JJ II J I - cscgt.orgcscgt.org/2009conf/ppt/葛根年.pdf · Applications: The designs of missile...

Introduction

Inequalities for Perfect . . .

Additive Sequences of . . .

PDFs with “holes” and . . .

Direct Constructions . . .

Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 1 of 56

Go Back

Full Screen

Close

Quit

http://www.math.zju.edu.cn/~ggn/

Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 2 of 56

Go Back

Full Screen

Close

Quit

NSFC, Grant No. 10825103 and No. 10771193.31th, July, 2009· Zhejiang University

Perfect Difference Families, PerfectDifference Matrices, and Related

Combinatorial Structures

Gennian Ge

Department of MathematicsZhejiang University

Hangzhou 310027, Zhejiang, [email protected]

(Joint with Ying Miao and Xianwei Sun)


mailto:[email protected]

Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 3 of 56

Go Back

Full Screen

Close

Quit

Outlines

Part I Introduction

Part II Inequalities for Perfect Systems of Difference Sets

Part III Additive Sequences of Permutations

Part IV PDFs with “holes” and PDMs with “holes”

Part V Direct Constructions for PDMs and PDPs

Part VI Recursive Constructions for PDFs and PDPs

Part VII Concluding Remarks


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 4 of 56

Go Back

Full Screen

Close

Quit

1 IntroductionIn order to measure various spatial frequencies rela-tive to some area of the sky,n movable antennas areused inm successive configurations. The distancesbetween antennas within configurations determinewhich frequencies are obtained. Thesem successiveconfigurations ofn antennas can give up tom

(n2

)spacings. It is thus required that these distances be-tween antennas within configurations are successivemultiples of a given increment, without hole or re-dundancy.

1. F. Biraud, E. J. Blum and J. C. Ribes, “On optimum synthetic lineararrays,”IEEE Trans. Antennas Propagation, vol. AP 22, pp. 108–109, 1974.

2. F. Biraud, E. J. Blum and J. C. Ribes, “Some new possibilities of op-timum synthetic linear arrays for radioastronomy,”Astronomy andAstrophysics, vol. 41, pp. 409–413, 1975.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 5 of 56

Go Back

Full Screen

Close

Quit

Perfect Systems of Difference Sets

Let k ≥ 3 and v be positive integers. For a givenk-subsetB = b1, b2, . . . , bk of the setIv = 0, 1, . . . , v − 1 of non-negativeintegers, let∆B = bj − bi | 1 ≤ i < j ≤ k be thelist of directeddifferences fromB.

Let c, m, p1, . . . , pm be positive integers, and letS = S1, S2, . . . , Sm,whereSi = si1, si2, . . . , sipi

, 0 ≤ si1 < si2 < · · · < sipi, and

all sij ’s are integers. We say that∆S = ∆S1, ∆S2, . . . , ∆Sm isa perfect system of difference sets forc (or starting with c, orwith threshold c), or briefly, an (m, p1, p2, . . . , pm, c)-PSDS, if∆S = c, c + 1, . . . , c− 1 +

∑1≤i≤m

(pi

2

). An (m, p1, p2, . . . , pm, c)-

PSDS isregular if p1 = p2 = · · · = pm = p. As usual, a regular(m, p, c)-PSDS is abbreviated to(m, p, c)-PSDS.

1. J. -C. Bermond, A. Kotzig and J. Turgeon, “On a combinatorial problem ofantennas in radioastronomy,”Colloq. Math. Soc. Janos Bolyai, vol. 18, pp.135–149, 1978.

2. A. Kotzig and J. Turgeon, “Regular perfect systems of difference sets,”Dis-crete Math., vol. 20, pp. 249–254, 1977.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 6 of 56

Go Back

Full Screen

Close

Quit

Perfect Difference Family∼= (t, k, 1)-PSDS

Let k ≥ 3, t ≥ 1, andv = k(k−1)t+1 be positive in-tegers. For a given collectionB = B1, B2, . . . , Btof k-subsets ofIv, let ∆B = ∪1≤i≤t∆Bi be thelist ofdirected differences fromB.

If ∆B = 1, 2, . . . , (v − 1)/2, thenB is called aperfect difference family, or briefly, a(v, k, 1)-PDF.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 7 of 56

Go Back

Full Screen

Close

Quit

Examples

A (12× 1 + 1, 4, 1)-PDF:

0, 1, 4, 6 =⇒ 1, 4, 3, 6, 5, 2

A (12× 4 + 1, 4, 1)-PDF:

0, 5, 22, 24 =⇒ 5, 22, 17, 24, 19, 2

0, 7, 13, 23 =⇒ 7, 13, 6, 23, 16, 10

0, 3, 14, 18 =⇒ 3, 14, 11, 18, 15, 4

0, 1, 9, 21 =⇒ 1, 9, 8, 21, 20, 12


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 8 of 56

Go Back

Full Screen

Close

Quit

Related Combinatorial Structures and Codes

1.Cyclic Difference Families

2.Optical Orthogonal Codes

3.Difference Triangle Sets

4.Convolutional Codes

5.Radar Arrays

6.Graceful Labeling of Graphs


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 9 of 56

Go Back

Full Screen

Close

Quit

Cyclic Difference Families

Let Zv be the group of residues modulov. A (v, k, 1)difference family in Zv, briefly a cyclic (v, k, 1)-DF, is a collectionF of k-subsets ofZv such that∆F ∪ (−∆F) = Zv \ 0.

A cyclic (v, k, 1)-DF =⇒ A (v, k, 1)-BIBD.

A (v, k, 1)-PDF=⇒ A cyclic (v, k, 1)-DF.

R. J. R. Abel and M. Buratti, “Difference families,” inHandbookof Combinatorial Designs, Second Edition,C. J. Colbourn and J. H.Dinitz, Eds., Boca Raton, FL: Chapman & Hall/CRC, pp. 392–410,2007.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 10 of 56

Go Back

Full Screen

Close

Quit

Optical Orthogonal Codes

Let v, k, λ be positive integers. A(v, k, λ optical orthogonal code,or briefly (v, k, λ)-OOC,C, is a family of (0, 1)-sequences (called thecodewords) of length v and weight k satisfying the following twoproperties:

(1) (The Auto-Correlation Property )∑0≤t≤v−1xtxt+i ≤ λ for anyx = (x0, x1, . . . , xv−1) ∈ C and any

integeri 6≡ 0 modv;

(2) (The Cross-Correlation Property)∑0≤t≤v−1xtyt+i ≤ λ for any x = (x0, x1, . . . , xv−1) ∈ C, y =

(y0, y1, . . . , yv−1) ∈ C with x 6= y, and any integeri.

1 0 0 0 0 0 0 1 0 1 0 0 0 0 01 1 0 0 1 0 0 0 0 0 0 0 0 0 0

0 1 2 3 4 5 6 7 8 9 a b c d e

(15, 3, 1)−OOC : 0, 7, 9; 0, 1, 4


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 11 of 56

Go Back

Full Screen

Close

Quit

Optical Orthogonal Codes

From a set-theoretic perspective, a(v, k, 1) optical orthogonal code,or (v, k, 1)-OOC, is a collectionC of k-subsets ofZv such that∆F ∪ (−∆F) does not have repeated elements inZv \ 0 andthe set-wise stabilizer of eachk-subset ofC is the subgroup0 of Zv.A (v, k, 1)-OOC isoptimal if its size reaches the upper boundb v−1

k(k−1)c.

A (v = k(k − 1)t + 1, k, 1)-PDF =⇒ An optimal (v + i, k, 1)-OOCwith size exactly equal tob v−1+i

k(k−1)c = v−1k(k−1) = t for 0 ≤ i ≤ k(k−1)−1.

Motivation: Optical code-division multiple-access communicationsystems, mobile radio, frequency-hopping spread-spectrum com-munications, radar, sonar signal design, collision channel withoutfeedback, neuromorphic networks, etc.

T. Helleseth, “Optical orthogonal codes”, inHandbook of Combina-torial Designs, Second Edition,C. J. Colbourn and J. H. Dinitz, Eds.,Boca Raton, FL: Chapman & Hall/CRC, pp. 321–322, 2007.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 12 of 56

Go Back

Full Screen

Close

Quit

Difference Triangle Sets

An (n, k)-difference triangle set, or briefly (n, k)-D∆S, is a set∆ = ∆1, ∆2, . . . , ∆n, where∆i = ai0, ai1, . . . , aik, 1 ≤ i ≤ n, is aset of non-negative integers such that0 = ai0 < ai1 < · · · < aik

in the real number system, and the differencesail − aij with1 ≤ i ≤ n and 0 ≤ j < l ≤ k are all distinct. Letm(∆) = maxaik | 1 ≤ i ≤ n be thescopeof ∆, so the scope is atleast equal ton

(k+1

2

). An (n, k)-D∆S is regular if m(∆) = nk(k+1)

2 .Let M(n, k) = minm(∆) | ∆ is an(n, k)-D∆S. If m(∆) = M(n, k),then∆ is said to beoptimal. Clearly, any regular D∆S is optimal.

A regular(n, k)-D∆S⇐⇒ A (2n(k+1

2

)+ 1, k + 1, 1)-PDF.

Applications: The designs of missile guidance codes, convolutionalself-orthogonal codes, optical orthogonal codes, the allocation radiofrequencies to reduce intermodulation interference, and in variousother areas.

J. B. Shearer, “Difference triangle sets”, inHandbook of CombinatorialDesigns, Second Edition,C. J. Colbourn and J. H. Dinitz, Eds., BocaRaton, FL: Chapman & Hall/CRC, pp. 436–440, 2007.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 13 of 56

Go Back

Full Screen

Close

Quit

Convolutional Codes

Let F [x] denote the ring of polynomials in the indeterminatex over afinite field F , andF [x]n the set of alln-tuplesf = (f1(x), . . . , fn(x))of polynomials fromF [x]. A convolutional codeC is defined to be asubset ofF [x]n, which is (a) closed under componentwise addition andsubtraction, and (b) closed under multiplication by elements ofF [x].

Applications: Digital radio, mobile phones and satellite links.

A construction for convolutional self-orthogonal codes (CSOC’s) basedon D∆S’s was introduced by Robinson and Bernstein. The CSOC cor-responding to the(I, J)–D∆S has generator polynomials

gi(x) = ΣJj=1x

aij , 1 ≤ i ≤ I.

J. P. Robinson and A. J. Bemstein, “A class of binary recurrent codeswith limited error propagation,0IEEE Trans. Inform. Theory, vol.IT-13, pp. 106-113, 1967.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 14 of 56

Go Back

Full Screen

Close

Quit

Radar Arrays

Let N andM be two positive integers. Aradar array R is anN ×M(0, 1)-array with a single1 per column, such that the horizontal auto-correlation function only has values0, 1, andM .

The radar array problem is tofind the maximum number of columnssuch that any out-of-phase horizontally shifted copy has at most asingle1 in common.

S. W. Golomb and H. Taylor, “Two-dimensional synchronization pat-terns for minimum ambiguity,”IEEE Trans. Inform. Theory, vol. 28,pp. 600–604, 1982.

A (v, k, 1)-PDF=⇒ Radar Arrays, See

G. Ge, A. C. H. Ling and Y. Miao, “A system construction for radararrays,”IEEE Trans. Inform. Theory, vol. 54, pp. 410–414, 2008.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 15 of 56

Go Back

Full Screen

Close

Quit

Graceful Labeling of Graphs

Let G = (V, E) be an undirected finite graph without loops or multipleedges. Agraceful labeling (or β-labeling) of a graphG = (V, E) withn vertices andm edges is a one-to-one mappingΨ of the vertex setV (G) into the set0, 1, 2, . . . ,m with this property: if we define, forany edgee = (u, v) ∈ E(G), the valueΩ(e) = |Ψ(u) − Ψ(v)|, thenΩis a one-to-one mapping of the setE(G) onto the set1, 2, . . . ,m. Agraph is calledgraceful if it has a graceful labeling.

The concept of a graceful labeling has been introduced by Rosa as ameans of attacking the famous conjecture of Ringel thatK2n+1 can bedecomposed into2n + 1 subgraphs that are all isomorphic to a giventree withn edges.

A (2n(k2

)+ 1, k, 1)-PDF is equivalent to a graceful labeling of a graph

with n connected components all isomorphic to the complete graph onk vertices.

A. Rosa, “On certain valuations of the vertices of a graph,” inTheoriedes graphes, journees internationales d’etudes,Rome, 1966, Dunod,Paris, 1967, pp. 349–355.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 16 of 56

Go Back

Full Screen

Close

Quit

Known Existence Results

A simple numerical necessary condition for the existence of a( v−1k(k−1), k, 1)-PSDS, i.e. a(v, k, 1)-PDF, isv ≡ 1 (mod k(k − 1)).

Kotzig and Turgeon proved that:

Theorem 1.1For k = 3, the existence problem has been completelysettled: a(v, k, 1)-PDF exists if and only ifv ≡ 1, 7 (mod 24).

Bermond, Kotzig and Turgeon proved that:

Theorem 1.2Perfect difference families cannot exist fork ≥ 6.

1. A. Kotzig and J. Turgeon, “Regular perfect systems of differencesets,”Discrete Math., vol. 20, pp. 249-254, 1977.

2. J. -C. Bermond, A. Kotzig and J. Turgeon, “On a combinatorialproblem of antennas in radioastronomy,”Colloq. Math. Soc. JanosBolyai, vol. 18, pp. 135–149, 1978.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 17 of 56

Go Back

Full Screen

Close

Quit

Known Existence Results for PDFs withk = 4, 5

Theorem 1.3 1. Let m = 12t + 1. Then(m, 4, 1)-PDFs exist for thefollowing values oft < 50: 1, 4− 33, 36, 41.

2. Suppose a(12t + 1, 4, 1)-PDF exists. Then(m, 4, 1)-PDFs exist form = 60t+13, 156t+13, 228t+49, 276t+61, 300t+61 and300t+73.

3. Let m = 20t + 1. Then(m, 5, 1)-PDFs are known fort = 6, 8, 10but for no other small values oft.

4. There are no(m, k, 1)-PDFs for the following values:(a) k = 4, m ∈ 25, 37,(b) k = 5, m ≡ 21 (mod 40) or m ∈ 41, 81.

1. R. Mathon, “Constructions for cyclic Steiner2-designs,”Ann. DiscreteMath., vol. 34, pp. 353–362, 1987.

2. J. H. Huang and S. S. Skiena, “Gracefully labeling prisms,”Ars Combin.,vol. 38, pp. 225–236, 1994.

3. R. J. R. Abel and M. Buratti, “Difference families,” inHandbook of Com-binatorial Designs, Second Edition,C. J. Colbourn and J. H. Dinitz, Eds.,Boca Raton, FL: Chapman & Hall/CRC, pp. 392–410, 2007.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 18 of 56

Go Back

Full Screen

Close

Quit

2 Inequalities for Perfect Systems of Dif-ference SetsWe consider an(m, N, c)-PSDS∆S = ∆S1, ∆S2, . . . , ∆Sm, whereSi = si1, si2, . . . , sipi

andN = p1, . . . , pm.

Theorem 2.1There is no(m, N, c)-PSDS ifpi > 5 for all pi ∈ N.

Next, we consider the average sizep of N , wherep = 1m

∑mi=1 pi = 1

mS.Suppose that there are exactlya even elements inN .

Theorem 2.2(1) If p ≥ 4 +√

6, then there is no(m, N, c)-PSDS.

(2) If a ≤ m/7 and p ≥ 6, or a = m and p ≥ 6, then there is no(m, N, c)-PSDS.

Lemma 2.3In an (m, 5, 7, c)-PSDS, the number of blocks of size5

is at least18m+5(2c−1)−√

∆14 , where∆ = 18(2m− 2c + 1)2 + 7(2c− 1)2.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 19 of 56

Go Back

Full Screen

Close

Quit

3 Additive Sequences of PermutationsLet X(1) = (x

(1)1 , . . . , x

(1)m ) be an ordered set of distinct integers. For

j = 2, . . . , n, let X(j) = (x(j)1 , . . . , x

(j)m ) be a permutation of distinct inte-

gers inX(1). Then the ordered set(X(1), X(2), . . . , X(n)) is called anadditivesequence of permutationsof length n andorder m, ASP(n,m) in short, if forevery subsequence of consecutive permutations of the ordered setX(1), theirvector-sum is again a permutation ofX(1). The setX(1) is usually called thebasisof the additive sequence of permutations. In the application of additivesequences of permutations to perfect systems of difference sets, one often con-siders ASP(n,m) with basisX(1) = (−r,−r + 1, . . . ,−1, 0, 1, . . . , r − 1, r),wherem = 2r + 1 andr a positive integer, unless otherwise stated.

Example 3.1An ASP(4, 5):

X(1) = (−2,−1, 0, 1, 2), X(2) = (0, 1, 2,−2,−1),

X(3) = (1,−2, 0, 2,−1), X(4) = (1, 0,−1,−2, 2).−2 −1 0 1 2−2 0 2 −1 1−1 −2 2 1 0

0 −2 1 −1 2

.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 20 of 56

Go Back

Full Screen

Close

Quit

ASP⇔ PSDS

Kotzig and Turgeon discovered that arbitrarily large perfect systemsof difference sets can be constructed from smaller ones via additivesequences of permutations. On the contrary, certain perfect systemsof difference sets can also be used to construct additive sequences ofpermutations.

1. A. Kotzig and J. Turgeon, “Perfect systems of difference sets andadditive sequences of permutations,”Congress. Numer., vol. 23–24, pp. 629–636, 1979.

2. J. M. Turgeon, “An upper bound for the length of additive sequencesof permutations,”Utilitas Math., vol. 17, pp. 189–196, 1980.

3. J. M. Turgeon, “Construction of additive sequences of permutationsof arbitrary lengths,”Ann. Discrete Math., vol. 12, pp. 239–242,1982.

4. J. Abrham, “Perfect systems of difference sets – A survey,”ArsCombin., vol. 17A, pp. 5–36, 1984.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 21 of 56

Go Back

Full Screen

Close

Quit

PSDS⇒ ASP

Theorem 3.2 1. If there exists an(m, 4, c)-PSDS, then one can con-struct explicitly an ASP(3, 12m) with the basis containing the ele-ments±c,±(c + 1), . . . ,±(6m + c − 1). Furthermore, ifc = 1,then one can also construct explicitly an ASP(3, 12m + 1) with thebasis containing the elements0 ∪ ±1,±2, . . . ,±6m.

2. If there exists an(m, 5, c)-PSDS, then one can construct ex-plicitly an ASP(4, 20m) with the basis containing the elements±c,±(c + 1), . . . ,±(10m + c − 1). Furthermore, ifc = 1, thenone can also construct explicitly an ASP(4, 20m + 1) with the basiscontaining the elements0 ∪ ±1,±2, . . . ,±10m.

J. Abrham, “Perfect systems of difference sets – A survey,”Ars Com-bin., vol. 17A, pp. 5–36, 1984.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 22 of 56

Go Back

Full Screen

Close

Quit

Known Existence Results on Additive Sequencesof Permutations

Turgeon proved the following:

Theorem 3.3If there is an ASP(n, m) with basisX(1) = (−r,−r +1, . . . ,−1, 0, 1, . . . , r−1, r), wherem = 2r+1 andr a positive integer,then the conditionn ≤ m− 1 must be satisfied.

Theorem 3.4Let (X(j) = (x(j)1 , . . . , x

(j)2r+1) | j = 1, . . . , n) and(Y (j) =

(y(j)1 , . . . , y

(j)2s+1) | j = 1, . . . , n) be an ASP(n, 2r+1) and an ASP(n, 2s+

1), respectively. Then(Z(j) = (z(j)1 , . . . , z

(j)(2r+1)(2s+1)) | j = 1, . . . , n),

where

z(j)(i−1)(2s+1)+h = (2s + 1)x

(j)i + y

(j)h , 1 ≤ i ≤ 2r + 1, 1 ≤ h ≤ 2s + 1

is an ASP(n, (2r + 1)(2s + 1)).

J. M. Turgeon, “An upper bound for the length of additive sequences ofpermutations,”Utilitas Math., vol. 17, pp. 189–196, 1980.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 23 of 56

Go Back

Full Screen

Close

Quit

Known Existence Results on Additive Sequencesof Permutations

Lemma 3.51.There is no ASP(3, m) for m = 9, 11.

2.There exists an ASP(4, 5i) for any positive integeri.

3.There exists an ASP(3, m) for any m =5i57i713i1315i1517i1719i1945i45121i121161i161201i201,whereij, j = 5, 7, 13, 15, 17, 19, 45, 121, 161, 201,are non-negative integers, not all equal to0.

J. Abrham, “Perfect systems of difference sets – Asurvey,”Ars Combin., vol. 17A, pp. 5–36, 1984.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 24 of 56

Go Back

Full Screen

Close

Quit

Perfect Difference Matrix

An n × m matrix D = (dij) with entries fromIm = 0, 1, . . . ,m− 1 is called aperfect differencematrix , denoted by PDM(n,m), if the entries ofeach row ofD comprise all the elements ofIm, andDts = −m−1

2 , . . . ,−1, 0, 1, . . . , m−12 holds for any

0 ≤ s < t ≤ n− 1.

Example 3.6A PDM(4, 5): 0 1 2 3 40 2 4 1 31 0 4 3 22 0 3 1 4

,

−2 −1 0 1 2−2 0 2 −1 1−1 −2 2 1 0

0 −2 1 −1 2

.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 25 of 56

Go Back

Full Screen

Close

Quit

PDM ∼= ASP

Theorem 3.7For any odd m, a PDM(n, m) is equivalent to anASP(n, m) based on−m−1

2 , . . . ,−1, 0, 1, . . . , m−12 .

Proof. Let Di, 0 ≤ i ≤ n−1, be theith row of a PDM(n, m) D = (dij). Define

X(1) = D0 −m− 1

2= (d00 −

m− 1

2, . . . , d0,m−1 −

m− 1

2),

X(1) + X(2) = D1 −m− 1

2= (d10 −

m− 1

2, . . . , d1,m−1 −

m− 1

2),

...

X(1)+ · · ·+X(n) = Dn−1−m− 1

2= (dn−1,0−

m− 1

2, . . . , dn−1,m−1−

m− 1

2).

Then (X(1), X(2), . . . , X(n)) is an ASP(n,m) based on−m−1

2 , . . . ,−1, 0, 1, . . . , m−12 .

Conversely, let (X(1), X(2), . . . , X(n)) be an ASP(n, m) based on−m−1

2 , . . . ,−1, 0, 1, . . . , m−12 . For all1 ≤ i ≤ n, let

X(1)+· · ·+X(i)+m− 1

2= (x

(1)1 +· · ·+x

(i)1 +

m− 1

2, . . . , x(1)

m +· · ·+x(i)m +

m− 1

2)

be the(i− 1)th row of ann×m matrixD. ThenD is a PDM(n, m).


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 26 of 56

Go Back

Full Screen

Close

Quit

4 PDFs with “holes” and PDMs with“holes”ASP⇒ PDF

Theorem 4.1Let Di = 0, ai, bi, ci, i = 1, . . . , t, be the blocks of a(12t + 1, 4, 1)-PDF, and letX(1), X(2), X(3) be an ASP(3, m), wherem = 2r + 1, r ≥ 2. Then fori = 1, . . . , t andj = 1, . . . ,m, the6tmdirected differences from the family

∆m(i−1)+j = 0, mai + αj, mbi + βj, mci + γj

cover the set [r + 1, 6tm + r]. Here α = (α1, . . . , αm),β = (β1, . . . , βm) and γ = (γ1, . . . , γm) are the m-vectorsX(1), X(1) + X(2), X(1) + X(2) + X(3), respectively.

R. Mathon, “Constructions for cyclic Steiner2-designs,”Ann. DiscreteMath., vol. 34, pp. 353–362, 1987.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 27 of 56

Go Back

Full Screen

Close

Quit

Adjoining Additional Blocks

In order to utilize Theorem4.1 for constructing newperfect difference families, we need to finds addi-tional blocks with directed differences covering theset[1, r] and possibly the set[6mt + r + 1, 6mt + 6s]for some integers ≥ 0. We first estimate this numbers.

Lemma 4.2Let s be the number of the additionalblocks whose directed differences cover the set[1, r]and possibly the set[6mt + r + 1, 6mt + 6s]. Then

r/6 ≤ s ≤ r/2.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 28 of 56

Go Back

Full Screen

Close

Quit

Perfect Difference Packings

Let B = B1, B2, . . . , Bh, where Bi =bi1, bi2, . . . , biki

, 1 ≤ i ≤ h, be a collection ofhsubsets ofIv = 0, 1, . . . , v − 1 calledblocks, andK = k1, k2, . . . , kh. If the list of directed differ-ences∆B = bij−bil | i = 1, . . . , h, 1 ≤ l < j ≤ kicovers each element of the set1, 2, . . . , v−1

2 \L ex-actly once, whereL ⊆ 1, 2, . . . , v−1

2 , then we callBa (v, K, 1) perfect difference packing, or (v, K, 1)-PDP, withdifference leaveL.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 29 of 56

Go Back

Full Screen

Close

Quit

Semi-perfect Group Divisible Designs

Let m be an odd integer,n a positive integer, andKa subset of positive integers. LetF be a collectionof subsets (calledblocks) in In × Im of sizes fromK such that any block intersects anyi × Im, i ∈In, in at most one point. For any two distincti, j ∈In, define the(i, j)-differences∆ijF =

∑B∈F∆ijB,

where∆ijB = a<b | (i, a), (j, b) ∈ B. F is calleda semi-perfect group divisible design, denotedK-SPGDD of typemn if, for any two distincti, j ∈ In,∆ijF = −m−1

2 , . . . ,−1, 0, 1, . . . , m−12 . As usual, a

k-SPGDD will be abbreviated to ak-SPGDD.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 30 of 56

Go Back

Full Screen

Close

Quit

PDF+ SPGDD⇒ PDF

Theorem 4.3If there exist a(v, K, 1)-PDP with difference leaveL and aK ′-SPGDD of typemk for anyk ∈ K, then there exists an(mv,K ′, 1)-PDP withdifference leave[1, m−1

2 ] ∪ L, whereL = ml − m−12 , . . . ,ml − 1, ml,ml +

1, . . . ,ml + m−12 | l ∈ L.

Proof. LetB be the collection of blocks of a given(v, K, 1)-PDP with differenceleaveL. For eachB = b0, b1, . . . , bk−1 ∈ B, by hypothesis, we know thatk ∈ K and there exists aK ′-SPGDD of typemk, F(B), overIk × Im. We canthen construct a collectionA(B) of new blocks with sizes fromK ′ as follows:

A(B) = mbi1 + x1, mbi2 + x2, . . . ,mbik′ + xk′ |

(i1, x1), (i2, x2), . . . , (ik′, xk′) ∈ F(B).In this way, asB ranges over all blocks ofB, we obtain a collection of newblocksA = ∪B∈BA(B), each having size fromK ′, and each element of thenew blocks not exceedingm(v − 1) + (m− 1) = mv − 1. For any two distincti, j ∈ Ik, since∆ijF(B) = −m−1

2 , . . . ,−1, 0, 1, . . . , m−12 , the list of direct

differences fromA = ∪B∈BA(B) is then

∆A = ∪B∈B∆A(B) = m× 1− m− 1

2, . . . ,m× v − 1

2+

m− 1

2 \ L.

ThereforeA is the collection of blocks of the desired(mv,K ′, 1)-PDP.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 31 of 56

Go Back

Full Screen

Close

Quit

Corollary

Theorem 4.4Let l be a non-negative integer. LetDi = 0, ai, bi, ci, i = 1, . . . , t, be the blocks of a(12t + 2l + 1, 4, 1)-PDP with difference leaveL =[1, l], and letX(1), X(2), X(3) be an ASP(3, m), wherem = 2r + 1 and r ≥ 2. Then fori = 1, . . . , t andj = 1, . . . ,m, the6tm directed differences from thefamily

∆m(i−1)+j = 0, mai + αj, mbi + βj, mci + γj

cover the set[(2l + 1)r + l + 1, 6tm + (2l + 1)r + l],where α = (α1, . . . , αm), β = (β1, . . . , βm) andγ = (γ1, . . . , γm) are the m-vectorsX(1), X(1) +X(2), X(1) + X(2) + X(3), respectively.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 32 of 56

Go Back

Full Screen

Close

Quit

Incomplete Perfect Difference Matrix

An n × (m − h) matrix ∆ = (δij) with entries fromIm \ (rIh + r′),where rIh + r′ = ri + r′ | i ∈ Ih ⊆ Im for somer, r′ ∈ Z,is called an incomplete perfect difference matrix with a regu-lar hole H = −h−1

2 r, . . . ,−r, 0, r, . . . , h−12 r, denoted briefly by

IPDM(n, m; h, r), if the entries of each row of∆ comprise all the ele-ments ofIm\(rIh+r′), and for all0 ≤ s < t ≤ n−1, the lists of differ-ences∆ts = δtj− δsj | 0 ≤ j ≤ m−h− 1 are all identical and∆ts =−m−1

2 , . . . ,−1, 0, 1, . . . , m−12 \r·−h−1

2 , . . . ,−1, 0, 1, . . . , h−12 holds

for any0 ≤ s < t ≤ n− 1.

Note that here we are not interested in the parameterr′ since the differ-ences are invariant under any translation of the matrix byr′J , whereJis the all-one matrix. Whenh = 1, we can drop the letterr from the no-tation IPDM(n, m; 1, r) since for anyr ∈ Z, we always haveH = 0.Clearly, by adding the column vector(r′, . . . , r′)T to an IPDM(n, m; 1),we immediately obtain a PDM(n, m).


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 33 of 56

Go Back

Full Screen

Close

Quit

Orthogonal Arrays

An orthogonal array OA(n, k) is ann × k2 array,A, with entries from a setX of k elements such that,within any two rows ofA, every ordered pair of ele-ments fromX occurs in exactly one column ofA.

An orthogonal arrayA is idempotent if it containsthek distinctn× 1 vectors(x, x, . . . , x)T | x ∈ Xas columns ofA.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 34 of 56

Go Back

Full Screen

Close

Quit

PDP+Idempotent OA(n, k) ⇒ IPDM

Theorem 4.5Leth be a non-negative integer. If thereexist a(t−1

2 , h)-regular (m, K, 1)-PDP and an idem-potent OA(n, k) for eachk ∈ K, then there exists anIPDM(n− 1, m; t, h).

Furthermore, if there exist an IPDM(n − 1, t; 1) anda PDM(n − 1, t), then so do an IPDM(n − 1, m; 1)and a PDM(n− 1, m), respectively.

Corollary 4.6 Suppose that there exists an(m, K, 1)-PDF. If for eachk ∈ K, there exists an idempotentOA(n, k), then there also exists an IPDM(n−1, m; 1).


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 35 of 56

Go Back

Full Screen

Close

Quit

Proof for “PDP+Idempotent OA (n, k) ⇒ IPDM”

For each block Bi of size ki of the (m, K, 1)-PDP, B =B1, B2, . . . , Bh, we form an idempotent orthogonal arrayOi withn rows andki

2 columns over theki points ofBi, and then remove theidempotent part ofOi. In this way, we obtain a small “incomplete”orthogonal arrayO′

i with n rows andki(ki − 1) columns without theidempotent part for each block of the(m, K, 1)-PDP. Concatenatingthese small “incomplete” orthogonal arraysO′

i, we obtain a large “in-complete” orthogonal arrayO′ without the idempotent part

O′ = ( O′1 O′

2 · · · O′h ) .

Take out one row fromO′, subtract this row from all other rows term byterm, then we obtain an(n−1)× (

∑1≤i≤hki(ki−1)) arrayO′′. Adding

m−12 to each element ofO′′, we obtain an IPDM(n− 1,

∑1≤i≤hki(ki −

1) + t; t, h) with a regular hole−t−12 h, . . . ,−h, 0, h, . . . , t−1

2 h.From the definition of a perfect difference packing, we know that∑

1≤i≤hki(ki − 1) = m− t.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 36 of 56

Go Back

Full Screen

Close

Quit

PDF+IPDM(3, m; 1) ⇒ PDP

Theorem 4.7Let u, r, m be integers such that1 ≤u ≤ r andm = 2r + 1. Suppose that there exist anIPDM(3, m; 1), a (12t+1, 4, 1)-PDF, a(24t+1, 4, 1)-PDF. Then there exists a(12t(m + 1) + 2r + 1, 4, 1)-PDP with difference leave[1, r].


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 37 of 56

Go Back

Full Screen

Close

Quit

Proof for “PDF+IPDM (3, m; 1) ⇒ PDP”

Let Di = 0, ai, bi, ci, i = 1, . . . , t, be the blocks of a(12t + 1, 4, 1)-PDF. Then fori = 1, . . . , t andj = 1, . . . ,m−1, the6t(m−1) directeddifferences from the family

∆′m(i−1)+j = 0, (m + 1)ai + αj, (m + 1)bi + βj, (m + 1)ci + γj

cover the set[r + 1, 6t(m + 1) + r] \ (r + 1)[1, 12t]. Hereα = (α1, . . . , αm−1), β = (β1, . . . , βm−1) and γ = (γ1, . . . , γm−1)are the row vectors of a3 × (m − 1) matrix obtained by subtractingrfrom each element of the IPDM(3, m; 1), respectively.

Let El = 0, dl, el, fl, l = 1, 2, . . . , 2t, be the blocks of a(24t+1, 4, 1)-PDF. Then the12t directed differences from the family

Θl = 0, (r + 1)dl, (r + 1)el, (r + 1)fl

cover the set(r + 1)[1, 12t].


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 38 of 56

Go Back

Full Screen

Close

Quit

5 Direct Constructions for PDMs andPDPs

Example: An IPDM(3, 23; 1).

(7 9 4 6 5 10 2 1 8 3 11

18 6 9 15 13 4 12 3 1 2 714 17 15 10 3 1 5 11 2 4 16

).

For eachm, we list only (m − 1)/2 columns ofan IPDM(3, m; 1). The other(m − 1)/2 non-zerocolumns are given by subtracting each element of thelisted columns fromm.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 39 of 56

Go Back

Full Screen

Close

Quit

More Examples:

Example: An IPDM(3, 29; 5, 5).

(7 9 2 3 4 1 13 8 12 6 11 14

−1 −4 13 −9 7 −8 6 14 11 2 −3 128 3 14 −1 −7 −12 4 11 −2 13 9 6

).

We list only m−h2 columns of an IPDM(3, m; h, r). Multiply each ele-

ment of the listedm−h2 columns by−1 to obtainm−h

2 new columns. LetD be the matrix consisting of thesem − h columns. It is then readilychecked that the matrix obtained by addingm−1

2 to each element ofDis an IPDM(3, m; h, r).

Furthermore, if there exists a PDM(3, h), then subtracth−12 from each

element of this PDM(3, h), multiply each resultant element byr, andthen include the new3×h matrix intoD to form a3×m matrix. Finally,addm−1

2 to each element of this3×m matrix. It is easily checked thatthis matrix is a PDM(3, m).


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 40 of 56

Go Back

Full Screen

Close

Quit

PDP+Idempotent OA(n, k) ⇒ IPDM

Lemma 5.1There exists a regular(m, K, 1)-PDPwith difference leaveL for any m, K and L listedbelow:

m K L m K L m K L

51 4, 5 7, 14, 21 53 4 3, 6 55 4 4, 8, 1263 4, 5 5, 10, 15 65 4 13, 26 67 4 4, 8, 1271 4, 5 6, 12, 18 73 4 6, 12, . . . , 36 75 4, 5 6, 12, 18


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 41 of 56

Go Back

Full Screen

Close

Quit

List of IPDM (3, m; 1) and PDM(3, m) withm < 200

Theorem 5.21.There exists an IPDM(3, m; 1)for any odd integer5 ≤ m < 200 exceptfor m = 9, 11 and except possibly form ∈5, 7, 15, 19, 21, 27, 29, 35, 37, 43, 47, 51, 53, 55, 59,63, 67, 71, 75, 79, 83, 87, 95.

2.There exists a PDM(3, m) for any odd integer5 ≤m < 200 except form = 9, 11 and except possiblyfor m = 59.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 42 of 56

Go Back

Full Screen

Close

Quit

Direct Constructions for Perfect Difference Pack-ings

Lemma 5.3There exists a(v, 4, 1)-PDP with difference leaveL for anyv andL listed below:

v L v L v L v L v L v L

63 1 65 1, 2 75 1 77 1, 2 87 1 91 [1, 3]

99 1 101 1, 2 103 [1, 3] 111 1 113 1, 2 115 [1, 3]

123 1 125 1, 2 127 [1, 3] 135 1 137 1, 2 139 [1, 3]

141 [1, 4] 147 1 149 1, 2 151 [1, 3] 153 [1, 4] 159 1163 [1, 3] 165 [1, 4] 171 1 173 1, 2 177 [1, 4] 179 [1, 5]

183 1 185 1, 2 187 [1, 3] 189 [1, 4] 191 [1, 5] 195 1197 1, 2 199 [1, 3] 201 [1, 4] 203 [1, 5] 207 1 209 1, 2211 [1, 3] 213 [1, 4] 215 [1, 5] 219 1 221 1, 2 223 [1, 3]

225 [1, 4] 227 [1, 5] 231 1 233 1, 2 235 [1, 3] 237 [1, 4]

239 [1, 5] 243 1 245 1, 2 247 [1, 3] 249 [1, 4] 251 [1, 5]

255 1 257 1, 2 259 [1, 3] 261 [1, 4] 263 [1, 5] 271 [1, 3]

273 [1, 4] 275 [1, 5] 283 [1, 3] 285 [1, 4] 295 [1, 3]


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 43 of 56

Go Back

Full Screen

Close

Quit

Direct Constructions for PDPs

1. PDF ⇒ (2x + 12s + 1, 4, 1)-PDP with difference leaveL =[r + 1, x + r]

2. PDF ⇒ (2x + 12s + 3, 4, 1)-PDP with difference leaveL =1 ∪ [r + 1, x + r]

3. PDF ⇒ (2x + 12s + 5, 4, 1)-PDP with difference leaveL =1, 2 ∪ [r + 1, x + r]

4. PDF ⇒ (2x + 12s + 7, 4, 1)-PDP with difference leaveL =[1, 3] ∪ [r + 1, x + r]

5. PDF ⇒ (2x + 12s + 9, 4, 1)-PDP with difference leaveL =[1, 4] ∪ [r + 1, x + r]


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 44 of 56

Go Back

Full Screen

Close

Quit

Direct Constructions for PDPs

1. PDP with Difference LeaveL = 1 ⇒ (2x + 12s + 2u + 1, 4, 1)-PDP with difference leaveL = [1, u] ∪ [3r + 2, x + 3r + 1]

2. PDP with Difference LeaveL = 1, 2 ⇒ (2x + 12s + 2u + 1, 4, 1)-PDP with difference leaveL = [1, u] ∪ [5r + 3, x + 5r + 2]

3. PDP with Difference LeaveL = 1, 2, 3⇒ (2x+12s+2u+1, 4, 1)-PDP with difference leaveL = [1, u] ∪ [7r + 4, x + 7r + 3]

4. PDP with Difference LeaveL = 1, 2, 3, 4 ⇒ (2x + 12s + 2u +1, 4, 1)-PDP with difference leaveL = [1, u] ∪ [9r + 5, x + 9r + 4]

5. PDP with Difference LeaveL = 1, 2, 3, 4, 5 ⇒ (2x + 12s + 2u +1, 4, 1)-PDP with difference leaveL = [1, u]∪ [11r + 6, x + 11r + 5]


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 45 of 56

Go Back

Full Screen

Close

Quit

6 Recursive Constructions for PDFs andPDPsPDFs⇒ PDFs

Theorem 6.1If there exists a(12t + 1, 4, 1)-PDF, then there exists a(12(mt + s) + 1, 4, 1)-PDF for anym ands listed below:

m s m s m s m s

25 5, 6 27 5, 6 29 6, 7 31 6, 7

33 6, 7, 8 35 7, 8 37 7, 8, 9 39 7, 8, 9

41 8, 9, 10 43 8, 9, 10 45 8, 9, 10, 11 47 9, 10, 11

49 4, 9, 10, 11, 12 51 10, 11, 12 53 10, 11, 12, 13 55 10, 11, 12, 13

57 11, 12, 13, 14 61 5, 11, 12, 13, 14, 15 63 12, 13, 14, 15 65 12, 13, 15, 16

67 12, 13, 14, 15 69 14, 15, 16, 17 71 14, 16 73 6, 14, 15, 16

75 15, 16, 17 77 16, 17, 18 79 16 81 16, 17

83 17 85 7, 17, 18, 19 87 20 89 19, 20

95 20 97 8, 21 99 20 103 21


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 46 of 56

Go Back

Full Screen

Close

Quit

PDPs⇒ PDFs

Theorem 6.2If there exists a(12t + 3, 4, 1)-PDP with difference leaveL = 1, then there exists a(12(mt + s) + 1, 4, 1)-PDF for anym ands listed below:

m s m s m s m s m s

13 7, 8, 9 15 8, 9, 10, 11 17 10, 11, 12 19 11, 12, 13, 14 21 12, 13, 14, 15

23 14, 15, 16, 17 25 15, 16, 17 27 16, 17 29 20

Theorem 6.3If there exists a(12t + 5, 4, 1)-PDP with difference leaveL = [1, 2], then there exists a(12(mt + s) + 1, 4, 1)-PDF for anym ands listed below:

m s m s m s m s m s m s

5 5, 6 7 7, 8 13 12, 13, 15, 16 15 15, 16, 17 17 17, 18, 19 19 20


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 47 of 56

Go Back

Full Screen

Close

Quit

PDPs⇒ PDFs


m s m s

5 7, 8 7 4, 9, 10, 11, 12


m s m s

5 8, 9, 10, 11 7 12, 13, 14, 15


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 48 of 56

Go Back

Full Screen

Close

Quit

PDFs⇒ PDPs

Theorem 6.6 If there exists a(12t + 1, 4, 1)-PDF, then we have the following assertions.

(1) There exists a(12(mt + s) + 3, 4, 1)-PDP with difference leaveL = 1 for anym and slisted below:

m s m s m s m s m s

31 7 35 8 39 9 43 10 47 10, 11

(2) If there exists an(m, 4, 1)-PDP with difference leave1, then there also exists a((12t +1)m, 4, 1)-PDP with difference leave1.


(1) There exists a(12(mt + s) + 5, 4, 1)-PDP with difference leaveL = [1, 2] for anym andslisted below:

m s m s m s m s m s

5 0 25 5 35 7 37 8 39 8

41 9 43 8, 9 45 9, 10 47 9, 10 49 10, 11

51 10, 11 53 12 55 11, 12 63 13 65 13, 14

(2) If there exists an(m, 4, 1)-PDP with difference leave[1, 2], then there also exists a((12t +1)m, 4, 1)-PDP with difference leave[1, 2].


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 49 of 56

Go Back

Full Screen

Close

Quit

PDFs⇒ PDPs

Theorem 6.8Suppose that both a(12t + 1, 4, 1)-PDF and a PDM(3, m) exist. Then we have thefollowing assertions.



7 0 35 7 39 8 43 9 45 9 47 10

49 10 51 11 53 11 55 12 59 13 75 15

(2) If there exists an(m, 4, 1)-PDP with difference leave[1, 3], then there also exists a((12t +1)m, 4, 1)-PDP with difference[1, 3].



m s m s m s

45 9 55 11 75 15

(2) If there exists an(m, 4, 1)-PDP with difference leave[1, 4], then there also exists a((12t +1)m, 4, 1)-PDP with difference[1, 4].


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 50 of 56

Go Back

Full Screen

Close

Quit

PDPs⇒ PDPs

Theorem 6.10If there exists a(12t+3, 4, 1)-PDP with difference leaveL = 1, then there exists a(12(mt + s) + 2u + 1, 4, 1)-PDP withdifference leave[1, u] for anym, s andu listed below:

m s u m s u m s u m s u

13 9 1 13 8 2 15 9 2 15 10 2

21 13 2 13 8 3 17 11 3 15 9 4

Theorem 6.11If there exists a(12t+5, 4, 1)-PDP with difference leaveL = [1, 2], then there exists a(12(mt + s) + 2u + 1, 4, 1)-PDP withdifference leave[1, u] for anym, s andu listed below:

m s u m s u m s u m s u

7 8 1 5 5 2 7 7 2 13 14 2

7 7 3 15 15 3 15 15 4


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 51 of 56

Go Back

Full Screen

Close

Quit

PDPs⇒ PDPs


m s u m s u m s u m s u m s u

5 8 1 5 7 2 7 10 2 7 11 2 7 10 3


m s u m s u

5 9 4 7 13 2

Theorem 6.14If there exists a(12t + 11, 4, 1)-PDP with differenceleaveL = [1, 5], then there exists a(12(mt + s) + 2u + 1, 4, 1)-PDPwith difference leave[1, u] for anym, s andu listed below:

m s u m s u m s u

5 11 3 5 12 3 5 11 4


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 52 of 56

Go Back

Full Screen

Close

Quit

PDFs+IPDM(3, m; 1) ⇒ PDFs

Theorem 6.15If there exist both a(12t + 1, 4, 1)-PDF and a(24t +1, 4, 1)-PDF, then there exists a(12((m+1)t+s)+1, 4, 1)-PDF for anym ands listed below:

m s m s m s m s

13 1 25 5, 6 31 6, 7 33 6, 7, 8

39 7, 8, 9 41 8, 9, 10 45 8, 9, 10, 11 49 4, 9, 10, 11, 12

57 11, 12, 13, 14 61 5, 11, 12, 13, 14, 15 65 6, 12, 13, 14, 15, 16 69 14, 15, 16, 17

73 6, 14, 15, 16 77 7, 16, 17, 18 81 16, 17 85 7, 17, 18, 19

89 8, 19, 20 97 8, 21 99 20 103 21

109 9 121 10 133 11 145 12

157 13 169 14 181 15 193 16


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 53 of 56

Go Back

Full Screen

Close

Quit

PDFs+IPDM(3, m; 1) ⇒ PDPs

Theorem 6.16If there exist both a(12t + 1, 4, 1)-PDF and a(24t +1, 4, 1)-PDF, then there exists a(12(t(m + 1) + s) + 3, 4, 1)-PDP withdifference leaveL = 1 for anym ands listed below:


31 7 39 9 99 8 111 9 123 10 135 11

147 12 159 13 171 14 183 15 195 16

Theorem 6.17If there exist both a(12t + 1, 4, 1)-PDF and a(24t +1, 4, 1)-PDF, then there exists a(12(t(m + 1) + s) + 5, 4, 1)-PDP withdifference leaveL = [1, 2] for anym ands listed below:


25 5 39 8 41 9 45 9, 10 49 10, 11 65 5, 13, 14

77 6 101 8 113 9 125 10 137 11 149 12

173 14 185 15 197 16


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 54 of 56

Go Back

Full Screen

Close

Quit

PDFs+IPDM(3, m; 1) ⇒ PDPs



39 8 45 9 49 10 91 7 103 8 115 9

127 10 139 11 151 12 163 13 187 15 199 16



45 9 141 11 153 12 165 13 177 14 189 15


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 55 of 56

Go Back

Full Screen

Close

Quit

7 Concluding Remarks

In this talk, by observing the equivalence between a PDM(n, m) andan ASP(n, m) and by using combinatorial structures with “holes”,we described various direct and recursive constructions for additivesequences of permutations and perfect difference families. As theirimmediate consequences, we showed that:

Theorem 7.1A (12t + 1, 4, 1)-PDF exists for anyt ≤ 1000 except fort = 2, 3, and an ASP(3, m) exists for any odd3 < m < 200 except form = 9, 11 and possibly form = 59.

However, we should remark that it is still far from the complete set-tlement of the existence problems of perfect difference families withblock sizek, k = 4, 5, and additive sequences of permutations of lengthn, n = 3, 4. Novel ideas are expected in the further research for com-plete solutions for these challenging problems.


Introduction





Recursive . . .

Concluding Remarks

Home Page

Title Page

JJ II

J I

Page 56 of 56

Go Back

Full Screen

Close

Quit

Thank You!


Date post:	19-Jun-2020
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

JJ II J I - cscgt.orgcscgt.org/2009conf/ppt/葛根年.pdf · Applications: The designs of missile...

Documents