Nonexistence results on Generalized BentFunctions

Ka Hin LeungNational University of Singapore

Bernhard SchmidtNanyang Technological University

Qi WangSouthern University of Science and Technology

K.T. Arasu, J.F. Dillon, K.H. Leung and S.L. Ma,Cyclic relative difference sets with classicalparameters, Journal of Combinatorial Theory SeriesA, 94(2001), 118-126.

K.T. Arasu, K.H. Leung, S.L. Ma, A. Nabavi andD.K. Ray-Chaudhuri, Circulant weighing matrices ofweight 22t, Designs, Codes and Cryptography,41(2006), 111-123.

K.T. Arasu, K.H. Leung, S.L. Ma, A. Nabavi andD.K. Ray-Chaudhuri, Determination of all possibleorders of weight 16 circulant weighing matrices,Finite Fields and their Applications, 12(2006),498-538.

Definition 1

A function f : Zn2 → Z2 is called a bent function if

Wf (y) :=∑x∈Zn

2

(−1)f (x)+y ·x = ±2n/2

for all y ∈ Zn2. Here y · x denotes the usual inner

product.

Wf (y) is called the Walsh transform of f .

O. Rothaus originally defined f to be a bentfunction if |Wf (y)| is a constant and he had shownthat |Wf (y)| = 2n/2. It is clear that n must be even.

Bent functions are a highly active research field dueto their numerous applications in informationtheory, cryptography and coding theory.

There are more than 25 different generalizations ofbent functions recorded in Tokareva’s 2015monograph.

We concentrate on two generalizations of bentfunctions.

Our objecitve is to introduce two number theorticmethods to obtain some non-existence results andwe believe our techniques can be applied to studyother related problems.

Generalized Bent Functions (I) Kumar, Scholtz and Welch

Definition 2

Let m and q be positive integers and let ζq be aprimitive complex qth root of unity. A functionf : Zm

q → Zq is called a generalized bentfunction (GBF) if∣∣∣∣∣∣

∑x∈Zm

q

ζ f (x)−v ·xq

∣∣∣∣∣∣2

= qm for all v ∈ Zmq . (1)

Here x · v denotes the usual dot product.

Note that we have simply replaced 2 by q inDefinition 1.

Kumar, Scholtz, and Welch showed that GBFs fromZmq to Zq exist whenever

m is even or q 6≡ 2 (mod 4).

However, not a single GBF from Zmq to Zq with m

odd and q ≡ 2 (mod 4) is known.

Conjecture There is no GBF from Zmq to Zq if m is

odd and q ≡ 2 (mod 4).

Standard techniques, such as using self conjugateproperty, work in some cases.

Some researchers tried to find directly the form ofX ∈ Z[ζq] such that |X |2 = qm. But suchtechniques often work in very specific cases.

In case m = 1 and q = 2pr where p is prime, theconjecture was recently solved in [LS 2019].

For m = 3, there are still a few unsolved cases.

Bent functions in term of group ring equations

Our approach is different from other researchers,and the first step is to use group ring elements.Instead of Zm

q , we use multiplicative notation Cmq .

Let G = Cmq and let f : G → Zq be any function.

We associate f with an element Df in Z[ζq][G ] via

Df =∑x∈G

ζ f (x)q x .

For any v ∈ G , we define

F (v) =∑x∈Zm

q

ζ f (x)−x ·vq .

and a character χv of G such that

χv(x) = ζ−v ·xq for all x ∈ G .

It well known that every complex character of G isequal to some χv for some v ∈ G .

Note that for all v ∈ G ,

χv(Df ) =∑x∈G

ζ f (x)q χv(x) =∑x∈G

ζ f (x)−v ·xq = F (v).

(2)From (11) and (2), we see that f is a GBF if andonly if

|χ(Df )|2 = qm for all χ ∈ G . (3)

Proposition 3

Let m and q be positive integers, let f : Cmq → Zq

be a function, and set Df =∑

x∈G ζf (x)q x . Then f is

a GBF if and only if

DfD(−1)f = qm. (4)

In view of the above equation, we may also view aGeneralized Bent function as a generalization ofweighing matrices.

We will only concern with the case q = 2pa where pis a prime. Write G = U · K with U = Cm

pa andK = Cm

2 .

Suppose f : G → Z2pa is a GBF. Let Df be definedas before. Then

DfD(−1)f = 2mpam.

Let χ : U → C∗ be any character of U . We extendχ to a ring homomorphism Z[ζ][G ]→ Z[ζ][K ] bylinearity and setting χ(g) = g for all g ∈ K .

Write Dχ = χ(Df ) =∑

h∈K xhh with xh ∈ Z[ζ] andΘ =

∑p−1x=1(xp)ζxp .

For any character τ on K , we have|τ(Dχ)|2 = 2mpam and thus τ(Dχ) ≡ 0 (mod Θam),we can then conclude

xh|K | =∑τ∈K

τ(Dχ)τ(h)−1 ≡ 0 mod (Θam)

for all h ∈ K .

As |K | = 2m and gcd(2,Θ) = 1 in Z[ζ], it followsthat xh ≡ 0 (mod Θam) for all h ∈ K .

Thus Eχ := Dχ/Θam is an element of Z[ζ][K ].

EχE(−1)χ = 2m, as |Θ|2 = p.

For any character τ on K , |τ(Eχ)|2 = 2m. Thisleads us to study X ∈ Z[ζ] with |X |2 = 2m.

Suppose X =∑n−1

i=0 aiζin and |X |2 = k for an integer

k ∈ Z.

X is not uniquely written in the form of∑n−1

i=0 aiζin.

Our first step is to represent X in a certain way thatis ‘unique’.

Definition 4

Let G = 〈g〉 be a cyclic group of order n. For

Z =∑n−1

i=0 aigi ∈ Z[G ], write Z (ζn) =

∑n−1i=0 aiζ

in.

We say that Z is minimal if

|supp(Z )| = min {|supp(Y )| : Y ∈ Z[G ], Y (ζn) = Z (ζn)} .

If X ∈ Z[ζn] and Z (ζn) = X , then we say Z is aalias of X .

We define the length of X to be |supp(Z )|, whereZ is a minimal alias of X and denoted it by σ(X ).

Theorem 5

Suppose |X |2 = n in Z[ζ] and Z =∑pa−1

i=0 aigi is a

minimal alias of X in Z[〈g〉] where ◦(g) = pa. Then

n ≥ 1

p − 1

((p − σ(X ))

pa−1∑i=0

a2i + σ(X ) max{0, σ(X )− p

2}

).

(5)In particular,

n ≥ max

{pσ(X )

2(p − 1),σ(X )(p − σ(X ))

p − 1

}. (6)

Definition 6 (M-function)

For X ∈ Z[ζn], let

M(X ) =1

ϕ(n)

∑σ∈Gal(Q(ζn)/Q)

(XX )σ,

where ϕ denotes the Euler totient function.

The notion of M-functions was introduced byCassels. Theorem 5 still holds if we replace n byM(X ).

Note that M(X ) ≥ 1 for all nonzero X ∈ Z[ζn] bythe inequality of geometric and arithmetic means,since ∏

σ∈Gal(Q(ζn)Q)

(XX )σ ≥ 1.

The following is a consequence of Cassel’s results.

Result 7

Let X ∈ Z[ζn], let q be a prime divisor of n, andwrite n = qn′ with (q, n′) = 1. ThenX =

∑q−1i=0 Xiζ

iq with Xi ∈ Z[ζn′] and

M(X ) =1

q − 1

q−1∑i<j

M(Xi − Xj). (7)

To make use of the inequality in Theorem 5, we alsoneed to find a bound for σ(X ).

Result 8

Let X ∈ Z[ζn], let q be a prime divisor of n, andwrite n = qbn′ with (q, n′) = 1. If b > 1, then

X =∑qb−1−1

i=0 Xiζi with Xi ∈ Z[ζqn′] and

M(X ) =

qb−1−1∑i=0

M(Xi). (8)

The notion of multiplier is then useful in finding abound for σ(X ) by the following:

Proposition 9

Let t be an integer with gcd(t, p) = 1 and let G bea cyclic group of order pa. Write Opa(t) = f andsuppose that f divides p − 1. Let σ be theautomorphism of Q(ζ) determined by ζσ = ζ t . IfX σ = X for X ∈ Z[ζ], then there is a minimal aliasZ ∈ Z[G ] of X with

Z (t) = Z .

Theorem 10

Let n be a nonsquare integer. Let G = 〈g〉 be acyclic group of order pa and let P be the subgroupof G of order p. Assume there is X =

∑aiζ

i ∈ Z[ζ]with |X |2 = n. Then if Z =

∑aig

i ∈ Z[G ] is aminimal alias of X , there is Y ∈ Z[G ] such that

ZZ (−1) = n + PY . (9)

Moreover, we have σ(X ) ≤ n and

p ≤ n2 + n + 1. (10)

Corollary 11

Let p be a an odd prime, and let s 6= p be a prime.If a (ps, p, ps, s) relative difference set exists in anabelian group G , then p ≤ s2 + s + 1.

It can be shown that the existence of such relativedifference set implies the existence of X ∈ Z[ζp]such that |X |2 = s.

Corollary 12

Suppose there exists X ∈ Z[ζ] with |X |2 = 2m.Write f = ordp(2). We have the following.

(a) p ≤ 22m + 2m + 1 and f < 2m+1 is odd.

(b) f < 2m or p ≤ f 2−2mf−2m .

(c) If p > 22(m−2) + 2m−2 + 1, thenX 6≡ 0 (mod 2).

(d) σ(X ) ∈ {uf , uf + 1} for some positive integer uand σ(X ) < 2m+1.

(e) p ≡ 7 (mod 8) orp ≡ 1, 9, 17, 25, 33, 41, 49, 57 (mod 64).

In particular, it shows that if a GBF function fromCm2pa to Z2pa exists,

then p = 1 when m = 1;and p = 7, 23, 31, 73.

To remove some more cases, we need to useanother property of GBF.

Proposition 13

Let m and q be positive integers, G = Cmq , let

f : G → Zq be a bent function, and set

Df =∑

x∈G ζf (x)q x . Then∑

τ∈G

τ(Df )τχ(Df ) = 0 for all χ ∈ G \ {χ0}.

The key is to consider the set

W = {w ∈ Z[ζ] : ww = 2m}.

Condition:

v + w 6≡ 0 (mod 2) for all v ,w ∈ W with w 6= ±v .

If the above condition is satisfied in Z[ζ], then byusing the convolution property, it can be shown thatno such GBF exists.

Theorem 14 (L & Schmidt 2019)

Suppose that m is odd. If a GBF from Zm2pa to Z2pa

exists, then the following hold.

m ≥ 3.

If m = 3, then p = 7.

If m = 5, then p ∈ {7, 23, 31, 73, 89}.If m = 7, then p ∈ {7, 23, 31, 47, 71, 73, 79, 89103, 223, 233, 337, 431, 601, 631, 881, 1103, 1801}.If m ≥ 7, then p ≤ 22m/9 orordp(2) ≤ (2m + 3)/5.

Generalized Bent Functions (II)

Definition 15

Let m and n be positive integers and let ζm be aprimitive complex mth root of unity. A functionf : Zn

2 → Zm is called an (m, n) generalized bentfunction (GBF) if∣∣∣∣∣∣

∑x∈Zn

2

ζ f (x)m (−1)v ·x

∣∣∣∣∣∣2

= 2n for all v ∈ Zn2. (11)

Here x · v denotes the usual dot product.

(m; n)-GBF exists if both m and n are even; or 4|m.It remains to study the following two cases:

(i) m is odd.

(ii) m ≡ 2 mod 4 and n is odd.

Notation

For any subset S of G , we denote the group ringelement

∑g∈S g by S as well.

Let t be an integer coprime to q. ForX =

∑g∈G agg ∈ Z[ζq][G ], we write

X (t) =∑

aσggt

where σ is the automorphism of Q(ζq) determinedby ζσq = ζ tq.

Instead of using just the additive group Z2 and Zm,we also use the multiplicative notation, C2 and Cm.

From now on, we write G = C n2 .

Bent functions in terms of group ring equations

Definition 16

Let f : G (= C n2 )→ Zm be a function. We define an

element Bf in the group ring Z[ζm][G ]corresponding to f by

Bf :=∑x∈G

ζ f (x)m x .

Let g be a generator of Cm. We define an elementDf in the group ring Z[Cm][G ] by

Df :=∑x∈G

g f (x)x .

Note that for any character τ that maps g to ζm,τ(Df ) = Bf .

It then follows that f is an (m, n)-GBF if and only if

|χ(Bf )|2 = 2n, (12)

for all character χ over G .

We now have the following characterization of(m, n)-GBFs.

Proposition 17

Let f be a function from G = C n2 to Zm. Then f is

an (m, n)-GBF if and only if

BfB(−1)f = 2n. (13)

Furthermore, if f (G ) = Cm′ ⊂ Cm, then f can beregarded as an (m′, n)-GBF, where m = 2m′ withm′ odd.

Observe that we may write

DfD(−1)f =

∑x∈G

∑y∈G

g f (y+x)g−f (y)x =∑x∈G

Exx ,

where Ex =∑

y∈G g f (y+x)g−f (y) ∈ Z[Cm].

Therefore, Bf is an (m, n)-GBF if for each characterτ of order m on Cm, τ(Ex) = 0 for all x 6= 1G .

Write Ex =∑

aigi . Note that all ai ’s are

non-negative. Clearly, we have∑aiτ(g i) = 0.

This leads us to study the notion of vanishing sums.

Vanishing sums

Notation: For any group H , we denote

{∑g∈H

agg : ag ∈ Z and ag ≥ 0} by N[H].

Definition 18

Let D =∑m−1

i=0 aigi ∈ N[Cm]. We say D is a v-sum

if there exists a character τ of order m such thatτ(D) = 0. We say D is minimal ifτ(∑m−1

i=0 bigi) 6= 0 whenever 0 ≤ bi ≤ ai for all i

and bj < aj for some j .

To study v-sum, we define the following:

Definition 19

Let S be a finite set. Suppose X =∑

i∈S aiµi whereµi ’s are distinct roots of unity and all ai ’s arenonzero positive integers. We define

(i) u is the exponent of X if u is the smallestpositive integer such that µui = 1 for all i .

(ii) k is the reduced exponent if k is the smallestpositive integer such that there exists j with(µiµ

−1j )k = 1 for all i .

For example, if p is a prime, then the exponent of∑p−1i=1 ζ3ζ

ip is 3p, whereas the reduced exponent is p.

Definition 20

Suppose that X =∑

i∈S aiµi = 0 where µi ’s aredistinct roots of unity and all ai ’s are nonzerointegers. We say that the relation X = 0 is minimal,if for any proper subset I ( S ,

∑i∈I aiµi 6= 0.

Proposition 21 (Conway, Jones)

Suppose that X =∑

i∈S aiµi = 0 is a minimalrelation with reduced exponent k and all ai ’snonzero. Then k is square free and

|S | ≥ 2 +∑

p∈P(k)

(p − 2).

Here P(k) denotes the set of all prime divisors of k .

Note that if D =∑

i∈S aigi is a minimal v-sum,

then τ(D) =∑

i∈S aiτ(g)i is a minimal relation. Wethus define the reduced exponent of D as follows:

Definition 22

Suppose D =∑m−1

i=0 digi is minimal v-sum in

N[Cm]. We define the reduced exponent k of D asthe reduced exponent of the vanishing sumτ(D) =

∑m−1i=0 diτ(g)i .

Note that the reduced exponent defined above doesnot depend on the choice of the character τ .

Lemma 23

If D ∈ N[Cm] is a minimal v-sum with reducedexponent k , then D = D ′h for some D ′ ∈ N[Ck ] andh ∈ Cm.

To deal with v-sum D ∈ N[Cm] which is notminimal, we first decompose it into sum of minimalv-sum. It is straight forward to prove the following:

Lemma 24

Let D ∈ N[Cm] be a v-sum. Then D =∑

Di whereDi ’s are minimal v-sums in N[Cm].

Definition 25

Suppose D =∑m−1

i=0 digi is a v-sum in N[Cm]. We

define the c-exponent of D to be the smallest ksuch that there exist positive integer r , minimalv-sums D1, . . . ,Dr in N[Cm] with D =

∑ri=1Di and

k = lcm(k1, . . . , kr) where ki is the reduced exponetof Di for i = 1, . . . , r .

Example

For example, if m = 10 and h is a generator of C10,then we have

D =9∑

i=1

hi =4∑

i=0

(1 + h5)hi and

D =9∑

i=1

hi =1∑

i=0

(1 + h2 + h4 + h6 + h8)hi .

Note that (1 + h5)hi and (1 + h2 + h4 + h6 + h8)hj

are both minimal v-sums. However, c-exponent is 2.

Lemma 26

Suppose D =∑m−1

i=0 aigi is a v-sum in N[Cm] with

c-exponent k . Write m =∏s

i=1 pαi

i andk =

∏ti=1 pi . Note that t ≤ s and pi ’s are distinct

primes. Then we have the followings:

(a) ||D|| ≥ 2 +∑t

i=1(pi − 2);

(b) D =∑t

i=1 PiEi , where Pi is the subgroup oforder pi and Ei ∈ Z[Cm] for all i ;

(c) Suppose that∏t

i=1 pαi

i |d and d |m. Ifφ : Z[Cm]→ Z[Cd ] is the natural projection,then χ(φ(D)) = 0 whenever o(χ) = d .

Recall that if τ is a character of order m, τ(Ex) is avanishing sum.

Lemma 27

The c-exponent kx of Ex is a square free integerthat divides m such that

(a) k = p1 · · · pt .(b) 2n ≥ 2 +

∑ti=1(pi − 2);

(c) Ex =∑t

i=1 PiEi , where Pi is the subgroup oforder pi ; Ei ∈ Z[Cm] for all i .

Proposition 28

Suppose that f is an (m, n)-GBF and m =∏r

i=1 pαi

i

where pi ’s are distinct prime. Let tx be thec-exponent of Ex (as defined in Definition 2.7) foreach 1G 6= x ∈ G . Set

I = {i : pi - tx ∀x ∈ G} and m′ =∏i /∈I

pαi

i .

Then there exists an (m′, n)-GBF. In particular, ifpi |m and pi > 2n, then there exists an(m/pi , n)-GBF.

Proposition 28 allows to eliminate all prime factorsof m greater than 2n while proving nonexistenceresults on (m, n)-GBF.

Using Lemma 27, we are able to study the structureof Ex .

Theorem 29

Suppose that m =∏s

i=1 pαi

i , where2 < p1 < p2 < · · · < ps are primes and αi ’s are allpositive integers.

(a) There is no (m, n)-GBF when s = 1.

(b) There is no (m, n)-GBF if s ≥ 2 and3p1 + p2 > 2n.

(c) There is no (m, n)-GBF if there is no(∏r

i=1 pαi

i , n)-GBF where pr+1 is the smallestprime such that p1 + pr+1 > 2n.

Theorem 30

Let n be odd and m = 2∏s

i=1 pαi

i , wherep1 < p2 < · · · < ps are primes.

(a) Suppose s = 1. Then there is no (m, n)-GBF ifp1 > 2n−2; or p1 is not a Mersene prime andp1 > 2n−3; or p1 ≡ 3, 5 (mod 8).

(b) Suppose s ≥ 2 and r is the least integer suchthat pr+1 + p1 > 2n + 2. Then there is no(m, n)-GBF if there is no (2

∏ri=1 p

αi

i , n)-GBF.In particular, there is no (m, n)-GBF ifp1 > 2n−2 and p1 + p2 > 2n + 2.

We now consider nonexistence results for a fixedn.

In view of Theorem 20 and Theorem 30, weconclude that there is no (m, n)-GBF if n = 1;and m odd for n = 2.

For n = 3, we need only to consider(2m; 3)-GBFs with m = 3a · 5b · 7c in view ofProposition 28.

We now consider nonexistence results for a fixedn.

In view of Theorem 20 and Theorem 30, weconclude that there is no (m, n)-GBF if n = 1;and m odd for n = 2.

For n = 3, we need only to consider(2m; 3)-GBFs with m = 3a · 5b · 7c in view ofProposition 28.

We now consider nonexistence results for a fixedn.

In view of Theorem 20 and Theorem 30, weconclude that there is no (m, n)-GBF if n = 1;and m odd for n = 2.

For n = 3, we need only to consider(2m; 3)-GBFs with m = 3a · 5b · 7c in view ofProposition 28.

Theorem 31

For any odd integer m′, there is no(2m′, 3)-generalized bent function.

We need to determine what Ex can be, giventhat the c-exponent kx divides 210.

To show 7 - kx , we need the following result in[Lam & L]

Theorem 31

For any odd integer m′, there is no(2m′, 3)-generalized bent function.

We need to determine what Ex can be, giventhat the c-exponent kx divides 210.

To show 7 - kx , we need the following result in[Lam & L]

Proposition 32

Let D ∈ N[Cm] be minimal v-sum with reducedexponent k . Then we have the followings:

(a) If k = p is prime and P is the subgroup of orderp, then D = Ph for some h ∈ Cm.

(b) If k =∏t

i=1 pi with t ≥ 2 andp1 < p2 < · · · < pt are prime, then t ≥ 3 and

||D|| ≥ (p1 − 1)(p2 − 1) + (p3 − 1).

Moreover, equality holds only ifD = (P∗1P

∗2 + P∗3 )h for some h ∈ Cm. Here

P∗i = Pi − {e}, and Pi is the subgroup of orderpi .

To determine Ex , we first express it as∑

Di

where each Di ∈ N[Cm] is a minimal v-sum.

Observe that for each minimal v-sum Di , thereduced exponent is either a prime or a productof 3 primes.

Since ||Ex || = 8, we can apply Proposition 31 toeliminate 7.

Then, we determine what each Di can be.

Finally, using some ad hoc calcuations, it can beshown that no much GBF exists.

To determine Ex , we first express it as∑

Di

where each Di ∈ N[Cm] is a minimal v-sum.

Observe that for each minimal v-sum Di , thereduced exponent is either a prime or a productof 3 primes.

Since ||Ex || = 8, we can apply Proposition 31 toeliminate 7.

Then, we determine what each Di can be.

Finally, using some ad hoc calcuations, it can beshown that no much GBF exists.

To determine Ex , we first express it as∑

Di

where each Di ∈ N[Cm] is a minimal v-sum.

Observe that for each minimal v-sum Di , thereduced exponent is either a prime or a productof 3 primes.

Since ||Ex || = 8, we can apply Proposition 31 toeliminate 7.

Then, we determine what each Di can be.

Finally, using some ad hoc calcuations, it can beshown that no much GBF exists.

To determine Ex , we first express it as∑

Di

where each Di ∈ N[Cm] is a minimal v-sum.

Observe that for each minimal v-sum Di , thereduced exponent is either a prime or a productof 3 primes.

Since ||Ex || = 8, we can apply Proposition 31 toeliminate 7.

Then, we determine what each Di can be.

Finally, using some ad hoc calcuations, it can beshown that no much GBF exists.

To determine Ex , we first express it as∑

Di

where each Di ∈ N[Cm] is a minimal v-sum.

Observe that for each minimal v-sum Di , thereduced exponent is either a prime or a productof 3 primes.

Since ||Ex || = 8, we can apply Proposition 31 toeliminate 7.

Then, we determine what each Di can be.

Finally, using some ad hoc calcuations, it can beshown that no much GBF exists.

Some open problems

(i) To study the case n = 4 and odd m.

(ii) To study the case when m = 2a · 3 · 5.

Reference

J. H. Conway and A. J. Jones. TrigonometricDiophantine equations (On vanishing sums ofroots of unity). Acta Arith., 30(3):229–240,1976.

P. V. Kumar, R. A. Scholtz, and L. R. Welch,Generalized bent functions and their properties.J. Combin. Theory Ser. A, 40(1):90–107, 1985.

Y. Jiang, Y. Deng: New results on nonexistenceof generalized bent functions. Des. CodesCryptogr. 75 (2015), 375–385.

P. V. Kumar, R. A. Scholtz, L. R. Welch:Generalized bent functions and their properties.J. Combin. Theory Ser. A 40 (1985), 90–107.

T. Y. Lam and K. H. Leung, On vanishing sumsof roots of unity. J. Algebra, 224(1):91–109,2000.

H. Liu, K. Feng, R. Feng: Nonexistence ofgeneralized bent functions from Zn

2 to Zm. Des.Codes Cryptogr. 82 (2017), 647–662.

O. S. Rothaus: On ’bent’ functions. J. Combin.Theory Ser. A 20 (1976), 300–305.

K.-U. Schmidt: Quaternary constant-amplitudecodes for multicode CDMA. IEEE Trans. Inf.Theory 55 (2009), 1824–1832.

N. Tokareva (2015): Bent functions: results andapplications to cryptography, Academic Press

Some recent results on Hadamard matrices

Theorem 1 (L and Momihara 2019)

Let Φ1 = {q2 : q ≡ 1 ( mod 4) is a prime power},Φ2 = {n4 ∈ N : n ≡ 1 ( mod 2)} ∪ {9n4 ∈ N : n ≡ 1 (mod 2)}, Φ3 = {5} and Φ4 = {13, 37}. Then, the followinghold:

(1) There exists a Hadamard matrix of order 4(2v + 1) forv ∈ Φ1 ∪ Φ2 ∪ Φ3 ∪ Φ4.

(2) There exists a Hadamard matrix of order 4(3v + 1) forv ∈ Φ1 ∪ Φ2 ∪ Φ3.

(3) There exists a Hadamard matrix of order 4(5v + 1) forv ∈ Φ1 ∪ Φ2 ∪ Φ3.

(4) There exists a Hadamard matrix of order 8(uv + 1) foru ∈ Φ1 ∪ Φ2 and v ∈ Φ1 ∪ Φ2 ∪ Φ3.

Using the above result, we obtain some Hadamard matrices oforder 4p, where p is an odd prime, such matrices are known tobe difficult to be constructed. The number of odd n < 1000such that 2n4 + 1 is a prime is 32, and such n < 100 are

1, 3, 21, 45, 63, 81, 105, 153, 177, 201, 219, 225, 249, 279, 297.

Furthermore, the number of odd n < 1000 such that2 · 9n4 + 1 is a prime is 74, and such n < 100 are

1, 3, 5, 31, 45, 55, 57, 71, 79, 89, 107, 109, 119, 123, 137,

141, 159, 167, 173, 181, 197, 217, 255, 275, 285, 295.

Let Bi , i = 1, 2, . . . , `, be ki -subsets of G andB = {Bi : i = 1, 2, . . . , `}.A family B is said to be a difference family withparameters (v ; k1, k2, . . . , k`;λ) in G if

∑i=1

BiB(−1)i = λG +

(∑i=1

ki − λ)· 0G .

For ` = 2, 4, 8, a difference family B is said to be oftype H∗` if

∑`i=1 ki − `(|G |+ 1)/4 = λ.

For ` = 4, B is said to be of type H if∑4i=1 ki − |G | = λ.

It is well known that if there is a difference family oftype H in G , then we have a Hadamard matrix oforder 4|G | by plugging the circulant (−1, 1)matrices obtained from its blocks into theGoethals-Seidel array.For any difference family of type H∗` in G for ` = 2or 4, we construct a Hadamard matrix of order`(|G |+ 1) by plugging the circulant (−1, 1)matrices obtained from its blocks into the Szekeresarray or the Wallis-Whiteman array, respectively.

Theorem 2 (L, Momihara and Xiang 2020)

Let q be a prime power of the formq = 12c2 + 4c + 3 with c an arbitrary integer, andlet n = q2. Then there exists a Hadamard matrix oforder 4(2n + 1).

There are 386 prime powers of the formq = 12c2 + 4c + 3 < 107 while there are 166181 prime powersq < 107 such that q ≡ 3 ( mod 8). The first 58 prime powersof the form q = 12c2 + 4c + 3 < 105 are listed below:

3, 11, 19, 43, 59, 179, 211, 283, 563, 619, 739, 1163, 1499, 1979, 2083,

2411, 3011, 3539, 4259, 4723, 7603, 8011, 8219, 10211, 11411,

12163, 14011, 14563, 14843, 17483, 20011, 23059, 25579, 26699,

28619, 29803, 30203, 33923, 36083, 36523, 41539, 49411, 54139,

55219, 55763, 59083, 60779, 63659, 65419, 69011, 70843, 75211, ,

80363, 81019, 82339, 83003, 88411, 93283.