Circulant q-Butson Hadamard matrices

0.12 MB·

by  Trevor Hyde

CIRCULANT q-BUTSON HADAMARD MATRICES
TREVORHYDEANDJOSEPHKRAISLER
7
1
0
2
ABSTRACT. Ifq=pnisaprimepower,thenad-dimensionalq-ButsonHadamardmatrixHisa
ar d×dmatrixwithallentriesqthrootsofunitysuchthatHH∗ =dId. Weusealgebraicnumber
M theory to prove a strong constraint on the dimension of a circulant q-Butson Hadamard matrix
whend=pmandthenexplicitlyconstructafamilyofexamplesinallpossibledimensions.These

5 resultsrelatetothelong-standingcirculantHadamardmatrixconjectureincombinatorics.
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1. INTRODUCTION
]
O
For a prime power q = pn, a q-Butson Hadamard matrix (q-BH) of dimension d is a d d
C ×
matrixH withallentriesqthrootsofunitysuchthat
.
h
HH∗ = dI ,
t d
a
m whereH∗ istheconjugate transpose ofH. Ad dmatrixH issaidtobecirculant if
×
[ H = f(i j)
ij
−
2
for some function f defined modulo d. In this paper we investigate circulant q-BH matrices of
v
1 dimension d= pm.
7
8 Theorem1. Ifq = pn isaprimepowerandd = pm,thenthereexistsad dcirculantq-Butson
×
8 Hadamardmatrixifandonlyifm 2n,withoneexception when(m,n,p) = (1,1,2).
0 ≤
. Ouranalysis ofcirculant q-BHmatricesledustotheusefulnotionoffibrousfunctions.
1
0
Definition2. Letd 0andq = pn beaprimepower,
7 ≥
1 (1) If X is a finite set, we say a function g : X Z/(q) is fibrous if the cardinality of the
v: fiber g−1(b) dependsonlyonb modpn−1. →
| |
i (2) Wesayafunction f : Z/(d) Z/(q)isδ-fibrous iffor each k 0 mod dthe function
X
→ 6≡
δ (x) = f(x+k) f(x)isfibrous.
r k −
a
When q = d = p are both prime, δ-fibrous functions coincide with the concept of planar func-
tions, which arise in the study of finite projective planes [2] and have applications in cryp-
tography [6]. Circulant q-BH matrices of dimension d are equivalent to δ-fibrous functions
f :Z/(d) Z/(q).
→
Theorem 3. Let q = pn be a prime power and ζ a primitive qth root of unity. There is a
correspondence betweend dcirculantq-ButsonHadamardmatricesH andδ-fibrousfunctions
×
f :Z/(d) Z/(q)givenby
→
H = ζf(i−j) .
ij
Werestateourmainresultinthelangua(cid:0)ge o(cid:1)fδ-fi(cid:0)brous fu(cid:1)nctions.
Date:March14th,2017.
1
2 TREVORHYDEANDJOSEPHKRAISLER
Corollary4. Ifq = pnisaprimepower,thenthereexistδ-fibrousfunctionsf : Z/(pm) Z/(q)
→
ifandonlyifm 2nwithoneexception when(m,n,p) = (1,1,2).
≤
It would be interesting to know if δ-fibrous functions have applications to finite geometry or
cryptography.
When q = 2, q-BH matrices are called Hadamard matrices. Hadamard matrices are usually
definedasd dmatricesH withallentries 1suchthatHHt = dI . BustonHadamardmatrices
d
× ±
wereintroduced in[1]asageneralization ofHadamardmatrices.
CirculantHadamardmatricesariseinthetheoryofdifference sets,combinatorial designs, and
synthetic geometry [7,Chap. 9]. Therearearithmetic constraints onthepossible dimension ofa
circulant Hadamardmatrix. Whenthedimension d= 2m isapoweroftwowehave:
Theorem5(Turyn[9]). Ifd= 2m isthedimensionofacirculantHadamardmatrix,thenm = 0
orm = 2.
Turyn’sproofusesalgebraicnumbertheory,morespecificallythefactthat2istotallyramifiedin
the2nthcyclotomicextension Q(ζ2n)/Q;anelementaryexposition maybefoundinStanley[8].
Conjecturally thisaccounts forallcirculant Hadamardmatrices[7].
Conjecture6. Therearenod-dimensional circulant Hadamardmatricesford> 4.
Circulant q-Butson Hadamard matrices provide a natural context within which to consider
circulant Hadamard matrices. A better understanding of the former could lead to new insights
onthe latter. Forexample, ourproof ofTheorem 1showsthat thetwopossible dimensions for a
circulant Hadamard matrix given by Turyn’s theorem belong to larger family of circulant q-BH
matriceswiththeomissionofd = 2beingadegenerate exception. Circulantq-BHmatriceshave
been studied in some specific dimensions [3], but overall seem poorly understood. Weleave the
existence ofq-BHmatriceswhenthedimension disnotapowerofpforfuturework.
Acknowledgements. TheauthorsthankPadraigCathainforpointerstotheliterature, inparticu-
larforbringingtheworkofdeLauney[4]toourattention. WealsothankJeffLagariasforhelpful
feedback onanearlierdraft.
2. MAIN RESULTS
Wealwaysletq =pn denoteaprimepower. Firstrecallthedefinitionsofq-ButsonHadamard
andcirculant matrices.
Definition 7. A d-dimensional q-Butson Hadamard matrix H (q-BH) is a d d matrix all of
×
whoseentriesareqthrootsofunitysatisfying
HH∗ = dI ,
d
whereH∗ istheconjugate transpose ofH.
Ad-dimensional circulantmatrixC isad dmatrixwithcoefficients inaringRsuchthat
×
C = f(i j)
ij
−
forsomefunction f :Z/(d) R.
→
Example8. Hadamardmatricesarethespecialcaseofq-BHmatriceswithq = 2. Theq-Fourier
matrix (ζij)whereζ isaprimitive qthroot ofunity isan example ofaq-BHmatrix (whichmay
CIRCULANTq-BUTSONHADAMARDMATRICES 3
also be interpreted as the character table of the cyclic group Z/(q).) When q = 3 and ω is a
primitive3rdrootofunitythisisthematrix:
1 1 1
1 ω ω2
 
1 ω2 ω4
 
Thisexampleisnotacirculant 3-BHmatrix. Thefollowingisacirculant 3-BHmatrix:
1 ω ω
ω 1 ω
 
ω ω 1
 
The remainder of the paper is divided into two sections: first we prove constraints on the
dimensiondofacirculant q-BHmatrixwhendisapowerofp;nextweintroduce theconceptof
δ-fibrousfunctionsandconstructexamplesofcirculantq-BHmatricesinallpossibledimensions.
Constraintsondimension. Theorem9usestheramificationoftheprimepintheqthcyclotomic
extension Q(ζ)/Qtodeducestrongconstraints onthedimension ofaq-BHmatrix.
Theorem 9. If q = pn is a prime power and H is a circulant q-Butson Hadamard matrix of
dimensiond = pm+n,thenm n.
≤
Notethatourindexingofmhaschangedfromtheintroduction; thischoicewasmadetoimprove
notation inourproof. Weusethisindexingfortherestofthepaper.
ProofofTheorem9. SupposeH = (a )isacirculantq-ButsonHadamardmatrixofdimension
i−j
d = pm+n. FromH beingq-BHofdimension dwehave
HH∗ = dI ,
d
hence det(H) = dd/2 and each eigenvalue α of H has absolute value α = √d = p(m+n)/2.
± | |
Ontheotherhand,sinceH = (a )iscirculant, ithaseigenvalues
i−j
α = a ζjk. (1)
k j
j<d
X
forζ aprimitivedthrootofunitywithcorresponding eigenvector
ut = 1,ζk,ζ2k,...,ζ(d−1)k .
k
Theseobservations combinetogivetw(cid:0)owaysofcomputing de(cid:1)t(H).
α = det(H) = p(m+n)d/2. (2)
k
±
k
Y
Theidentity(2)istheessentialinteraction betweenthecirculantandq-BHconditions onH. The
prime p is totally ramified in Q(ζ), hence there is a unique prime ideal p Z[ζ] over (p) Z.
⊆ ⊆
Sinceallα Z[ζ],itfollowsfrom(2)that(α )= pvk asidealsofZ[ζ]forsomev 0andfor
k k k
∈ ≥
each k. Soeither α /α orα /α isanelement ofZ[ζ]. Sayα /α istheintegral quotient. We
0 1 1 0 0 1
noted α = p(m+n)/2 for each k, hence α /α = 1. The only integral elements of Z[ζ]with
k 0 1
| | | |
absolute value1arerootsofunity,henceα /α = ζr forsomer 0,hence
0 1
± ≥
α = ζrα . (3)
0 1
±
4 TREVORHYDEANDJOSEPHKRAISLER
By(1)wehave
α = a α = a ζj.
0 j 1 j
j<d j<d
X X
Eachj < d= pm+n hasaunique expression asj = j +j pm wherej < pm andj < pn. Let
0 1 0 1
ω = ζpm beaprimitiveqthrootofunity. Then
ζj = ζj0+j1pm = ωj1ζj0.
Writingα inthelinearbasis 1,ζ,ζ2,...,ζpm−1 ofQ(ζ)/Q(ω)wehave
1
(cid:8) α1 = ajζj =(cid:9) bj′ζj′,
j<d j′<pm
X X
where bj′ is a sum of pn complex numbers each with absolute value 1. Now (3) says α0 =
ωj0ζ−j1α forsomej andj ,thus
1 0 1
±
aj = α0 = ωj0ζ−j1α1 = ωj0bj′ζj′−j1.
± ±
j<d j′<pm
X X
Comparingcoefficientsweconclude that
α = ωj0b ,
0 ± j1
which is to say that α is the sum of pn complex numbers each with absolute value 1, hence
0
α pn. Onthe other hand wehave α = p(m+n)/2. Thusm+n 2n = 0 m nas
0 0
|desi|re≤d. | | ≤ ⇒ ≤ ≤ (cid:3)
Remark. Themainimpedimenttoextending thisresultfromq = pn toageneralintegerq isthat
we no longer have the total ramification of the primes dividing the determinant of H. It may be
possible to get some constraint in certain cases from acloser analysis of the eigenvalues α and
k
ramification, butwedonotpursuethis.
δ-Fibrousfunctionsandconstructionofcirculantq-BHmatrices. Recallthenotionoffibrous
functions fromtheintroduction:
Definition10. Letd 0andq = pn beaprimepower,
≥
(1) IfX isafiniteset,wesayg : X Z/(q)isfibrousifthecardinalityofthefibers g−1(b)
dependsonlyonb mod pn−1. → | |
(2) Wesayafunction f : Z/(d) Z/(q)isδ-fibrous iffor each k 0 mod dthe function
→ 6≡
x f(x+k) f(x)isfibrous.
7→ −
Lemma11isacombinatorial reinterpretation ofthecyclotomicpolynomials Φpn(x).
Lemma11. Letpbeaprimeandζ aprimitivepnthrootofunity.
(1) If b ζk = 0withb Q,thenb dependsonlyonk modpn−1.
k<pn k k ∈ k
(2) IfX isafinitesetandg :X Z/(pn)isafunction, theng isfibrous iff
P →
ζg(x) = 0.
x∈X
X
Proof. (1) Suppose b ζk = 0 for some b Q. Then r(x) = b xk Q[x] is
k<pn k k ∈ k<pn k ∈
a polynomial with degree < pn such that r(ζ) = 0. So there is some s(x) Q[x] such that
P P ∈
r(x)= s(x)Φpn(x)where
Φpn(x) = xjpn−1
j<p
X
CIRCULANTq-BUTSONHADAMARDMATRICES 5
is the pnth cyclotomic polynomial—the minimal polynomial of ζ over Q. Since degΦpn(x) =
pn pn−1,itfollowsthatdegs(x) < pn−1. Let
−
s(x) = a xi
i
i<pn−1
X
forsomeai Q. Expanding s(x)Φpn(x)wehave
∈
r(x)= s(x)Φpn(x) = aixi+jpn−1.
i<Xpn−1
j<p
Comparingcoefficientsyields
bk = bi+jpn−1 = ai,
whichistosay,b dependsonlyi k modpn−1.
k
≡
(2)Supposeg isfibrous. Foreachi< pn−1,leta = g−1(i). Then
i
| |
ζg(x) = aiζi+jpn−1 = Φpn(ζ) aiζi = 0.
xX∈X i<Xpn−1Xj<p i<Xpn−1
Conversely, foreachk < pn letc = g−1(k). Then
k
| |
0 = ζg(x) = c ζk,
k
x∈X k<pn
X X
and(1)impliesc depends onlyonk modpn−1. Henceg isfibrous. (cid:3)
k
Theorem 12establishes theequivalence between q-BHmatrices ofdimension dandδ-fibrous
functions f :Z/(d) Z/(q).
→
Theorem 12. Let q = pn be a prime power. There is a correspondence between circulant q-
Butson Hadamard matrices H ofdimension dand δ-fibrous functions f : Z/(d) Z/(q)given
→
by
H = ζf(i−j) .
i,j
Proof. Suppose f is δ-fibrous. Defin(cid:0)e the(cid:1)mat(cid:0)rix H =(cid:1) (H ) by H = ζf(i−j) where ζ is
ij ij
a primitive qth root of unity. H is plainly circulant and has all entries qth roots of unity. It
remains to show that HH∗ = dI , which is to say that the inner product r of column j
d j,j+k
and column j +k is 0 for each j and each k 0mod d. For each k 0 mod d the function
6≡ 6≡
δ (x) = f(x+k) f(x)isfibrous. Thenwecompute
k
−
r = ζf(i−j+k)−f(i−j) = ζδk(i−j) = 0,
j,j+k
i<d i<d
X X
wherethelastequalityfollowsfromLemma11(2).
Conversely, suppose H = (H ) is a circulant q-Butson Hadammard matrix. Then H =
i,j i,j
ζf(i−j)forsomefunctionf : Z/(d) Z/(q). SinceHH∗ = dI wehaveforeachk 0mod d,
d
→ 6≡
0 = r = ζf(i−j+k)−f(i−j).
j,j+k
i<d
X
Lemma11(2)thenimpliesδ (x) =f(x+k) f(x)isfibrous. Thereforef isδ-fibrous. (cid:3)
k
−
Lemma13checksthataffinefunctions arefibrous. WeusethisinourproofofTheorem14.
6 TREVORHYDEANDJOSEPHKRAISLER
Lemma 13. If q = pn is a prime power, then for all a 0mod q and arbitrary b, the function
6≡
f(x)= ax+bisfibrous.
Proof. Sincea 0 modq,
6≡
ζf(j) = ζaj+b = ζb (ζa)j = 0.
j<q j<q j<q
X X X
Thus,byLemma11(2)weconclude thatf isfibrous. (cid:3)
Theorem14. Ifq = pnisaprimepower,thenthereexistsacirculantq-ButsonHadamardmatrix
ofdimension d= pm+n = pmq foreachm nunless(m,n,p) = (0,1,2).
≤
Our construction in the proof of Theorem 14 misses the family (m,n,p) = (0,n,2) for each
n 1. Lemma 15 records a quick observation that circumvents this issue for n > 1, as our
con≥struction doesgivecirculant 2n−1-BHmatricesofdimension 2n.
Lemma15. Ifq = pn isaprimepowerandthereexistsacirculantq-BHmatrixofdimension d,
thenthereexistsacirculantpkq-BHmatrixofdimension dforallk 0.
≥
Proof. Everyqthrootofunity isalsoapkqthrootofunity, hence wemayviewacirculant q-BH
matrixH ofdimension dasacirculant pkq-BHforallk 0. (cid:3)
≥
ProofofTheorem14. Ourstrategyistofirstconstruct asequence offunctions
δ : Z/(pm) Z/(q) Z/(q)
k
× →
which are fibrous for each k < pmq. If i : Z/(pm) Z/(q) Z/(pmq) is the bijection
× →
i(x,y) = x + pmy, we define f : Z/(pmq) Z/(q) such that f(z + k) f(z) = δ (x,y)
k
→ −
whenz = i(x,y). Hencef isδ-fibrousandTheorem12impliestheexistenceofacorresponding
circulant q-BHmatrixofdimension pmq.
Nowforeachk 0defineδ by
k
≥
δ (x,y) = ky+ S (x), (4)
k j
j<k
X
where
1 x 1mod pm,
S (x) = χ(x+i), χ(x) = ≡ −
j
(0 otherwise.
i<j
X
ObservethatS (x)countstheintegersintheinterval[x,x+j)congruentto 1mod pm (which
j
−
onlydependsonx modpm.) Anyintervaloflengthpmj containspreciselyj integerscongruent
1 1
to 1 modpm. Foreachj < pmq,writej = j +pmj withj < pm andj < q,then
0 1 0 1
−
S (x) = χ(x+i) = χ(x+i)+j = S (x)+j . (5)
j 1 j0 1
i<j0X+pmj1 iX<j0
We show that δ is fibrous when k < pmq. If k 0 mod q, then for each x = x the function
k 0
δ (x ,y)isaffinehence fibrous byLemma13. S6≡oδ (x,y)isfibrous. Nowsuppose k = k′q for
k 0 k
somek′ < pm. Using(5)wereduce(4)to
δ (x,y) = S (x) = S (x)+j = ℓ S (x)+pm ℓ , (6)
k j j0 1 j0 2
jX<k′q j0+pXmj1<k′q j0X<pm (cid:0) (cid:1)
CIRCULANTq-BUTSONHADAMARDMATRICES 7
where k′q = pm(k′pn−m) = pmℓ. Here we use our assumption m n. The definition of χ
≤
implies
0 j < pm x
0
S (x) = − = S (x) =x,
j0 (1 j0 ≥ pm−x, ⇒ j0X<pm j0
whenceδ (x,y) = ℓx+pm ℓ . Sincek′ < pmitfollowsthatℓ = k′pn−m < pn = q,soδ (x,y)
k 2 k
isaffinehencefibrousbyLemma13.
(cid:0) (cid:1)
Define f : Z/(pmq) Z/(q) by f(k) = δ (0,0). For this to be well-defined, it suffices to
k
→
checkthatδk+pmq(x,y) = δk(x,y)witharbitrary k. By(4),
δk+pmq(x,y) = (k+pmq)y+ Sj(x)
j<k+pmq
X
= ky+ S (x)+ S (x)
j j+k
j<k j<pmq
X X
= δ (x,y)+ S (x).
k j+k
j<pmq
X
Theargument leadingto(6)gives
S (x)= qx+pm q = pm q .
j+k 2 2
j<pmq
X (cid:0) (cid:1) (cid:0) (cid:1)
Finally, pm q 0 modq unless (m,n,p) = (0,n,2). Lemma 15 implies that constructing
2 ≡
an example for (1,n 1,2) implies the existence of example for (0,n,2), hence we proceed
(cid:0) (cid:1) −
under the assumption that either p = 2 or p = 2 and m > 0. The case (m,n,p) = (0,1,2)
6
is an exception as one can check explicitly that there are no 2-dimensional Hadamard matrices.
Hence it follows that f is well-defined. Let i : Z/(pm) Z/(q) Z/(pmq) be the bijection
× →
i(x,y) = x+pmy. Tofinishtheconstruction wesupposez = i(x,y),show
f z+k f z = δ (x,y),
k
−
(cid:0) (cid:1) (cid:0) (cid:1)
andthenourproofthatδ isfibrousforallk < pmqimpliesf isδ-fibrous. Theorem12translates
k
thisintotheexistence ofacirculant q-BHmatrixofdimensionpm+n. Now,
f(z+k) f(z)= δ (0,0) δ (0,0) = S (0) = χ(i)
z+k z j
− −
z≤j<z+k j−z<ki−z<j−z
X X X
= χ(x+i′)= Sj′(x) = δk(x,y).
j′<ki′<j′ j′<k
X X X
(cid:3)
Remark. PadraigCathainbroughttheworkofdeLauney[4]toourattention afterreadinganini-
tial draft. Thereone finds aconstruction ofcirculant q-Butson Hadamard matrices ofdimension
q2 forallprimepowersq whichappearstobecloselyrelatedtoourconstruction inTheorem14.
8 TREVORHYDEANDJOSEPHKRAISLER
Example16. Weprovide two low dimensional examples to illustrate our construction. First we
havean8dimensional circulant 4-BHmatrix.
1 1 i 1 1 1 i 1
− −
1 1 1 i 1 1 1 i
− − 
i 1 1 1 i 1 1 1
− −
 1 i 1 1 1 i 1 1
 − − 
 1 1 i 1 1 1 i 1
 − − 
 1 1 1 i 1 1 1 i
 − − 
 i 1 1 1 i 1 1 1
 − − 
 1 i 1 1 1 i 1 1
− − 
Letω beaprimitive3rdrootofunity. Thefollowingisa9dimensional circulant 3-BHmatrix.
1 ω2 ω 1 1 1 1 ω ω2
ω2 1 ω2 ω 1 1 1 1 ω
 
ω ω2 1 ω2 ω 1 1 1 1
1 ω ω2 1 ω2 ω 1 1 1 
 
1 1 ω ω2 1 ω2 ω 1 1 
 
1 1 1 ω ω2 1 ω2 ω 1 
 
1 1 1 1 ω ω2 1 ω2 ω 
 
ω 1 1 1 1 ω ω2 1 ω2
 
ω2 ω 1 1 1 1 ω ω2 1 
 
 
Corollary17isanimmediateconsequence ofourmainresultsbyTheorem12.
Corollary 17. If q = pn is a prime power and d = pm+n, then there exists a δ-fibrous function
f :Z/(pm+n) Z/(q)iffm n,withtheoneexception of(m,n,p) = (0,1,2).
→ ≤
Closing remarks. Our analysis focused entirely on the existence of circulant pn-BH matrices
with dimension d a power of p. The number theoretic method of Theorem 9 cannot be imme-
diately adapted to the case where d is not a power of p, although as we noted earlier, it may be
possible togetsomeconstraint withacloseranalysisoftheeigenvalues ofacirculant matrixand
theramificationovertheprimesdividing dinthedthcyclotomicextension Q(ζ)/Q.
Thefamily of examples constructed in Theorem 14 was found empirically. It would be inter-
estingtoknowiftheconstruction extendstoanydimensions whicharenotpowersofp.
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CIRCULANTq-BUTSONHADAMARDMATRICES 9
DEPT.OFMATHEMATICS,UNIVERSITYOFMICHIGAN,ANNARBOR,MI48109-1043,
E-mailaddress:[email protected]
DEPT.OFMATHEMATICS,UNIVERSITYOFMICHIGAN,ANNARBOR,MI48109-1043,
E-mailaddress:[email protected]