Existence of (q, k, 1) DifferenceFamilies With q a Prime Powerand k = 4, 5

K. Chen, L. ZhuDepartment of Mathematics, Suzhou University, Suzhou 215006, China

Received January 1, 1997; accepted July 24, 1997

Abstract: The existence of a (q, k, 1) difference family in GF(q) has been completely solvedfor k = 3. For k = 4, 5 partial results have been given by Bose, Wilson, and Buratti. In thisarticle, we continue the investigation and show that the necessary condition for the existence ofa (q, k, 1) difference family in GF(q), i.e., q ≡ 1 (mod k(k − 1)) is also sufficient for k = 4, 5.For general k, Wilson’s bound shows that a (q, k, 1) difference family in GF(q) exists wheneverq ≡ 1 (mod k(k − 1)) and q > [k(k − 1)/2]k(k−1). An improved bound on q is also presented.c© 1999 John Wiley & Sons, Inc. J Combin Designs 7: 21–30, 1999

Keywords: difference family; block design

1. INTRODUCTION

Let F = {B1, B2, . . . , Bt} be a family of k-subsets of an additive group G of order v. F iscalled simple difference family [briefly denoted (v, k, 1)-DF] if any nonzero element of Gcan be represented in a unique way as a difference of two elements lying in some memberof F .

For k = 3, it has been known in Netto [7] that for any prime power q = 6t + 1 a(q, 3, 1)-DF in GF(q) always exists. For k = 4, 5 the existence of a (q, k, 1)-DF in GF(q)has been investigated by Bose [2], Wilson [8], and Buratti [3], [4], the known results canbe summarized as follows:

Correspondence to: Dr. ZhuContract grant sponsor: NSFC; contract grant number: 19231060-2c© 1999 John Wiley & Sons, Inc. CCC 1063 8539/99/010021-10

21

22 CHEN AND ZHU

Lemma 1.1 ([3]). (i) For any prime p ≡ 1(mod 12) there exists a (p, 4, 1)-DF in GF(p)if p < 106 or p > 612. (ii) For any (nonprime) prime power q ≡ 1(mod 12) there exists a(q, 4, 1)-DF in GF(q).

Lemma 1.2 ([3], [4]). For any prime power q ≡ 1(mod 20) there exists a (q, 5, 1)-DFin GF(q) if q > 1020, or q is a prime and q < 104.

Let q = pn be a power of a prime p. The congruence q ≡ 1 (mod 20) is equivalent tothe following cases:

p ≡ 1 (mod 20) and n arbitrary.

p ≡ 9, 11, 19 (mod 20) and n even.

p ≡ 2, 3 (mod 5), p odd and n ≡ 0 (mod 4).

For these cases partial results have been presented in [4]. With slight modification wepresent them as follows.

Lemma 1.3 ([4]). Let p be prime. We have:

(i) For p ≡ 1(mod 20) a (pn, 5, 1)-DF in GF(pn) exists if a (p, 5, 1)-DF in GF(p) exists.(ii) For p ≡ 9(mod 20) a (p2n, 5, 1)-DF in GF(p2n) exists if a (p2, 5, 1)-DF in GF(p2)

exists.(iii) For p ≡ 2, 3(mod 5) a (p4n, 5, 1)-DF in GF(p4n) always exists.

In this article we shall improve the known results in Lemmas 1.1 to 1.3 and show thatfor k = 4, 5 a (q, k, 1)-DF in GF(q) exists whenever q ≡ 1 (mod k(k − 1)). Specifically,we shall prove the following.

Theorem 1.4. For any prime power q ≡ 1(mod 12) there exists a (q, 4, 1)-DF in GF(q).

Theorem 1.5. For any prime power q ≡ 1(mod 20) there exists a (q, 5, 1)-DF in GF(q).

For general background on difference families and related block designs see [1]. Forrecent results on difference families see [6]. For more constructive results of (q, 4, 1) and(q, 5, 1) difference families see [2] and [4].

2. IMPROVED BOUNDS

In this section, we shall improve the bound p > 612 for k = 4 and the bound q > 1020 fork = 5 in Lemmas 1.1 and 1.2. Respectively, they can be lowered to be p ≥ 14441 and q ≥28139101. We mainly follow the approach in [8] and [9]. We need the following lemma.

Lemma 2.1 ([9]). Let α1, . . . , αn be real numbers, α = (α1 + · · · + αn)/n theirmean, and V = (1/n)

∑1≤i≤n(αi − α)2 their variance. If we put N = α1 + · · · + αh,

0 ≤ h ≤ n, then

|N − hα|2 ≤ h(n − h)V ≤ n2V/4.

Let q = me + 1 be a prime power and F = GF(q). The cyclic multiplicative group ofF has a unique subgroup C0 of index m (and order e). The multiplicative cosets Ci, i ∈

EXISTENCE OF (q, k, 1) DIFFERENCE FAMILIES 23

Zm = {0, 1, . . . , m − 1} are the cyclotomic classes of index m. For a set X, Xr denotesthe set of all r-tuples (x1, . . . , xr) of elements of X , and X(r) denotes the subset of Xr

consisting of (x1, . . . , xr) with x1, . . . , xr distinct. Thus if |X| = n, then |Xr| = nr and|X(r)| = n(r) = n(n − 1) · · · (n − r + 1).

Let (i) = (i1, . . . , ir) ∈ (Zm)r and (a) = (a1, . . . , ar) ∈ F (r). Let E(i)(a) denote thenumber of field elements x ∈ F such that

x − a1 ∈ Ci1 , x − a2 ∈ Ci2 , . . . , x − ar ∈ Cir.

Lemma 2.2 ([9]). The mean value of E(i)(a) over the nr = mrq(r) choices of (i) ∈(Zm)r and (a) ∈ F (r) is αr = (q − r)/mr, and the variance Vr of these nr quantities is

Vr = q(q − 1)[(q − m − 1)/m](r)/nr + αr − (αr)2 < αr.

Let q = me + 1 be a prime power, where m = k(k − 1)/2. For 0 ≤ r ≤ k, defineMr to be the set of (a1, . . . , ar) ∈ F (r) such that the differences aj − ai(1 ≤ i < j ≤ r)represent r(r − 1)/2 distinct cyclotomic classes of index m. Let Mr = |Mr|. ThusM0 = 1, M1 = q, M2 = q(q − 1). If Mk > 0, then there is (a1, . . . , ar) ∈ F (k) such thatthe differences aj − ai(1 ≤ i < j ≤ k) represent m distinct cyclotomic classes of indexm. Thus we may rewrite Wilson’s theorem in [8] as follows.

Lemma 2.3 ([8]). Let q = me + 1 be a prime power, where m = k(k − 1)/2. IfMk > 0, then there exists a (q, k, 1)-DF in GF(q).

Let hr = [m− r(r −1)/2](r)Mr. From the proof of Theorem 2.2 in [9], Mr+1 is a sumof hr of the quantities E(i)(a). By Lemmas 2.1 and 2.2, we have

Lemma 2.4. Let q = me + 1 be a prime power, where m = k(k − 1)/2. Then

|Mr+1 − hrαr|2 ≤ hr(nr − hr)Vr < hr(nr − hr)αr, (1)

|Mr+1 − hrαr|2 ≤ (nr)2Vr/4 < (nr)2αr/4. (2)

Lemma 2.5. If hr ≥ nr/(αr + 1), then Mr+1 > 0 for 2 ≤ r ≤ k − 1.

Proof. From (1) in Lemma 2.4, we have

Mr+1 > hrαr −√

hrαr(nr − hr) = (√

hrαr −√

nr − hr)√

hrαr.

When hr ≥ nr/(αr + 1), we have hrαr ≥ nr − hr. It follows that Mr+1 > 0.

Lemma 2.6. For 2 ≤ r ≤ k − 1, hr+1 > [m − r(r + 1)/2](r+1)[hrαr − ( 12 )nr

√αr].

Proof. By definition,

hr+1 = [m − r(r + 1)/2](r+1)Mr+1.

From (2) in Lemma 2.4, we have

Mr+1 > hrαr − ( 12 )nr

√αr.

Combining these two expressions gives the conclusion.

24 CHEN AND ZHU

Lemma 2.7. For any prime power q ≡ 1(mod 12) there exists a (q, 4, 1)-DF in GF(q)if q ≥ 14441.

Proof. Let k = 4. By Lemma 2.3 we need only to show that M4 > 0 if q ≥ 14441.By Lemma 2.5, what we need to show is h3 ≥ n3/(α3 + 1). By Lemma 2.6, this can beaccomplished if we can show the following

[m − 3](3)[h2α2 − ( 1

2

)n2

√α2

] ≥ n3/(α3 + 1). (3)

Since m = k(k − 1)/2 = 6,

h2 = [m − 1](2)M2 = 20q(q − 1),α2 = (q − 2)/62, α3 = (q − 3)/63,

n2 = 62q(q − 1), n3 = 63q(q − 1)(q − 2),

(3) becomes

6q(q − 1)[20(q − 2)/62 − 3√

q − 2] ≥ 63q(q − 1)(q − 2)/[(q − 3)/63 + 1]. (4)

Simplifying (4) we get

20(q − 2) ≥ 108√

q − 2 + 67(q − 2)/[q − 3 + 63].

That is

1 ≥ 108/[20√

q − 2] + 67/20[q − 3 + 63]. (5)

It is readily checked that (5) holds when q = 14441. It is also clear that the right-handside of (5) is a decreasing function in q. So, (5) holds for all q ≥ 14441. This completesthe proof.

Let

f(q, m, r) =q(r+1)

mr(r+1)/2

r+1∏

i=3

mim(3) − 1

2

r∑i=3

r+1∏j=i

mjm(i−1)(i+1)/2√

q − i + 1

− 12

mr+1mr(r+2)/2

√q − r

](6)

where m = k(k − 1)/2 and mi = [m − i(i − 1)/2](i), 3 ≤ i ≤ r + 1.

Lemma 2.8. For 3 ≤ r ≤ k − 2, hr+1 > f(q, m, r).

Proof. By Lemma 2.6, we have

h3 > m3(h2α2 − ( 1

2

)n2

√α2

). (7)

By (7) and Lemma 2.6, we get

h4 > m4(h3α3 − ( 1

2

)n3

√α3

)> m4m3h2α2α3 − ( 1

2

)m4m3n2α3

√α2 − ( 1

2

)m4n3

√α3. (8)

By induction, we obtain

hr+1 >r+1∏i=3

mi

r∏j=2

αjh2 − 12

r∑i=3

r+1∏j=i

mj

r∏t=i

αtni−1√

αi−1 − 12mr+1nr

√αr. (9)

EXISTENCE OF (q, k, 1) DIFFERENCE FAMILIES 25

Note that in (9), αj = (q − j)/mj , h2 = [m − 1](2)q(q − 1), ni = miq(i). The right-handside of (9) becomes the right-hand side of (6). Then the conclusion follows from (9).

Let

g(q, m, r) =12

r∑i=3

r+1∏j=i

mjm(i−1)(i+1)/2√

q − i + 1+

12

mr+1mr(r+2)/2

√q − r

+m(r+1)(r+4)/2

q − (r + 1) + mr+1 .

Lemma 2.9. Let q = me + 1 be a prime power, m = k(k − 1)/2 and k ≥ 5. ThenMk > 0 if

k−1∏i=3

mim(3) ≥ g(q, m, k − 2).

Proof. By Lemmas 2.8 and 2.5, we need only to show that

f(q, m, k − 2) ≥ nk−1/(αk−1 + 1).

Notice that nk−1 = mk−1q(k−1) and αk−1 = (q − k + 1)/mk−1, this is equivalent to theinequality given in the lemma.

Lemma 2.10. For any prime power q ≡ 1(mod 20) there exists a (q, 5, 1)-DF in GF(q)if q ≥ 28139101.

Proof. Let k = 5. By Lemma 2.3 we need only to show that M5 > 0 if q ≥ 28139101.Since

m = k(k − 1)/2 = 10, m3 = [m − 3](3) = 210,

m4 = [m − 6](4) = 24, m(3) = 720,

g(q, 10, 3) =12

210 · 24 · 104√

q − 2+

12

24 · 1015/2√

q − 3+

1014

q − 4 + 104 .

It is easily checked that for q = 28139101 we have

4∏i=3

mim(3) ≥ g(q, 10, 3).

It is also clear that g(q, 10, 3) is a decreasing function in q. So, this inequality holds for allq ≥ 28139101. Then the conclusion follows from Lemma 2.9.

3. MAIN RESULTS

We can now prove Theorem 1.4.

Theorem 1.4. For any prime power q ≡ 1(mod 12) there exists a (q, 4, 1)-DF in GF(q).

Proof. By Lemma 1.1 we need only to deal with primes p ≡ 1 (mod 12) in [106, 612]. Thisinterval has been covered by Lemma 2.7.

26 CHEN AND ZHU

In order to prove Theorem 1.5, by Lemma 2.10 we need only to discuss prime powersq ≡ 1 (mod 20) and q < 28139101. By Lemmas 1.2 and 1.3 we need only to consider thefollowing cases:

i. q ≡ 1 (mod 20) is a prime, q ∈ [104, 28139101];iia. p ≡ 9, 11, 19 (mod 20) is a prime, q = p2 and p < 5305;iib. p ≡ 11, 19 (mod 20) is a prime, q = p4 and p ∈ {11, 19, 31, 59, 71}, or q = 116.

By Lemma 2.3, it suffices to find a 5-tuple (a1, . . . , a5) in GF(q) such that the differencesaj −ai(1 ≤ i < j ≤ 5) represent 10 distinct cyclotomic classes of index 10. In fact, in mostcases we take (a1, . . . , a5) = (0, 1, b, b2, b3). The values of b are determined by a computerprogram. Let e = (q − 1)/10. It is easy to see that two elements x and y are in the samecyclotomic class of index 10 if and only if xe = ye. What we actually did in the programis to search for b such that the 10 differences have their eth powers all distinct. Since thevalue of e may be quite large, we express e in its binary form so that the computation canbe reduced to square and multiplication in GF(q).

Because of the following known theorem in [2] we need only to consider those q suchthat 5 is a 4th power in GF(q).

Theorem 3.1 ([2]). For any prime power q ≡ 1(mod 20) there exists a (q, 5, 1)-DF inGF(q) if 5 is not a 4th power in GF(q).

Remark. One of the referees pointed out that an improved version of this theorem (seeBuratti [4]) can further reduce the problem to only consider prime powers q = 20t + 1such that (11 + 5

√5)/2 is not a 2e+1th power in GF(q), where 2e is the largest power of

2 dividing t. Although we think that Theorem 3.1 is more convenient to implement, wethank the referee for his comment that some numbers in our original Tables I and II can bedeleted if the improved version is used.

Lemma 3.2. For any prime q ≡ 1(mod 20) and q ∈ [104, 28139101], there exists a(q, 5, 1)-DF in GF(q).

Proof. With the aid of a computer an element b of GF(q) satisfying the properties mentionedabove has been found for any prime q ≡ 1 (mod 20) and q ∈ [104, 28139101], where 5 is a

TABLE I. Pairs (q, b) for 104 < q < 2 × 104

q b q b q b q b q b

10301 −129 10501 163 10781 189 11161 127 11821 −8911981 82 12101 53 12281 −421 12301 63 12401 −19012421 −525 12541 103 12601 −40 12641 −26 12781 −20112821 −7 12941 −21 13721 −23 13781 378 13921 −26414341 −339 14741 −60 14821 −193 15061 78 15101 −715241 72 15541 131 15581 119 16381 −8 16661 9016741 −148 16901 −274 16981 6 17021 −17 17341 −8918061 558 18301 106 18341 35 18661 −259 19141 819301 −119 19421 −52 19501 361 19541 119 19661 3819961 4

EXISTENCE OF (q, k, 1) DIFFERENCE FAMILIES 27

TABLE II. Triples (p, m, b), p ≡ 9 (mod 20) and p < 5305

p m b p m b p m b p m b

29 2 x + 10 229 2 6x + 15 349 2 x + 17 509 2 x + 452709 2 x + 501 809 3 x + 81 1009 11 x + 161 1049 3 x + 448

1109 2 x + 22 1229 2 x + 23 1289 3 x + 129 1549 2 x + 1771669 2 x + 43 1709 2 x + 448 1789 2 x + 384 2029 2 x + 4552069 2 x + 259 2389 2 x + 293 2729 3 x + 316 3049 11 x + 1573389 2 x + 302 3469 2 x + 258 3929 3 x + 131 3989 2 x + 954229 2 x + 455 4549 2 x + 125 4649 3 x + 1153 4789 2 x + 1335209 11 x + 645

4th power in GF(q). Here, we list the pairs (q, b) in Table I for 104 < q < 2 × 104, whichhas been reduced by the above remark.

Lemma 3.3. For any prime p ≡ 9, 11, 19(mod 20) and p < 5305, there exists a(p2, 5, 1)-DF in GF(p2).

Proof. First, we take a nonsquare element m in GF(p) and take f(x) = x2 − m as theirreducible polynomial to construct a GF(p2). With the aid of a computer an element bof GF(p2) satisfying the properties mentioned above has been found for any prime p ≡9, 11, 19 (mod 20) and p < 5305. We list the triples (p, m, b) in Tables II, III, and IV. InTable II the numbers have been reduced by the above remark.

In Table III, there are two missing cases where p = 11 and 31. To construct a (112, 5, 1)-DF and a (312, 5, 1)-DF in GF(p2), we take f(x) = x2 − 2 or x2 − 3, and (a1, . . . , a5) =(0, 1, 3, x + 4, 7x + 10), or (0, 1, 3, x + 8, 4x + 23), respectively. It is readily checked thatthe differences aj − ai(1 ≤ i < j ≤ 5) represent 10 distinct cyclotomic classes of index10 in GF(p2).

Lemma 3.4. For any prime p ∈ {11, 19, 31, 59, 71}, there exists a (p4, 5, 1)-DF inGF(p4). There exists also a (116, 5, 1)-DF in GF(116).

Proof. For p ∈ {11, 19, 31, 59, 71}, we take f(x) as follows to construct GF(p4). In eachcase we take b = x + 2.

p = 11, f(x) = x4 + x + 2.

p = 19, f(x) = x4 + x + 8.

p = 31, f(x) = x4 + x + 1.

p = 59, f(x) = x4 + x + 1.

p = 71, f(x) = x4 + x + 11.

For GF(116), we take f(x) = x6 + x + 2, and b = 3. It is readily checked that thedifferences aj − ai(1 ≤ i < j ≤ 5) of (a1, . . . , a5) = (0, 1, b, b2, b3) represent 10 distinctcyclotomic classes of index 10.

We can now prove Theorem 1.5.

Theorem 1.5. For any prime power q ≡ 1(mod 20) there exists a (q, 5, 1)-DF in GF(q).

28 CHEN AND ZHU

TABLE III. Triples (p, m, b), p ≡ 11 (mod 20) and p < 5305

p m b p m b p m b p m b

11 2 no 31 3 no 71 7 2x + 4 131 2 2x + 130151 3 2x + 18 191 7 x + 64 211 2 x + 135 251 2 x + 72271 3 3x + 139 311 11 x + 89 331 2 2x + 153 431 7 x + 257491 2 x + 88 571 2 x + 140 631 3 x + 14 691 2 x + 567751 3 x + 16 811 2 x + 361 911 7 x + 550 971 2 x + 473991 3 x + 621 1031 7 x + 417 1051 2 x + 278 1091 2 x + 133

1151 13 x + 124 1171 2 x + 145 1231 3 x + 133 1291 2 x + 12311451 2 x + 167 1471 3 x + 646 1511 11 x + 29 1531 2 x + 601571 2 x + 36 1811 2 x + 252 1831 3 x + 69 1871 7 x + 641931 2 x + 85 1951 3 x + 423 2011 2 x + 169 2111 7 x + 662131 2 x + 44 2251 2 x + 46 2311 3 x + 448 2351 13 x + 1482371 2 x + 213 2411 2 x + 996 2531 2 x + 31 2551 3 x + 322591 7 x + 46 2671 3 x + 340 2711 7 x + 242 2731 2 x + 1362791 3 x + 454 2851 2 x + 370 2971 2 x + 31 3011 2 x + 4303191 11 x + 74 3251 2 x + 419 3271 3 x + 107 3331 2 x + 93371 2 x + 384 3391 3 x + 219 3491 2 x + 525 3511 3 x + 3493571 2 x + 3 3631 3 x + 28 3671 13 x + 83 3691 2 x + 5563581 2 x + 58 3911 13 x + 98 3931 2 x + 407 4051 2 x + 1124091 2 x + 509 4111 3 x + 13 4211 2 x + 297 4231 3 x + 6244271 7 x + 247 4391 7 x + 198 4451 2 x + 774 4591 3 x + 144651 2 x + 34 4691 2 x + 178 4751 13 x + 328 4831 3 x + 1674871 11 x + 116 4931 2 x + 48 4951 3 x + 70 5011 2 x + 75051 2 x + 218 5171 2 x + 48 5231 7 x + 11

Proof. By Lemmas 1.2, 1.3, and 2.10 we need only to deal with the cases (i), (iia), and(iib). They have been taken care of by Lemmas 3.2, 3.3, and 3.4, respectively.

4. REMARKS

For general k, Wilson [8] gave a bound q > [k(k − 1)/2]k(k−1) for which a (q, k, 1)-DFalways exists in GF(q) whenever q is a prime power and q ≡ 1 (mod k(k −1)). By Lemma2.9 we can get an improved bound. In fact, since

k−1∏i=3

mim(3) = m!,

k−1∏j=i

mj = [m − i(i − 1)/2]!,

the following inequality implies the inequality in Lemma 2.9

m! ≥ 12

k−2∑i=3

(m − i(i − 1)/2)!m(i−1)(i+1)/2√

q − (k − 2)+

12

mk−1mk(k−2)/2√

q − (k − 2)+

m(k−1)(k+2)/2

q − (k − 2).

EXISTENCE OF (q, k, 1) DIFFERENCE FAMILIES 29

TABLE IV. Triples (p, m, b), p ≡ 19 (mod 20) and p < 5305

p m b p m b p m b p m b

19 2 8x + 13 59 2 2x + 41 79 3 6x + 69 139 2 x + 125179 2 7x + 132 199 3 x + 144 239 7 x + 211 359 7 x + 20379 2 x + 11 419 2 2x + 383 439 3 x + 234 479 13 x + 183499 2 x + 369 599 7 x + 274 619 2 x + 170 659 2 x + 38719 11 x + 85 739 2 x + 108 839 11 x + 652 859 2 x + 765919 3 x + 260 1019 2 x + 38 1039 3 x + 296 1259 2 x + 333

1279 3 x + 29 1319 13 x + 138 1399 3 x + 171 1439 7 x + 5601459 2 x + 220 1499 2 x + 660 1559 17 x + 122 1579 2 x + 2011619 2 x + 71 1699 2 x + 98 1759 3 x + 128 1879 3 x + 1761979 2 x + 238 1999 3 x + 13 2039 7 x + 26 2099 2 x + 3472179 2 x + 274 2239 3 x + 422 2339 2 x + 99 2399 11 x + 42459 2 x + 334 2539 2 x + 75 2579 2 x + 141 2659 2 x + 942699 2 x + 336 2719 3 x + 175 2819 2 x + 46 2879 7 x + 852939 2 x + 148 2999 17 x + 141 3019 2 x + 168 3079 3 x + 1093119 7 x + 38 3259 2 x + 1063 3299 2 x + 105 3319 3 x + 7003359 11 x + 176 3499 2 x + 304 3539 2 x + 511 3559 3 x + 573659 2 x + 341 3719 7 x + 248 3739 2 x + 150 3779 2 x + 4463919 3 x + 315 4019 2 x + 220 4079 11 x + 210 4099 2 x + 964139 2 x + 26 4159 3 x + 201 4219 2 x + 239 4259 2 x + 134339 2 x + 38 4519 3 x + 543 4639 3 x + 132 4679 11 x + 154759 3 x + 131 4799 7 x + 351 4919 13 x + 243 4999 3 x + 3155039 11 x + 152 5059 2 x + 117 5099 2 x + 90 5119 3 x + 825179 2 x + 1278 5279 7 x + 48

This inequality is further implied by the following inequality, which gives the explicit boundwe want

q ≥ k − 2 +

[B +

√B2 + 4AC

2A

]2

,

where A = m!, C = m(k−1)(k+2)/2 and

B =12

[mk−1m

k(k−2)/2 +k−2∑i=3

(m − i(i − 1)/2)!m(i−1)(i+1)/2

].

Note added in revision. The existence of (q, 6, 1) difference families has also been estab-lished recently by the present authors for any prime powers q = 30t+1 with one exceptionof q = 61, where the method of character sums is used (see [5]).

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[2] R. C. Bose, On the construction of balanced incomplete block designs, Ann. Eugenics 9 (1939),353–399.

[3] M. Buratti, Constructions for (q, k, 1) difference families with q a prime power and k = 4, 5,Discrete Math. 138 (1995), 169–175.

30 CHEN AND ZHU

[4] , Improving two theorems of Bose on difference families, J. Combin. Designs 3 (1995),15–24.

[5] K. Chen and L. Zhu, Existence of (q, 6, 1) difference families with q a prime power, preprint.

[6] C. J. Colbourn and J. H. Dinitz, CRC Handbook of combinatorial designs, CRC Press, NewYork, 1996.

[7] E. Netto, Zur theorie der triplesysteme, Math. Ann. 42 (1893), 143–152.

[8] R. M. Wilson, Cyclotomy and difference families in elementary abelian groups, J. NumberTheory 4 (1972), 17–47.

[9] , Constructions and uses of pairwise balanced designs, Math. Centre Tracts 55 (1974),18–41.