.
t
a
m
AMS subject classifications. 05B20,15A15,65F40,65F05.
[

1
v 1. Introduction. A Hadamard matrix H of order n is an n×n matrix with
2
elements ±1 and HHT = nI. A Hadamard matrix is said to be normalized if it
3
3 has its first row and column all 1′s. We can always normalize a Hadamard matrix
4 by multiplying rows and columns by −1. These matrices must have order 1, 2 or a
.
1 multipleof4. ItisconjecturedthatHadamardmatricesexistforeveryn≡0(mod4).
0
Althoughnoproofofthisfactisknown,thereismuchevidenceaboutitsvalidity(see
2
1 [18] and the references there cited).
:
v Two Hadamard matrices H and H are called equivalent (or Hadamard equiv-
1 2
i
X alent, or H-equivalent) if one can be obtained from the other by a sequence of row
r and/orcolumn interchangesandrowand/or columnnegations. The questionof clas-
a
sifying Hadamard matrices of order n ≥ 32 remains unanswered and only partial
results are known [19].
ProblemsinvolvingHadamardmatricessoundveryeasy,buttheyarenotoriously
difficult to solve. One interesting open problem, among others, is the question of
the largest pivot encountered during the process of Gaussian Elimination (GE) with
completepivotingforann×nHadamardmatrixH (thesocalled“growthfactor”for
H). Traditionally, backward error analysis for Gaussian Elimination (GE), see e.g.
[7], on a matrix A= a(1) is expressed in terms of the growth factor
ij
h i
∗DepartmentofAppliedMathI,UniversityofSeville,Avda. ReinaMercedess/n,41012Seville,
Spain({valvarez,armario,mdfrau,gudiel}@us.es).
1
2 V.A´lvarez,J.A.Armario,M.D.Frau,andF.Gudiel
max |a(k)|
i,j,k ij
g(n,A)=
max |a(1)|
i,j ij
whichinvolvesalltheelementsa(k), k=1,2,...,n,thatoccurduringtheelimination
ij
for a choice of pivoting strategy given. Matrices with the property where no row and
columnexchangesareneededduringGEwithcompletepivotingarecalledcompletely
pivoted (CP) or feasible. In other words, at each step of the elimination the element
oflargestmagnitude(the“pivot”,denotedbyp )islocatedatthetopleftpositionof
k
every appearing submatrix during the process. If A(k) denotes the absolute value of
k×k principal minor of A, then mathematically A being CP means (or is equivalent
to) that for each k, we have that A(k) is greater than or equal to the absolute value
of any other k×k determinant that includes the first k−1 rows and columns. This
isnotnecessarilythe maximumk×k minorofA,butonlythe maximumk×k minor
of A when its first k−1 rows and columns are fixed.
For a CP matrix A we have
max{p ,p ,...,p }
1 2 n
g(n,A)= .
|a(1)|
11
If a matrix is notinitially CP,by applyingrow andcolumnoperationswith complete
pivoting we can always bring it in CP form.
The following lemma gives a useful relation between pivots and minors.
Lemma 1.1. [6]LetAbeaCP matrix. Themagnitude ofthepivots which appear
after application of GE operations to A is given by
A(j)
p = , j =1,2,...,n, A(0)=1.
j
A(j−1)
In1969,Cryer[6]conjecturedthatifAisarealn×nmatrixsuchthat|a |≤1,
i,j
theng(n,A)≤n,withequalityiffAisaHadamardmatrix. In1991Gould[14]proved
thatthefirstpartoftheconjectureisnottrue. Hefoundmatriceswithgrowthbigger
than their orders. Thus, the following remains open:
Conjecture(Cryer) The growth of a Hadamard matrix is its order.
Thisconjecturehasonlybeenprovenforn=4,8,12and16(see[6,9,22]). Great
difficulty arisesinthe study ofthis problembecauseH-equivalenceoperationsdonot
preservepivots,i.e. theH-equivalentmatricesdonotnecessarilyhavethesamepivot
pattern. For instance, for n = 16 there are 34 pivot patterns although there are
EmbeddingcocyclicD-optimaldesignsincocyclicHadamardmatrices 3
only 5 equivalence classes of Hadamard matrices for this order. Furthermore, many
pivot patterns can be observed by permuting the rows and columns of any 20 by 20
Hadamard matrix and there are just 3 inequivalent matrices.
However,the existence ofD-optimaldesigns(andotherspecific submatriceswith
concrete determinants) that exist embedded in a Hadamard matrix have provided
some clues on the pivot patterns (see [26, 24]).
A D-optimal design of order n is an n×n (1,−1)-matrix having maximal deter-
minant. Hereandthroughoutthispaper,forconvenience,wheneveradeterminantor
minor is mentioned, we mean its absolute value. The question of finding the deter-
minant of a D-optimal design of order n is an old one which remains unanswered in
general.
In 1893 Hadamard proved in [15] that for every (−1,1)-matrix M,
n
(1.1) det(M)≤n2.
We recallthatthe originalinterestin Hadamardmatricesstemmedfromthe fact
that these matrices are the only ones that satisfy equality in (1.1).
Thishasledtofurtherstudyandtighterboundsforthemaximaldeterminantfor
all (−1,1)-matrices of order n 6= 0 (mod 4) have been found (see [5, 10, 11, 28, 20]).
For instance, when n ≡ 2 (mod 4), Ehlich in [10] and independently Wojtas in [28]
proved that
n−2
(1.2) det(M)≤(2n−2)(n−2) 2 .
In order for equality to hold, it is required that there exists a (−1,1)-matrix M of
L 0
ordernsuchthatMMT = ,whereL=(n−2)In+2Jn. Here,asusual,In
0 L 2 2
(cid:20) (cid:21)
denotes the identity matrix of order n, and J denotes the n×n matrix all of whose
n
entries are equal to one. In these circumstances, it may be proven that, in addition,
2n−2 is the sum of two squares,a condition which is believed to be sufficient (order
138 is the lowest for which the question has not been settled yet, [13]).
In the early 90s, a surprising link between homological algebra and Hadamard
matrices[16]ledtothestudyofcocyclicHadamardmatrices[17]. Hadamardmatrices
of many types are revealed to be (equivalent to) cocyclic matrices [8, 18]. Among
them, Sylvester Hadamard matrices, Williamson Hadamard matrices, Ito Hadamard
matrices and Paley Hadamard matrices. Furthermore, the cocyclic construction is
the most uniform construction technique for Hadamard matrices currently known,
and cocyclic Hadamard matrices may consequently provide a uniform approach to
the famous Hadamard conjecture.
4 V.A´lvarez,J.A.Armario,M.D.Frau,andF.Gudiel
The main advantages of the cocyclic framework concerning 4t by 4t Hadamard
matrices may be summarized in the following facts:
• ThetesttodecidewhetheracocyclicmatrixisHadamardrunsinO(t2)time,
better thanthe O(t3)algorithmfor usual(notnecessarilycocyclic)matrices.
• Thesearchspaceisreducedtothe setofcocyclicmatricesoveragivengroup
(that is, 2s matrices, provided that a basis for cocycles over G consists of s
generators),insteadofthe wholesetof216t2 matricesoforder4t withentries
in {−1,1}.
In [1] was shown that the cocyclic technique can certainly be extended to handle
the maximal determinant problem at least when n ≡ 2 (mod 4). More concretely,
the study focused on cocyclic matrices over the dihedral group of 2t elements with t
odd. Based on exhaustive and heuristic searches, three algorithms for constructing
cocyclic matrices with large determinants were provided.
In this paper we are interested in embedding (cocyclic) submatrices of orders
2t with large determinants in certain cocyclic Hadamard matrices of orders 4t. If
these determinants attain the largest possible value, we are embedding D-optimal
designs. Also,we discuss the relationbetween the existence of these submatrices and
the growth factor for these Hadamard matrices.
In Section 2, an algebraic formalism (in terms of cocycles) to describe two com-
binatorial operations on a matrix (eliminate and add certain rows and columns) is
provided. As a consequence of this formalism a method arises for embedding (co-
cyclic)submatricesoforders2twithlargedeterminantsincertaincocyclicHadamard
matrices of orders 4t. In Section 3, we connect the existence of specific matrices em-
bedded in cocyclic Hadamard matrices of order 20 with the values of the pivots that
appear when we perform Gaussianelimination with complete pivoting on them. The
last section is devoted to conclusions and future work.
Notation. Throughout this paper we use − for −1 and 1 for +1. We write H for a
Hadamard matrix and D for a D-optimal design of order j. The notation D ∈ H
j j
means D is embedded in H.
j
2. Cocyclic D-optimal designs embedded in Cocyclic Hadamard ma-
trices. Assume throughout that G = {g = 1, g ,...,g } is a multiplicative group,
1 2 n
not necessarily abelian. Functions ψ:G×G→h−1i∼=Z which satisfy
2
(2.1) ψ(g ,g )ψ(g g ,g )=ψ(g ,g )ψ(g ,g g ), ∀g ,g ,g ∈G
i j i j k j k i j k i j k
arecalled(binary)cocycles(overG)[21]. Acocycleisacoboundary∂φifitisderived
from a set mapping φ:G→h−1i by ∂φ(a,b)=φ(a)φ(b)φ(ab)−1.
Acocycleψ isnaturallydisplayedasacocyclic matrix (or G-matrix)M ;thatis,
ψ
EmbeddingcocyclicD-optimaldesignsincocyclicHadamardmatrices 5
theentryinthe(i,j)thpositionofthecocyclicmatrixisψ(g ,g ),forall1≤i,j ≤n.
i j
A cocycle ψ is normalized if ψ(1,g ) = ψ(g ,1) = 1 for all g ,g ∈ G. The
j i i j
cocyclic matrix coming from a normalized cocycle is called normalized as well. Each
unnormalized cocycle ψ determines a normalized one −ψ, and vice versa. Therefore,
we may reduce, without loss of generality, to the case of normalized cocycles.
The set of cocycles forms an abelian groupZ(G) under pointwise multiplication,
and the coboundaries form a subgroup B(G). A basis B for cocycles over G consists
of some elementary coboundaries ∂ and some representative cocycles, so that every
i
cocyclic matrix admits a unique representation as a Hadamard (pointwise) product
M =M ◦...◦M ◦R,in terms ofsome coboundary matrices M anda matrix
∂i1 ∂iw ∂ij
R formed from representative cocycles.
Recallthateveryelementary coboundary∂ isconstructedfromthecharacteristic
d
set map δ :G→{−1,1} associated with an element g ∈G, so that
d d
−1 g =g ,
d i
∂ (g ,g )=δ (g )δ (g )δ (g g ) for δ (g )=
d i j d i d j d i j d i
1 g 6=g .
(cid:26) d i
Remark 2.1. ([2, Lemma 1])
In particular, for d6=1, every row s∈/ {1,d} in M contains precisely two −1s,
∂d
which are located at the positions (s,d) and (s,e), for g = g−1g . Furthermore, the
e s d
first row is always formed by 1s, while the d-th row is formed all by −1s, excepting
the positions (d,1) and (d,d).
Although the elementary coboundaries generate the set of all coboundaries,they
might not be linearly independent (see [3] for details).
Let G (M) (resp. G (M)) be the Gram matrix of the rows (resp. columns) of
r c
M,
G (M)=MMT, (resp.G (M)=MTM).
r c
The Gram matrices of a cocyclic matrix can be calculated as follows.
Proposition 2.2. ([18, lemma 6.6])
Let M be a cocyclic matrix,
ψ
(2.2) [G (M )] =ψ(g g−1,g ) ψ(g g−1,g),
r ψ ij i j j i j
g∈G
X
(2.3) [G (M )] =ψ(g ,g−1g ) ψ(g,g−1g ).
c ψ ij i i j i j
g∈G
X
6 V.A´lvarez,J.A.Armario,M.D.Frau,andF.Gudiel
If a cocyclic matrix M is Hadamard, we say that the cocycle involved, ψ, is
ψ
orthogonal and M is a cocyclic Hadamard matrix. The cocyclic Hadamard test
ψ
asserts that a normalized cocyclic matrix is Hadamard if and only if every row sum
(apart from the first) is zero [17]. In fact, this is a straightforward consequence of
Proposition 2.2.
Analyzing this relation from a new perspective, one could think of normalized
cocyclic matrices meeting the bound (1.1) as normalized cocyclic matrices for which
every row sum is zero. Could it be possible that such a relation translates somehow
to the case n ≡ 2 (mod 4)? We now prove that, in fact, the answer to this question
is affirmative.
A natural way to measure if the rows of a normalizedcocyclic matrix M =[m ]
ij
are close to sum zero, is to define an absolute row excess function RE, such that
n n
RE(M)= m .
ij
(cid:12) (cid:12)
i=2(cid:12)j=1 (cid:12)
X(cid:12)X (cid:12)
(cid:12) (cid:12)
Thisisanaturalextensionoftheusualnotion(cid:12)ofexces(cid:12)sofaHadamardmatrix,E(H),
(cid:12) (cid:12)
which consists in the summation of the entries of H.
With this definition at hand, it is evident that a cocyclic matrix M is Hadamard
if and only if RE(M) = 0. That is, a cocyclic matrix M meets (1.1) if and only if
RE(M) is minimum. This condition may be generalized to the case n≡2(mod4).
For the remainder of the paper t denotes an odd positive integer.
Proposition 2.3. [1] Let M be a normalized cocyclic matrix over G of order
n=2t. Then
RE(M)≥2t−2.
But we may go even further. Having the minimum possible value 2t−2 is a
necessary condition for a cocyclic matrix M to meet the bound (1.2).
Proposition 2.4. [1] If a cocyclic matrix M of order n = 2t meets the bound
(1.2), then RE(M)=2t−2.
Unfortunately, although having minimum absolute row excess is a necessary and
sufficient condition for meeting the bound (1.1), it is just a necessary (but not suffi-
cient, in general, see [1, Table 5] ) condition for meeting the bound (1.2). But there
aresome empiricalevidences thatmatriceshavingminimum absoluterowexcesscor-
respond with matrices having large determinants, see Table 2.1., page 11.
Fromnowon,wefixG=D asthedihedralgroupwithpresentationha,b: am =
2m
EmbeddingcocyclicD-optimaldesignsincocyclicHadamardmatrices 7
b2 = (ab)2 = 1i, with ordering {1,a,...,am−1,b,ab,...,am−1b} and indexed as
{1,...,2m}. A basis for cocycles over D consists in (see [1, 2]):
2m
• Let m be an odd positive integer.
B ={∂ ,...,∂ ,β }.
2 2m−1 2
• Let m be an even positive integer.
B ={∂ ,...,∂ ,β ,β ,γ}.
2 2m−2 1 2
Here ∂ denotes the coboundary associated with the ith-element of the dihedral
i
groupD2m,thatisai−1(modm)b⌊i−m1⌋. Andβ1,β2 andγ aretherepresentativecocycles
in cohomology, i.e. the cocyclic matrices coming from inflation are M = J ⊗
β1 m
1 1 1 1
andM = ⊗J . WeuseA⊗B fordenotingtheusualKronecker
1 − β2 1 − m
(cid:20) (cid:21) (cid:20) (cid:21)
product of matrices, that is, the block matrix whose blocks are a B.
ij
A A
m m
The transgression cocyclic matrix M is M = for the m × m
γ γ
B B
(cid:20) m m (cid:21)
matrices A =(a ) and B =(b ) where
m ij m ij
−1 i+j >m+1 −1 i<j
a = , and b = ;
ij ij
1 otherwise 1 otherwise
(cid:26) (cid:26)
IthasbeenobservedthatcocyclicHadamardmatricesoverthedihedralgroupmostly
useM ◦M anddonotuseM (see[3,12]). Inthesequel,weconsideronlycocyclic
β2 γ β1
Hadamard matrices of this form M ◦...◦M ◦M ◦M .
∂i1 ∂iw β2 γ
In what follows, the goal is to provide an algebraic formalism (in terms of co-
cycles) to describe two combinatorial operations on a matrix: the first consisting in
eliminating and the second in adding certain rows and columns.
Remark 2.5. D istriviallyembeddedasasubgroupofD ,the dihedralgroup
2t 4t
of 4t elements. Concretely, if D4t = ha,b: a2t =b2 = (ab)2 = 1i then D2t ∼= ha2,bi⊂
D .
4t
Proposition2.6. LetM beacocyclic matrixoverD ,thenthe2tby2tmatrix
ψ 4t
obtained by eliminating from M the rows and columns indexed with an even number
ψ
is a cocyclic matrix over D and we denote it as M˜ .
2t ψ
Proof. On the one hand, taking into account the ordering fixed above, D =
4t
{1,a,...,a2t−1,b,ab,...,a2t−1b}, the rows and columns in M indexed with an odd
ψ
number correspond with {1,a2,...,a2t−2,b,a2b,...,a2t−2b}=ha2,bi.
On the other hand, if ψ denotes the restriction of ψ to the subgroupha2,bi,
|ha2,bi
8 V.A´lvarez,J.A.Armario,M.D.Frau,andF.Gudiel
then ψ|ha2,bi satisfies (2.1) for G = ha2,bi ∼= D2t since ψ satisfies (2.1) for G = D4t.
In other words, ψ is a cocycle for G=ha2,bi and M˜ =M .
|ha2,bi ψ ψ|ha2,bi
Lemma 2.7. Let M and M be cocyclic matrices over D then
ψ1 ψ2 4t
M˜ =M˜ ◦M˜ ,
ψ1·ψ2 ψ1 ψ2
where M =M ◦M .
ψ1·ψ2 ψ1 ψ2
Proof. It is a straightforwardconsequence of the pointwise multiplication.
From now on, B ={∂ ,...,∂ ,β ,β ,γ} and B˜={∂˜ ,...,∂˜ ,β˜ } denote a
2 4t−2 1 2 2 2t−1 2
basis of cocycles for D and D , respectively.
4t 2t
Lemma 2.8. The following identities hold:
M˜ = J2t ieven , M˜ = J2t i=1
∂i ( M∂˜i+21 iodd βi ( M∂˜β2 i=2
and
t−1
2
M˜ = M ◦M .
γ ∂˜2i ∂˜2t−2i+1
i=1
Y
Proof. The identities above follow by direct inspection.
Given M a D -matrix. The following result describes, in terms of cocycles, the
ψ 4t
unique D -matrix obtained by eliminating from M the rows and columns indexed
2t ψ
with an even number.
Theorem 2.9. Given a D -matrix
4t
M =Mα2 ◦...◦Mα4t−2 ◦Mk1 ◦Mk2 ◦Mk3
ψ ∂2 ∂4t−2 β1 β2 γ
where (α ,...,α ,k ,k ,k ) denotes a concrete 4t-upla with entries 0 or 1. Then
2 4t−2 1 2 3
M˜ψ =M˜∂α22 ◦...◦M˜∂α44tt−−22 ◦M˜βk11 ◦M˜βk22 ◦M˜γk3
2t−2 t−21 k3
= Mα2j+1 ◦Mk2 ◦ M ◦M .
∂˜j+1 β˜2  ∂˜2i ∂˜2t−2i+1
j=1 i=1
Y Y
 
Proof. It follows from Lemmas 2.7 and 2.8.
EmbeddingcocyclicD-optimaldesignsincocyclicHadamardmatrices 9
Now, fixed M a D -matrix. The whole set of D -matrices constructed by
ψ˜ 2t 4t
adding certain rows and columns to M is provided in the next theorem.
ψ˜
Theorem 2.10. Given a D -matrix
2t
M =M ◦...◦M ◦M .
ψ˜ ∂˜i1 ∂˜iw β˜2
Then any D -matrix of the form:
4t
2t−1 t−21 k2
M = Mαi◦Mk1◦Mk2◦ M ◦M ◦M ◦...◦M ◦M ,
φ ∂2i β1 γ  ∂4i−1 ∂4t−4i+1 ∂2i1−1 ∂2iw−1 β2
i=1 i=1
Y Y
 
with α ,k and k taking values from {0,1}, satisfies that M˜ =M .
i 1 2 φ ψ˜
Proof. Using Theorem 2.9, it is easy to check that M˜ =M .
φ ψ˜
Using our algebraic formalism we are able to give a method for embedding a
D -matrixintherowsandcolumnsindexedwithanoddnumberofaD -Hadamard
2t 4t
matrix whenever it is possible. Although, theoretically this method provides a solu-
tion, from the practical perspective it is only appropriate for numerical calculations
in low orders, because in the worst case it needs to check 22t possibilities. Hence,
properties providing some cut down in complexity and some heuristic are generally
needed.
Algorithm 2.11. Embedding D -matrices in D -Hadamard matrices.
2t 4t
Input: a D -matrix. M =M ◦...◦M ◦M .
2t ψ˜ ∂˜i1 ∂˜iw β˜2
Output: a D - Hadamard matrix M which contains embedded M (that is, M˜ =
4t φ ψ˜ φ
M ), if such matrix exists.
ψ˜
Step 1. Calculate M =M ◦...◦M ◦M .
ψ ∂2i1−1 ∂2iw−1 β2
Step 2. Calculate all possible combinations
k
2t−1 t−21
M = Mαi ◦Mk◦ M ◦M ◦ M
φ ∂2i γ  ∂4i−1 ∂4t−4i+1 ψ
i=1 i=1
Y Y
 
where k and α may take the values 0 or 1.
i
Step 3. If there exists a (α ,...,α ,k) such that RE(M ) = 0, then M
1 2t−1 φ φ
is Hadamard. Otherwise, such matrix does not exist for M .
ψ˜
Verification: By construction, M is a cocyclic Hadamard matrix over D and
φ 4t
M˜ =M (see Theorem 2.10).
φ ψ˜
10 V.A´lvarez,J.A.Armario,M.D.Frau,andF.Gudiel
Inthe sequel,wedescribeanalgorithmlookingforD -matriceswithlargedeter-
2t
minant embedded in a Hadamard matrix. Actually, the output M is a Hadamard
ψ
matrix where the D -matrix, M , is embedded in a such a way that M˜ = M .
2t ψ˜ ψ ψ˜
We give two strategies that are both based on exhaustive searches. The first one
needs to calculateD -matriceswithlargedeterminant(bythe approachgivenin[1])
2t
and use Algorithm 2.11 for detecting if it is embedded in a Hadamard matrix. The
secondone needs to calculate D -Hadamardmatrices, M , (using [3]) and check the
4t ψ
determinant of M˜ .
ψ
Algorithm 2.12. Search for D -Hadamard matrices where a D -matrix with
4t 2t
large determinant exists embedded in it.
Input: an odd positive integer t.
Output: aD -HadamardmatrixM where det(M˜ψ) ≥κ,ifsuchmatrixexists,
4t ψ (4t−2)(2t−2)t−1
with 0.85≤κ≤1.
Approach 1
Ω←∅
S ← The list of D -matrices satisfying that det(Mψ˜) ≥κ
2t (4t−2)(2t−2)t−1
while S is not empty {
1. Choose a matrix M in S.
ψ˜
2. S ←S\{M }.
ψ˜
3. Check (using Algorithm 2.11) whether exists a D -Hadamardmatrix M
4t ψ
such that M˜ =M . If not, go to 1; otherwise Ω←M , S =∅.
ψ ψ˜ ψ
4. End while.
} Ω
Approach 2
Ω←∅
S ← The list of D -Hadamard matrices
4t
while S is not empty {
1. Choose a matrix M in S.
ψ
2. S ←S\{M }.
ψ
3. Check whether det(M˜ψ) ≥ κ. If not, go to 1; otherwise Ω ← M ,
(4t−2)(2t−2)t−1 ψ
S =∅.
4. End while.