The document discusses difference systems of sets (DSS), which are sets with certain distance properties. It provides examples of perfect and regular DSS, and discusses approaches to constructing optimal DSS, including using cyclic difference packings, flats in projective geometry, and cyclotomic classes. Optimal constructions are given using finite fields and cyclotomic cosets.

1. the NSFC Grant 10771051
2008 7 ·
Difference Systems of Sets and Related Designs
1 38
leijg1964@yahoo.com.cn

2. 2 38

3. 1
1.1.
n
· · · x1 · · · xn y1 · · · yn · · ·
· · · x1 · · · xi xi+1 · · · xn y1 · · · yi · · · yn · · ·
3 38
· · · x1 · · · xi xi+1 · · · xn y1 · · · yi · · · yn · · ·

4. x = x1 · · · xn y = y1 · · · yn :
Ti (x, y) = xi+1 · · · xn y1 · · · yi , 1 ≤ i ≤ n − 1.
:
n
?
C ⊆ F n)
n ( ( 4 38
C
) . comma-free .

5. 1.2. comma-free
: C ⊆ Fqn , Fq = {0, 1, . . . , q − 1}.
n q
C ⊆ Fqn ρ(C):
comma-free
ρ(C) = min d(z, Ti (x, y)),
x, y, z ∈ C i = 1, . . . , n − 1.
d Hamming ,
(ρ(C) − 1)/2
ρ(C) > 0 comma-free
ρ(C) ( ). 5 38
comma-free .

6. 2
(Levenshtein,1971)
{Qi : i = 0, . . . , q − 1} |Qi | = τi , i =
Zn q ,
0, . . . , q − 1. s, s = 1, . . . , n − 1,
Zn
x − y ≡ s (mod n)
ρ x, y,
x ∈ Qi , y ∈ Qj , i, j = 0, . . . , q − 1, i = j.
6 38
{Qi : i = 0, . . . , q − 1} (n, τ0 , . . . , τq−1 , ρ)
DSS(n, {τ0 , . . . , τq−1 }, q, ρ).
(difference system of sets).

7. Qi (i = 0, . . . , q − 1) m,
DSS DSS(n, m, q, ρ).
regular,
s, 1 ≤ s ≤ n − 1, x − y ≡ s (mod n) ρ ,
DSS perfect.
DSS (redundancy):
q−1
|Qi |.
r=
i=0
n, q, ρ, DSS ,
(optimal). 7 38
DSS n, q, ρ, rq (n, ρ)
: .

8. . Z13 :
D0 = {0, 7, 8, 11}, D1 = {4, 10, 12} , D2 = {1, 2, 3, 5, 6, 9}
regular DSS(13, {4, 3, 6}, 3, 9).
perfect,
. Z37 :
{1, 10, 26}, {3, 4, 30} , {9, 12, 16}
{11, 27, 36}, {7, 33, 34} , {21, 25, 28}
8 38
perfect DSS(37, 3, 6, 6).
regular,

9. DSS
2.1. comma-free
Fqn
q .
DSS
comma-free 0.
ρ>0
comma-free .
q−1
(n, τ0 , . . . , τq−1 , ρ) DSS{Q0 , . . . , Qq−1 }, Zn Qi
i=0
(information symbols), 0,
C ⊆ Fqn .
n−r
C C :
C = C + (h1 , . . . , hn ),
j ∈ Qi (0 ≤ i ≤ q − 1).
hj = i
9 38
ρ(C ) ≥ ρ.
. n = 9, q = 2, ρ = 1 : Q0 = {1, 2}, Q1 = {3, 5}.
h = (001 ∗ 1 ∗ ∗ ∗ ∗).

10. rq (n, ρ)
2.2.
: (Levenshtein, 1971)
qρ(n−1)
rq (n, ρ) ≥ q−1
DSS perfect regular.
, square-
free .
: (Wang, 2007)

 qρ(n−1) + 1, qρ(n−1)
square-free ;
q−1 q−1
rq (n, ρ) ≥ 10 38
 qρ(n−1) , .
q−1
DSS Qi Qj
perfect,
1.

11. . Z31 :
Q0 = {1, 2, 4, 8, 16}, Q1 = {5, 9, 10, 18, 20}, Q2 = {7, 14, 19, 25, 28}.
perfect, regular DSS(31, 5, 3, 5), optimal.
: Levenshtein (1971)
√
2(n − 1) r2 (n, 2) = 2 n − 1 .
r2 (n, 1) =
q ≥ 3, 11 38
.

12. 3
3.1.
2-(n, K, λ) :
{Bi : Bi ⊆ Zn , |Bi | ∈ K}
j ∈ Zn , Bi + j (mod n)
Bi ,
x, y ∈ Zn , x = y λ .
B1 , . . . , Bs (Bi ⊆ Zn ) Zn 2-
(n, K, λ)
{x − y (mod n) : x, y ∈ Bi , i = 1, . . . , s} 12 38
s ∈ Zn ρ .
. 2-(13,3,1) :
: B1 = {1, 3, 9}, B2 = {2, 5, 6}.

13. (cyclic difference packing) m-DP (v, K, λ):
m
P = {Bi : Bi ⊆ Zn , |Bi | ∈ K, 0 ≤ i ≤ m − 1}
d ∈ Zv {0}, d ≡ b − b (mod v) (b, b ) ∈
Bi × Bi B0 , . . . , Bm−1 λ.
d ∈ Zv {0}, B0 , . . . , Bm−1 λ
: ,
P m-DF (v, K, λ).
, 13 38

14. partition-type
P = {B0 , . . . , Bm−1 } m-DP (v, K, λ), Zv
P , .
partition-type
.
B0 = {0}, B1 = {1, 4}, B2 = {2, 3}
partition-type 3-DF (5, {1, 2, 2}, 1).
Z5
14 38

15. 3.2.
q ,t P G(2t + 1, q),
,
2t+2
−1
q
Zv ,v= q−1 .
k-flat: P G(2t + 1, q) k .
GF (q 2t+2 ) GF (q t+1 ) t-flat
q 2t+2 −1 q t+1 −1
S = {0, m, 2m, . . . , (k − 1)m}, m= k=
q t+1 −1 , q−1 .
W GF (q 2t+2 ) F0 ∈ W.
S (t + 1)-flat ,
P G(2t + 1, q) S :
15 38
σ : x → x + m.
, (t + 1)-flat Fi = σ i F0 (0 ≤ i ≤ k − 1) |W| = k,
S.
W = {Fi : 0 ≤ i ≤ k − 1}.

16. . a GF (26 ) a6 + a + 1 = 0.
,
, v = 63, k = 7, m = 9. Ai = Fi S,
P G(5, 2) , Z63
S = {0, 9, 18, 27, 36, 45, 54},
A0 = {1, 6, 8, 14, 38, 48, 49, 52},
A1 = {10, 15, 17, 23, 47, 57, 58, 61},
A2 = {3, 4, 7, 19, 24, 26, 32, 56},
A3 = {2, 12, 13, 16, 28, 33, 35, 41},
A4 = {11, 21, 22, 25, 37, 42, 44, 50},
A5 = {20, 30, 31, 34, 46, 51, 53, 59}, 16 38
A6 = {5, 29, 39, 40, 43, 55, 60, 62}.

17. 4
4.1.
B = {x0 , . . . , xq−1 } (n, q, ρ) ,
Q0 = {x0 }, Q1 = {x1 }, . . . , Qq−1 = {xq−1 }
perfect regular DSS(n, 1, q, ρ).
: DSS .
(n, q, ρ) B, B
:
q−1 17 38
B= Qi ,
i=0
{Qi }q−1 optimal DSS?
i=0

18. (V.D. Tonchev, 2003)
B ⊆ Zn (n, q, λ) ,
q−1
B= Qi ,
i=0
{Qi }q−1
{Q0 , . . . , Qq−1 } q-DP (v, K, λ1 ), i=0
ρ = λ − λ1 DSS. DSS perfect
,
{Q0 , . . . , Qq−1 } 2-(n, K, λ1 ) .
18 38

19. (V.D. Tonchev, 2003)
n = mq + 1 ,α GF (n) .
Q0 = {αq , α2q , . . . , αmq },
Q1 = αQ0 , Q2 = α2 Q0 , . . . , Qq−1 = αq−1 Q0 .
perfect regular DSS(n, m, q, n − m − 1).
n = 19, q = 6, m = 3. DSS(19, 3, 6, 6).
:
{1, 7, 11}, {2, 14, 3}, {4, 9, 6}, {5, 16, 17}, {8, 18, 12}, {10, 13, 15}.
19 38
{1, 7, 11}, {2, 14, 3} perfect, regular DSS(19, 3, 2, 1).

20. (V.D. Tonchev, 2003)
n = 2mq + 1 ≡ 3 (mod 4) ,α GF (n) Dm =
.
{α2iq : 1 ≤ i ≤ m}.
Q0 = Dm , Q1 = α2 Dm , . . . , Qq−1 = α2(q−1) Dm .
perfect regular DSS(n, m, q, (n − 2m − 1)/4).
n = 31, q = 3, m = 5, α = 3.
:
Q0 = {16, 8, 4, 2, 1}, Q1 = {20, 10, 5, 18, 9}, Q2 = {25, 28, 14, 7, 19}.
20 38
perfect regular DSS(31, 5, 3, 5).

21. (Y. Mutoh, V.D. Yonchev, 2004)
n = emq + 1 .
eq eq eq
eq
Q0 = C0 , Q1 = Ce , Q2 = C2e , . . . , Qq−1 = C(q−1)e .
regular DSS(n, m, q, ρ),
q−1
ρ = min{(i, 0)e − (i + je, 0)eq : 0 ≤ i < e}.
j=0
q−1
i, (i, 0)e − (i + je, 0)eq ρ=
, ,
j=0
m(q − 1)/e perfect DSS. 21 38
, V.D. Tonchev (2005) .

22. 4.2.
• partition-type m-DP (v, K, λ) P =
Zv
:
{B0 , . . . , Bm−1 }, K = {|Bi | : 0 ≤ i ≤ m − 1}. Zv
v
v DSS(v, K, m, ρ = v − λ). DSS λ = m + δx ,
,
xvm ,

0, if x = 0, 1;
δx =
if 2 ≤ x ≤ m − 1.
0, 1,
.
22 38
B0 = {0}, B1 = {1, 4}, B2 = {2, 3}
perfect DSS(5, {1, 2, 2}, 3, 4).
Z5

23. • , 1 ≤ x ≤ v.
:x Zv partition-type
DSS(nv, K ∪{(n−1)x}, m+1, ρ),
m-DP (v, K, λ1 ), Znv
ρ = min{v − λ1 , 2x}, r = (n − 1)x + v.
{B0 , . . . , Bm−1 } Zv partition-type cyclic m-DP (v, K, λ1 ).
:
|X| = x
X Zv Znv
. :
∗
Q0 = nB0 , . . . , Qm−1 = nBm−1 , Qm = {a + nb : a ∈ Zn , b ∈ X}.
{Q0 , . . . , Qm−1 , Qm } DSS.
partition-type m-DP (v, K, λ1 )
: 23 38
2x = v − λ1 . DSS perfect.

24. {D0 , D1 } partition-type 2-DF (7, {3, 4}, 3),
Z7
.
D0 = {1, 2, 4}, D1 = {0, 3, 5, 6}.
Z7n :
B0 = {n, 2n, 4n}, B1 = {0, 3n, 5n, 6n},
B2 = {1, . . . , n − 1, n + 1, . . . , 2n − 1}.
perfect DSS(7n, {3, 4, 2n − 2}, 3, 4).
1≤n≤5 DSS
, , optimal. 24 38

25. • 1 ≤ x ≤ v.
: n, x ,n Zv
DSS(nv, K ∪
partition-type m-DP (v, K, λ1 ). Znv
{ n−1 x}, m + 1, ρ), n−1
ρ = min{v − λ1 , x}, r = 2x + v.
2
{B0 , . . . , Bm−1 } Zv partition-type cyclic m-DP (v, K, λ1 ).
:
|X| = x
X Zv , O n Zn .
Znv :
Q0 = nB0 , . . . , Qm−1 = nBm−1 , Qm = {a + nb : a ∈ On , b ∈ X}.
{Q0 , . . . , Qm−1 , Qm } DSS.
25 38
partition-type m-DP (v, K, λ1 )
.
x = v − λ1 . DSS perfect.

26. {D0 , D1 } partition-type 2-DF (7, {3, 4}, 3),
Z7
.
D0 = {1, 2, 4}, D1 = {0, 3, 5, 6}.
n Z7n
. :
B0 = {n, 2n, 4n}, B1 = {0, 3n, 5n, 6n},
n−3
B2 = 2i + 1, 2i + 1 + n, 2i + 1 + 2n, 2i + 1 + 3n : 0 ≤ i ≤ .
2
perfect DSS(7n, {3, 4, 2n − 2}, 3, 4).
n = 3, 5 , DSS
, optimal. 26 38

27. • 1 ≤ x ≤ v.
x
: ,

partition-typem-DP (v, K1 , λ1 ) 

x DSS(v, K2 , q, ρ ) 

DSS(2v, K1 ∪ K2 , m + q, ρ),
=⇒
ρ = min{v − λ1 + ρ , 2x}, r = x + v.
27 38

28. {D0 , D1 , D2 } Z5 partition-type 3-DP (5, {1, 2, 2},
.
1),
D0 = {0}, D1 = {1, 4}, D2 = {2, 3}.
{Q0 , Q1 } Z5 DSS(5, 2, 2, 2),
Q0 = {1, 4}, Q1 = {2, 3}.
Z10 :
B0 = {0}, B1 = {2, 8}, B2 = {4, 6}, B3 = {3, 9}, B4 = {5, 7}.
28 38
DSS(10, {1, 2, 2, 2, 2}, 5, 6).
qρ(n−1) 135
= = 9 = r, DSS optimal .
q−1 2

29. • 1 ≤ x ≤ v. 1 ≤ x ≤ v.
x x
. , ,

partition-typem-DP (v, K1 , λ1 ) 

x DSS(v, K2 , q, ρ ) 

DSS(3v, K1 ∪ K2 , m + q, ρ),
=⇒
ρ = min{v − λ1 + ρ , x}, r = x + v.
29 38

30. {D0 , D1 } Z7 partition-type 2-DP (7, {3, 4}, 3),
.
D0 = {1, 2, 4}, D1 = {0, 3, 5, 6}.
{Q0 , Q1 } Z7 DSS(7, 3, 2, 3),
Q0 = {1, 2, 4}, Q1 = {3, 5, 6}.
Z21 :
B0 = {3, 6, 12}, B1 = {0, 9, 15, 18}, B2 = {4, 7, 13}, B3 = {10, 16, 19}.
30 38
DSS(21, {3, 4, 3, 3}, 4, 6).
√
qρ(n−1)
= 160 = 13 = r, DSS optimal.
q−1

31. DSS, . :
• v
: , :
p ≡ 3 (mod 4)
1. v = p, ;
2. v = 2t − 1, t≥1 ;
3. v = p(p + 2), p p+2 .
Z3v
1. DSS(3v, { v+1 , v+1 , v−1 }, 3, v+1 );
2 2 2 2
2. DSS(3v, { v+1 , v−1 , v−1 }, 3, v−1 );
2 2 2 2
3. DSS(3v, { v−1 , v−1 , v+1 , v+1 }, 4, v).
2 2 2 2
• v
: , :
p ≡ 3 (mod 4)
1. v = 4p, ; 31 38
2. v = 4(2t − 1), t≥1 ;
3. v = 4p(p + 2), p p+2 .
DSS(3v, { v−2 , v , v+2 }, 3, v ).
Z3v 2 2 2 2

32. 4.3.
q 2t+2 −1 t+1
−1
k = q q−1 ,
• v= q−1 , q ,t
: .
perfect DSS(k, {s1 , s2 , . . . , sm }, m, ρ ),
Zk Zv
1) DSS(v, {s1 q t+1 , s2 q t+1 , . . . , sm q t+1 }, m, ρ),
ρ = ρ (q t+1 − q).
2) DSS(v, {k, s1 q t+1 , s2 q t+1 , . . . , sm q t+1 }, m + 1, ρ),
m
t+1
si , ρ q t+1 }.
− q) + 2
ρ = min{ρ (q
i=1
3) DSS(v, {k + s1 q t+1 , s2 q t+1 , . . . , sm q t+1 }, m, ρ),
32 38
m
ρ = min{ρ (q t+1 − q) + 2 si ), ρ q t+1 }.
i=2
DSS
: perfect, .

33. {D0 , D1 , D2 }
. P G(5, 2) , v = 63, k = 7. Z7 perfect
DSS(7, 2, 3, 4),
D0 = {1, 6}, D1 = {2, 5}, D2 = {3, 4}.
P G(5, 2) :
Q0 = A1 ∪ A6 = {5, 10, 15, 17, 23, 29, 39, 40, 43, 47, 55, 57, 58, 60, 61, 62},
Q1 = A2 ∪ A5 = {3, 4, 7, 19, 20, 24, 26, 30, 31, 32, 34, 46, 51, 53, 56, 59},
Q2 = A3 ∪ A4 = {2, 11, 12, 13, 16, 21, 22, 25, 28, 33, 35, 37, 41, 42, 44, 50},
regular DSS(63, 16, 3, 24).
√ 33 38
qρ(n−1)
= 2232 = 48 = r, DSS optimal.
q−1

34. q 2t+2 −1 t+1
−1
k = q q−1 ,
• v= q−1 , q ,t
: .
k perfect DSS(k, {s1 , s2 , . . . , sn }, n, ρ ),
Zk
Zv
1) DSS(v, {k − 1, s1 q t+1 , s2 q t+1 , . . . , sn q t+1 }, n + 1, ρ),
ρ = min{ρ (q t+1 − q) + 2k − 2, ρ q t+1 }.
2) DSS(v, {k − 1 + s1 q t+1 , s2 q t+1 , . . . , sn q t+1 }, n, ρ),
ρ = min{ρ (q t+1 − q) + 2k − 2 − 2s1 , ρ q t+1 }.
34 38
DSS
: perfect, .

35. {D0 , D1 , D2 , D3 }
P G(5, 2) , v = 63, k = 7. Z7
.
perfect DSS(7, {1, 2, 2, 2}, 4, 6),
D0 = {0}, D1 = {1, 6}, D2 = {2, 5}, D3 = {3, 4}.
P G(5, 2) :
Q0 = S ∗ ∪ A0 = {1, 6, 8, 9, 14, 18, 27, 36, 38, 45, 48, 49, 52, 54},
Q1 = A1 ∪ A6 = {5, 10, 15, 17, 23, 29, 39, 40, 43, 47, 55, 57, 58, 60, 61, 62},
Q2 = A2 ∪ A5 = {3, 4, 7, 19, 20, 24, 26, 30, 31, 32, 34, 46, 51, 53, 56, 59},
Q3 = A3 ∪ A4 = {2, 11, 12, 13, 16, 21, 22, 25, 28, 33, 35, 37, 41, 42, 44, 50},
DSS(63, {14, 16, 16, 16}, 4, 46). 35 38
qρ(n−1)
= 11408/3 = 62 = r, DSS optimal.
q−1

36. DSS, . :
2t+2
−1
: v = q q−1 ,
• q ,t>1 Zv
.
DSS :
t t
t
1. DSS(v, q t+1 , q −1 , (q −q)(q −1) );
q−1 q−1
2. DSS(v, q t+1 , q t , q t (q t − 1)(q − 1));
t+1
−1)(q t+1 −q)
t+1
3. DSS(v, q t+1 , q q−1 , (q
−1
);
q−1
t+1 t+1
t+1
DSS(v, q t+1 , q q−1 , (q −2q+1)(q −q) );
−q
4. q−1
t+2 t+1
−q −1
DSS(v, { q q−1 , q t+1 , . . . , q t+1 }, q q−1 , v − q t+1 − 3);
5.
t+2 t
t+2 t+1
DSS(v, { q q−1 , q t+1 , . . . , q t+1 }, q q−1 , (q +q)(q −1) );
−1 −1
6. q−1
t+2
−1
DSS(v, { q q−1 , q t+1 , . . . , q t+1 }, q t , ρ), ρ = q 2t (q − 1) + 36 38
q ≥ 3,
7.
q t (3 − q) − 2;
t+2 t+1
−1 −q
8. DSS(v, { q q−1 , q t+1 , . . . , q t+1 }, q q−1 , ρ), ρ = v − 3q t+1 + 2q − 3.

37. 5
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Ser. II 9(2003), 217-226.
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[8] V. D. Tonchev, Partitions of difference sets and code synchronization, Finite Fields Appl.
11(2005), 601-621.
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58(2006), 161-168.

38. !
38 38