Introduction Inequalities for Perfect... Additive Sequences of... PDFs with holes and... Direct Constructions... Recursive... Concluding Remarks

Transcription

1 Page 1 of 56

2 NSFC, Grant No and No th, July, 009 Zhejiang University Perfect Difference Families, Perfect Difference Matrices, and Related Combinatorial Structures Gennian Ge Department of Mathematics Zhejiang University Hangzhou 31007, Zhejiang, China (Joint with Ying Miao and Xianwei Sun) Page of 56

3 Outlines Part I 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 Page 3 of 56

4 1 In order to measure various spatial frequencies relative to some area of the sky, n movable antennas are used in m successive configurations. The distances between antennas within configurations determine which frequencies are obtained. These m successive configurations of n antennas can give up to m ( ) n spacings. It is thus required that these distances between antennas within configurations are successive multiples of a given increment, without hole or redundancy. 1. F. Biraud, E. J. Blum and J. C. Ribes, On optimum synthetic linear arrays, IEEE Trans. Antennas Propagation, vol. AP, pp , F. Biraud, E. J. Blum and J. C. Ribes, Some new possibilities of optimum synthetic linear arrays for radioastronomy, Astronomy and Astrophysics, vol. 41, pp , Page 4 of 56

5 Perfect Systems of Difference Sets Let k 3 and v be positive integers. For a given k-subset B = {b 1, b,..., b k } of the set I v = {0, 1,..., v 1} of non-negative integers, let B = {b j b i 1 i < j k} be the list of directed differences from B. Let c, m, p 1,..., p m be positive integers, and let S = {S 1, S,..., S m }, where S i = {s i1, s i,..., s ipi }, 0 s i1 < s i < < s ipi, and all s ij s are integers. We say that S = { S 1, S,..., S m } is a perfect system of difference sets for c (or starting with c, or with threshold c), or briefly, an (m, {p 1, p,..., p m }, c)-psds, if S = {c, c + 1,..., c 1 + pi 1 i m( ) }. An (m, {p1, p,..., p m }, c)- PSDS is regular if p 1 = p = = p m = p. As usual, a regular (m, {p}, c)-psds is abbreviated to (m, p, c)-psds. Page 5 of J. -C. Bermond, A. Kotzig and J. Turgeon, On a combinatorial problem of antennas in radioastronomy, Colloq. Math. Soc. János Bolyai, vol. 18, pp , A. Kotzig and J. Turgeon, Regular perfect systems of difference sets, Discrete Math., vol. 0, pp , 1977.

6 Perfect Difference Family = (t, k, 1)-PSDS Let k 3, t 1, and v = k(k 1)t+1 be positive integers. For a given collection B = {B 1, B,..., B t } of k-subsets of I v, let B = 1 i t B i be the list of directed differences from B. If B = {1,,..., (v 1)/}, then B is called a perfect difference family, or briefly, a (v, k, 1)-PDF. Page 6 of 56

7 Examples A ( , 4, 1)-PDF: {0, 1, 4, 6} = 1, 4, 3, 6, 5, A ( , 4, 1)-PDF: {0, 5,, 4} = 5,, 17, 4, 19, {0, 7, 13, 3} = 7, 13, 6, 3, 16, 10 {0, 3, 14, 18} = 3, 14, 11, 18, 15, 4 {0, 1, 9, 1} = 1, 9, 8, 1, 0, 1 Page 7 of 56

8 Related Combinatorial Structures and Codes 1. Cyclic Difference Families. Optical Orthogonal Codes 3. Difference Triangle Sets 4. Convolutional Codes 5. Radar Arrays 6. Graceful Labeling of Graphs Page 8 of 56

9 Cyclic Difference Families Let Z v be the group of residues modulo v. A (v, k, 1) difference family in Z v, briefly a cyclic (v, k, 1)- DF, is a collection F of k-subsets of Z v such that F ( F) = Z v \ {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, in Handbook of Combinatorial Designs, Second Edition, C. J. Colbourn and J. H. Dinitz, Eds., Boca Raton, FL: Chapman & Hall/CRC, pp , 007. Page 9 of 56

10 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 the codewords) of length v and weight k satisfying the following two properties: (1) (The Auto-Correlation Property) 0 t v 1 x tx t+i λ for any x = (x 0, x 1,..., x v 1 ) C and any integer i 0 mod v; () (The Cross-Correlation Property) 0 t v 1 x ty t+i λ for any x = (x 0, x 1,..., x v 1 ) C, y = (y 0, y 1,..., y v 1 ) C with x y, and any integer i a b c d e (15, 3, 1) OOC : {0, 7, 9}; {0, 1, 4} Page 10 of 56

11 Optical Orthogonal Codes From a set-theoretic perspective, a (v, k, 1) optical orthogonal code, or (v, k, 1)-OOC, is a collection C of k-subsets of Z v such that F ( F) does not have repeated elements in Z v \ {0} and the set-wise stabilizer of each k-subset of C is the subgroup {0} of Z v. A (v, k, 1)-OOC is optimal if its size reaches the upper bound v 1 k(k 1). A (v = k(k 1)t + 1, k, 1)-PDF = An optimal (v + i, k, 1)-OOC with size exactly equal to v 1+i k(k 1) = v 1 k(k 1) = t for 0 i k(k 1) 1. Motivation: Optical code-division multiple-access communication systems, mobile radio, frequency-hopping spread-spectrum communications, radar, sonar signal design, collision channel without feedback, neuromorphic networks, etc. T. Helleseth, Optical orthogonal codes, in Handbook of Combinatorial Designs, Second Edition, C. J. Colbourn and J. H. Dinitz, Eds., Boca Raton, FL: Chapman & Hall/CRC, pp. 31 3, 007. Page 11 of 56

12 Difference Triangle Sets An (n, k)-difference triangle set, or briefly (n, k)-d S, is a set = { 1,,..., n }, where i = {a i0, a i1,..., a ik }, 1 i n, is a set of non-negative integers such that 0 = a i0 < a i1 < < a ik in the real number system, and the differences a il a ij with 1 i n and 0 j < l k are all distinct. Let m( ) = max{a ik 1 i n} be the scope of, so the scope is at least equal to n ( ) k+1. An (n, k)-d S is regular if m( ) = nk(k+1). Let M(n, k) = min{m( ) is an (n, k)-d S}. If m( ) = M(n, k), then is said to be optimal. Clearly, any regular D S is optimal. A regular (n, k)-d S A (n ( ) k+1 + 1, k + 1, 1)-PDF. Applications: The designs of missile guidance codes, convolutional self-orthogonal codes, optical orthogonal codes, the allocation radio frequencies to reduce intermodulation interference, and in various other areas. J. B. Shearer, Difference triangle sets, in Handbook of Combinatorial Designs, Second Edition, C. J. Colbourn and J. H. Dinitz, Eds., Boca Raton, FL: Chapman & Hall/CRC, pp , 007. Page 1 of 56

13 Convolutional Codes Let F [x] denote the ring of polynomials in the indeterminate x over a finite field F, and F [x] n the set of all n-tuples f = (f 1 (x),..., f n (x)) of polynomials from F [x]. A convolutional code C is defined to be a subset of F [x] n, which is (a) closed under componentwise addition and subtraction, and (b) closed under multiplication by elements of F [x]. Applications: Digital radio, mobile phones and satellite links. A construction for convolutional self-orthogonal codes (CSOC s) based on D S s was introduced by Robinson and Bernstein. The CSOC corresponding to the (I, J) D S has generator polynomials g i (x) = Σ J j=1x a ij, 1 i I. J. P. Robinson and A. J. Bemstein, A class of binary recurrent codes with limited error propagation,0ieee Trans. Inform. Theory, vol. IT-13, pp , Page 13 of 56

14 Radar Arrays Let N and M be two positive integers. A radar array R is an N M (0, 1)-array with a single 1 per column, such that the horizontal autocorrelation function only has values 0, 1, and M. The radar array problem is to find the maximum number of columns such that any out-of-phase horizontally shifted copy has at most a single 1 in common. S. W. Golomb and H. Taylor, Two-dimensional synchronization patterns for minimum ambiguity, IEEE Trans. Inform. Theory, vol. 8, pp , 198. A (v, k, 1)-PDF = Radar Arrays, See G. Ge, A. C. H. Ling and Y. Miao, A system construction for radar arrays, IEEE Trans. Inform. Theory, vol. 54, pp , 008. Page 14 of 56

15 Graceful Labeling of Graphs Let G = (V, E) be an undirected finite graph without loops or multiple edges. A graceful labeling (or β-labeling) of a graph G = (V, E) with n vertices and m edges is a one-to-one mapping Ψ of the vertex set V (G) into the set {0, 1,,..., m} with this property: if we define, for any edge e = (u, v) E(G), the value Ω(e) = Ψ(u) Ψ(v), then Ω is a one-to-one mapping of the set E(G) onto the set {1,,..., m}. A graph is called graceful if it has a graceful labeling. The concept of a graceful labeling has been introduced by Rosa as a means of attacking the famous conjecture of Ringel that K n+1 can be decomposed into n + 1 subgraphs that are all isomorphic to a given tree with n edges. A (n ( k ) + 1, k, 1)-PDF is equivalent to a graceful labeling of a graph with n connected components all isomorphic to the complete graph on k vertices. A. Rosa, On certain valuations of the vertices of a graph, in Théorie des graphes, journées internationales d études, Rome, 1966, Dunod, Paris, 1967, pp Page 15 of 56

16 Known Existence Results A simple numerical necessary condition for the existence of a, k, 1)-PSDS, i.e. a (v, k, 1)-PDF, is v 1 (mod k(k 1)). ( v 1 k(k 1) Kotzig and Turgeon proved that: Theorem 1.1 For k = 3, the existence problem has been completely settled: a (v, k, 1)-PDF exists if and only if v 1, 7 (mod 4). Bermond, Kotzig and Turgeon proved that: Theorem 1. Perfect difference families cannot exist for k A. Kotzig and J. Turgeon, Regular perfect systems of difference sets, Discrete Math., vol. 0, pp , J. -C. Bermond, A. Kotzig and J. Turgeon, On a combinatorial problem of antennas in radioastronomy, Colloq. Math. Soc. János Bolyai, vol. 18, pp , Page 16 of 56

17 Known Existence Results for PDFs with k = 4, 5 Theorem Let m = 1t + 1. Then (m, 4, 1)-PDFs exist for the following values of t < 50: 1, 4 33, 36, 41.. Suppose a (1t + 1, 4, 1)-PDF exists. Then (m, 4, 1)-PDFs exist for m = 60t+13, 156t+13, 8t+49, 76t+61, 300t+61 and 300t Let m = 0t + 1. Then (m, 5, 1)-PDFs are known for t = 6, 8, 10 but for no other small values of t. 4. There are no (m, k, 1)-PDFs for the following values: (a) k = 4, m {5, 37}, (b) k = 5, m 1 (mod 40) or m {41, 81}. 1. R. Mathon, Constructions for cyclic Steiner -designs, Ann. Discrete Math., vol. 34, pp , J. H. Huang and S. S. Skiena, Gracefully labeling prisms, Ars Combin., vol. 38, pp. 5 36, R. J. R. Abel and M. Buratti, Difference families, in Handbook of Combinatorial Designs, Second Edition, C. J. Colbourn and J. H. Dinitz, Eds., Boca Raton, FL: Chapman & Hall/CRC, pp , 007. Page 17 of 56

18 Inequalities for Perfect Systems of Difference Sets We consider an (m, N, c)-psds S = { S 1, S,..., S m }, where S i = {s i1, s i,..., s ipi } and N = {p 1,..., p m }. Theorem.1 There is no (m, N, c)-psds if p i > 5 for all p i N. Next, we consider the average size p of N, where p = 1 m m i=1 p i = 1 m S. Suppose that there are exactly a even elements in N. Theorem. (1) If p 4 + 6, then there is no (m, N, c)-psds. () If a m/7 and p 6, or a = m and p 6, then there is no (m, N, c)-psds. Page 18 of 56 Lemma.3 In an (m, {5, 7}, c)-psds, the number of blocks of size 5 is at least 18m+5(c 1) 14, where = 18(m c + 1) + 7(c 1).

19 3 Additive Sequences of Permutations Let X (1) = (x (1) 1,..., x(1) m ) be an ordered set of distinct integers. For j =,..., n, let X (j) = (x (j) 1,..., x(j) m ) be a permutation of distinct integers in X (1). Then the ordered set (X (1), X (),..., X (n) ) is called an additive sequence of permutations of length n and order m, ASP(n, m) in short, if for every subsequence of consecutive permutations of the ordered set X (1), their vector-sum is again a permutation of X (1). The set X (1) is usually called the basis of the additive sequence of permutations. In the application of additive sequences of permutations to perfect systems of difference sets, one often considers ASP(n, m) with basis X (1) = ( r, r + 1,..., 1, 0, 1,..., r 1, r), where m = r + 1 and r a positive integer, unless otherwise stated. Example 3.1 An ASP(4, 5): X (1) = (, 1, 0, 1, ), X () = (0, 1,,, 1), X (3) = (1,, 0,, 1), X (4) = (1, 0, 1,, ) Page 19 of 56

20 ASP PSDS Kotzig and Turgeon discovered that arbitrarily large perfect systems of difference sets can be constructed from smaller ones via additive sequences of permutations. On the contrary, certain perfect systems of difference sets can also be used to construct additive sequences of permutations. 1. A. Kotzig and J. Turgeon, Perfect systems of difference sets and additive sequences of permutations, Congress. Numer., vol. 3 4, pp , J. M. Turgeon, An upper bound for the length of additive sequences of permutations, Utilitas Math., vol. 17, pp , J. M. Turgeon, Construction of additive sequences of permutations of arbitrary lengths, Ann. Discrete Math., vol. 1, pp. 39 4, J. Abrham, Perfect systems of difference sets A survey, Ars Combin., vol. 17A, pp. 5 36, Page 0 of 56

21 PSDS ASP Theorem If there exists an (m, 4, c)-psds, then one can construct explicitly an ASP(3, 1m) with the basis containing the elements {±c, ±(c + 1),..., ±(6m + c 1)}. Furthermore, if c = 1, then one can also construct explicitly an ASP(3, 1m + 1) with the basis containing the elements {0} {±1, ±,..., ±6m}.. If there exists an (m, 5, c)-psds, then one can construct explicitly an ASP(4, 0m) with the basis containing the elements {±c, ±(c + 1),..., ±(10m + c 1)}. Furthermore, if c = 1, then one can also construct explicitly an ASP(4, 0m + 1) with the basis containing the elements {0} {±1, ±,..., ±10m}. Page 1 of 56 J. Abrham, Perfect systems of difference sets A survey, Ars Combin., vol. 17A, pp. 5 36, 1984.

22 Known Existence Results on Additive Sequences of Permutations Turgeon proved the following: Theorem 3.3 If there is an ASP(n, m) with basis X (1) = ( r, r + 1,..., 1, 0, 1,..., r 1, r), where m = r +1 and r a positive integer, then the condition n m 1 must be satisfied. Theorem 3.4 Let (X (j) = (x (j) 1,..., x(j) (y (j) 1,..., y(j) s+1 r+1 ) j = 1,..., n) and (Y (j) = ) j = 1,..., n) be an ASP(n, r+1) and an ASP(n, s+ ) j = 1,..., n), 1), respectively. Then (Z (j) = (z (j) 1,..., z(j) where z (j) (i 1)(s+1)+h = (s + 1)x(j) i is an ASP(n, (r + 1)(s + 1)). (r+1)(s+1) + y (j) h, 1 i r + 1, 1 h s + 1 Page of 56 J. M. Turgeon, An upper bound for the length of additive sequences of permutations, Utilitas Math., vol. 17, pp , 1980.

23 Known Existence Results on Additive Sequences of Permutations Lemma There is no ASP(3, m) for m = 9, 11.. There exists an ASP(4, 5 i ) for any positive integer i. 3. There exists an ASP(3, m) for any m = 5 i 5 7 i 7 13 i i i i i i i i 01, where i j, j = 5, 7, 13, 15, 17, 19, 45, 11, 161, 01, are non-negative integers, not all equal to 0. Page 3 of 56 J. Abrham, Perfect systems of difference sets A survey, Ars Combin., vol. 17A, pp. 5 36, 1984.

24 Perfect Difference Matrix An n m matrix D = (d ij ) with entries from I m = {0, 1,..., m 1} is called a perfect difference matrix, denoted by PDM(n, m), if the entries of each row of D comprise all the elements of I m, and D ts = { m 1 m 1,..., 1, 0, 1,..., } holds for any 0 s < t n 1. Example 3.6 A PDM(4, 5): , Page 4 of 56

25 PDM = ASP Theorem 3.7 For any odd m, a PDM(n, m) is equivalent to an ASP(n, m) based on { m 1 m 1,..., 1, 0, 1,..., }. Proof. Let D i, 0 i n 1, be the ith row of a PDM(n, m) D = (d ij ). Define X (1) = D 0 m 1 X (1) + X () = D 1 m 1 X (1) + +X (n) = D n 1 m 1 = (d 00 m 1 = (d 10 m 1. = (d n 1,0 m 1,..., d 0,m 1 m 1 ),,..., d 1,m 1 m 1 ),,..., d n 1,m 1 m 1 ). Then (X (1), X (),..., X (n) ) is an ASP(n, m) based on { m 1 m 1,..., 1, 0, 1,..., }. Conversely, let (X (1), X (),..., X (n) ) be an ASP(n, m) based on { m 1 m 1,..., 1, 0, 1,..., }. For all 1 i n, let X (1) + +X (i) + m 1 = (x (1) 1 + +x(i) 1 +m 1,..., x (1) m + +x (i) m + m 1 ) be the (i 1)th row of an n m matrix D. Then D is a PDM(n, m). Page 5 of 56

26 4 PDFs with holes and PDMs with holes ASP PDF Theorem 4.1 Let D i = {0, a i, b i, c i }, i = 1,..., t, be the blocks of a (1t + 1, 4, 1)-PDF, and let X (1), X (), X (3) be an ASP(3, m), where m = r + 1, r. Then for i = 1,..., t and j = 1,..., m, the 6tm directed differences from the family m(i 1)+j = {0, ma i + α j, mb i + β j, mc i + γ j } cover the set [r + 1, 6tm + r]. Here α = (α 1,..., α m ), β = (β 1,..., β m ) and γ = (γ 1,..., γ m ) are the m-vectors X (1), X (1) + X (), X (1) + X () + X (3), respectively. Page 6 of 56 R. Mathon, Constructions for cyclic Steiner -designs, Ann. Discrete Math., vol. 34, pp , 1987.

27 Adjoining Additional Blocks In order to utilize Theorem 4.1 for constructing new perfect difference families, we need to find s additional blocks with directed differences covering the set [1, r] and possibly the set [6mt + r + 1, 6mt + 6s] for some integer s 0. We first estimate this number s. Lemma 4. Let s be the number of the additional blocks whose directed differences cover the set [1, r] and possibly the set [6mt + r + 1, 6mt + 6s]. Then r/6 s r/. Page 7 of 56

28 Perfect Difference Packings Let B = {B 1, B,..., B h }, where B i = {b i1, b i,..., b iki }, 1 i h, be a collection of h subsets of I v = {0, 1,..., v 1} called blocks, and K = {k 1, k,..., k h }. If the list of directed differences B = {b ij b il i = 1,..., h, 1 l < j k i } covers each element of the set {1,,..., v 1 } \ L exactly once, where L {1,,..., v 1 }, then we call B a (v, K, 1) perfect difference packing, or (v, K, 1)- PDP, with difference leave L. Page 8 of 56

29 Semi-perfect Group Divisible Designs Let m be an odd integer, n a positive integer, and K a subset of positive integers. Let F be a collection of subsets (called blocks) in I n I m of sizes from K such that any block intersects any {i} I m, i I n, in at most one point. For any two distinct i, j I n, define the (i, j)-differences ij F = B F ijb, where ij B = {a<b (i, a), (j, b) B}. F is called a semi-perfect group divisible design, denoted K- SPGDD of type m n if, for any two distinct i, j I n, ij F = { m 1 m 1,..., 1, 0, 1,..., }. As usual, a {k}-spgdd will be abbreviated to a k-spgdd. Page 9 of 56

30 PDF+ SPGDD PDF Theorem 4.3 If there exist a (v, K, 1)-PDP with difference leave L and a K - SPGDD of type m k for any k K, then there exists an (mv, K, 1)-PDP with difference leave [1, m 1 ] L, where L = {ml m 1,..., ml 1, ml, ml + 1,..., ml + m 1 l L}. Proof. Let B be the collection of blocks of a given (v, K, 1)-PDP with difference leave L. For each B = {b 0, b 1,..., b k 1 } B, by hypothesis, we know that k K and there exists a K -SPGDD of type m k, F(B), over I k I m. We can then construct a collection A(B) of new blocks with sizes from K as follows: A(B) = {{mb i1 + x 1, mb i + x,..., mb ik + x k } {(i 1, x 1 ), (i, x ),..., (i k, x k )} F(B)}. In this way, as B ranges over all blocks of B, we obtain a collection of new blocks A = B B A(B), each having size from K, and each element of the new blocks not exceeding m(v 1) + (m 1) = mv 1. For any two distinct i, j I k, since ij F(B) = { m 1 m 1,..., 1, 0, 1,..., }, the list of direct differences from A = B B A(B) is then A = B B A(B) = {m 1 m 1,..., m v 1 + m 1 } \ L. Therefore A is the collection of blocks of the desired (mv, K, 1)-PDP. Page 30 of 56

31 Corollary Theorem 4.4 Let l be a non-negative integer. Let D i = {0, a i, b i, c i }, i = 1,..., t, be the blocks of a (1t + l + 1, 4, 1)-PDP with difference leave L = [1, l], and let X (1), X (), X (3) be an ASP(3, m), where m = r + 1 and r. Then for i = 1,..., t and j = 1,..., m, the 6tm directed differences from the family m(i 1)+j = {0, ma i + α j, mb i + β j, mc i + γ j } cover the set [(l + 1)r + l + 1, 6tm + (l + 1)r + l], where α = (α 1,..., α m ), β = (β 1,..., β m ) and γ = (γ 1,..., γ m ) are the m-vectors X (1), X (1) + X (), X (1) + X () + X (3), respectively. Page 31 of 56

32 Incomplete Perfect Difference Matrix An n (m h) matrix = (δ ij ) with entries from I m \ (ri h + r ), where ri h + r = {ri + r i I h } I m for some r, r Z, is called an incomplete perfect difference matrix with a regular hole H = { h 1 h 1 r,..., r, 0, r,..., r}, denoted briefly by IPDM(n, m; h, r), if the entries of each row of comprise all the elements of I m \(ri h +r ), and for all 0 s < t n 1, the lists of differences ts = {δ tj δ sj 0 j m h 1} are all identical and ts = { m 1 m 1,..., 1, 0, 1,..., }\r { h 1 h 1,..., 1, 0, 1,..., } holds for any 0 s < t n 1. Note that here we are not interested in the parameter r since the differences are invariant under any translation of the matrix by r J, where J is the all-one matrix. When h = 1, we can drop the letter r from the notation IPDM(n, m; 1, r) since for any r Z, we always have H = {0}. Clearly, by adding the column vector (r,..., r ) T to an IPDM(n, m; 1), we immediately obtain a PDM(n, m). Page 3 of 56

33 Orthogonal Arrays An orthogonal array OA(n, k) is an n k array, A, with entries from a set X of k elements such that, within any two rows of A, every ordered pair of elements from X occurs in exactly one column of A. An orthogonal array A is idempotent if it contains the k distinct n 1 vectors {(x, x,..., x) T x X} as columns of A. Page 33 of 56

34 PDP+Idempotent OA(n, k) IPDM Theorem 4.5 Let h be a non-negative integer. If there exist a ( t 1, h)-regular (m, K, 1)-PDP and an idempotent OA(n, k) for each k K, then there exists an IPDM(n 1, m; t, h). Furthermore, if there exist an IPDM(n 1, t; 1) and a PDM(n 1, t), then so do an IPDM(n 1, m; 1) and a PDM(n 1, m), respectively. Page 34 of 56 Corollary 4.6 Suppose that there exists an (m, K, 1)- PDF. If for each k K, there exists an idempotent OA(n, k), then there also exists an IPDM(n 1, m; 1).

35 Proof for PDP+Idempotent OA(n, k) IPDM For each block B i of size k i of the (m, K, 1)-PDP, B = {B 1, B,..., B h }, we form an idempotent orthogonal array O i with n rows and k i columns over the k i points of B i, and then remove the idempotent part of O i. In this way, we obtain a small incomplete orthogonal array O i with n rows and k i(k i 1) columns without the idempotent part for each block of the (m, K, 1)-PDP. Concatenating these small incomplete orthogonal arrays O i, we obtain a large incomplete orthogonal array O without the idempotent part O = ( O 1 O O h ). Take out one row from O, subtract this row from all other rows term by term, then we obtain an (n 1) ( 1 i h k i(k i 1)) array O. Adding m 1 to each element of O, we obtain an IPDM(n 1, 1 i h k i(k i 1) + t; t, h) with a regular hole { t 1 t 1 h,..., h, 0, h,..., h}. From the definition of a perfect difference packing, we know that 1 i h k i(k i 1) = m t. Page 35 of 56

36 PDF+IPDM(3, m; 1) PDP Theorem 4.7 Let u, r, m be integers such that 1 u r and m = r + 1. Suppose that there exist an IPDM(3, m; 1), a (1t + 1, 4, 1)-PDF, a (4t + 1, 4, 1)- PDF. Then there exists a (1t(m + 1) + r + 1, 4, 1)- PDP with difference leave [1, r]. Page 36 of 56

37 Proof for PDF+IPDM(3, m; 1) PDP Let D i = {0, a i, b i, c i }, i = 1,..., t, be the blocks of a (1t + 1, 4, 1)- PDF. Then for i = 1,..., t and j = 1,..., m 1, the 6t(m 1) directed differences from the family m(i 1)+j = {0, (m + 1)a i + α j, (m + 1)b i + β j, (m + 1)c i + γ j } cover the set [r + 1, 6t(m + 1) + r] \ (r + 1)[1, 1t]. Here α = (α 1,..., α m 1 ), β = (β 1,..., β m 1 ) and γ = (γ 1,..., γ m 1 ) are the row vectors of a 3 (m 1) matrix obtained by subtracting r from each element of the IPDM(3, m; 1), respectively. Let E l = {0, d l, e l, f l }, l = 1,,..., t, be the blocks of a (4t+1, 4, 1)- PDF. Then the 1t directed differences from the family cover the set (r + 1)[1, 1t]. Θ l = {0, (r + 1)d l, (r + 1)e l, (r + 1)f l } Page 37 of 56

38 5 Direct Constructions for PDMs and PDPs Example: An IPDM(3, 3; 1). ( For each m, we list only (m 1)/ columns of an IPDM(3, m; 1). The other (m 1)/ non-zero columns are given by subtracting each element of the listed columns from m. ). Page 38 of 56

39 More Examples: Example: An IPDM(3, 9; 5, 5). ( We list only m h columns of an IPDM(3, m; h, r). Multiply each element of the listed m h columns by 1 to obtain m h new columns. Let D be the matrix consisting of these m h columns. It is then readily checked that the matrix obtained by adding m 1 to each element of D is an IPDM(3, m; h, r). Furthermore, if there exists a PDM(3, h), then subtract h 1 from each element of this PDM(3, h), multiply each resultant element by r, and then include the new 3 h matrix into D to form a 3 m matrix. Finally, add m 1 to each element of this 3 m matrix. It is easily checked that this matrix is a PDM(3, m). ). Page 39 of 56

40 PDP+Idempotent OA(n, k) IPDM Lemma 5.1 There exists a regular (m, K, 1)-PDP with difference leave L for any m, K and L listed below: m K L m K L m K L 51 {4, 5} {7, 14, 1} 53 {4} {3, 6} 55 {4} {4, 8, 1} 63 {4, 5} {5, 10, 15} 65 {4} {13, 6} 67 {4} {4, 8, 1} 71 {4, 5} {6, 1, 18} 73 {4} {6, 1,..., 36} 75 {4, 5} {6, 1, 18} Page 40 of 56

41 List of IPDM(3, m; 1) and PDM(3, m) with m < 00 Theorem There exists an IPDM(3, m; 1) for any odd integer 5 m < 00 except for m = 9, 11 and except possibly for m {5, 7, 15, 19, 1, 7, 9, 35, 37, 43, 47, 51, 53, 55, 59, 63, 67, 71, 75, 79, 83, 87, 95}.. There exists a PDM(3, m) for any odd integer 5 m < 00 except for m = 9, 11 and except possibly for m = 59. Page 41 of 56

42 Direct Constructions for Perfect Difference Packings Lemma 5.3 There exists a (v, 4, 1)-PDP with difference leave L for any v and L listed below: v L v L v L v L v L v L 63 {1} 65 {1, } 75 {1} 77 {1, } 87 {1} 91 [1, 3] 99 {1} 101 {1, } 103 [1, 3] 111 {1} 113 {1, } 115 [1, 3] 13 {1} 15 {1, } 17 [1, 3] 135 {1} 137 {1, } 139 [1, 3] 141 [1, 4] 147 {1} 149 {1, } 151 [1, 3] 153 [1, 4] 159 {1} 163 [1, 3] 165 [1, 4] 171 {1} 173 {1, } 177 [1, 4] 179 [1, 5] 183 {1} 185 {1, } 187 [1, 3] 189 [1, 4] 191 [1, 5] 195 {1} 197 {1, } 199 [1, 3] 01 [1, 4] 03 [1, 5] 07 {1} 09 {1, } 11 [1, 3] 13 [1, 4] 15 [1, 5] 19 {1} 1 {1, } 3 [1, 3] 5 [1, 4] 7 [1, 5] 31 {1} 33 {1, } 35 [1, 3] 37 [1, 4] 39 [1, 5] 43 {1} 45 {1, } 47 [1, 3] 49 [1, 4] 51 [1, 5] 55 {1} 57 {1, } 59 [1, 3] 61 [1, 4] 63 [1, 5] 71 [1, 3] 73 [1, 4] 75 [1, 5] 83 [1, 3] 85 [1, 4] 95 [1, 3] Page 4 of 56

43 Direct Constructions for PDPs 1. PDF (x + 1s + 1, 4, 1)-PDP with difference leave L = [r + 1, x + r]. PDF (x + 1s + 3, 4, 1)-PDP with difference leave L = {1} [r + 1, x + r] 3. PDF (x + 1s + 5, 4, 1)-PDP with difference leave L = {1, } [r + 1, x + r] 4. PDF (x + 1s + 7, 4, 1)-PDP with difference leave L = [1, 3] [r + 1, x + r] 5. PDF (x + 1s + 9, 4, 1)-PDP with difference leave L = [1, 4] [r + 1, x + r] Page 43 of 56

44 Direct Constructions for PDPs 1. PDP with Difference Leave L = {1} (x + 1s + u + 1, 4, 1)- PDP with difference leave L = [1, u] [3r +, x + 3r + 1]. PDP with Difference Leave L = {1, } (x + 1s + u + 1, 4, 1)- PDP with difference leave L = [1, u] [5r + 3, x + 5r + ] 3. PDP with Difference Leave L = {1,, 3} (x+1s+u+1, 4, 1)- PDP with difference leave L = [1, u] [7r + 4, x + 7r + 3] 4. PDP with Difference Leave L = {1,, 3, 4} (x + 1s + u + 1, 4, 1)-PDP with difference leave L = [1, u] [9r + 5, x + 9r + 4] 5. PDP with Difference Leave L = {1,, 3, 4, 5} (x + 1s + u + 1, 4, 1)-PDP with difference leave L = [1, u] [11r + 6, x + 11r + 5] Page 44 of 56

45 6 Recursive Constructions for PDFs and PDPs PDFs PDFs Theorem 6.1 If there exists a (1t + 1, 4, 1)-PDF, then there exists a (1(mt + s) + 1, 4, 1)-PDF for any m and s listed below: m s m s m s m s 5 5, 6 7 5, 6 9 6, , , 7, , , 8, , 8, , 9, , 9, , 9, 10, , 10, , 9, 10, 11, , 11, , 11, 1, , 11, 1, , 1, 13, , 11, 1, 13, 14, , 13, 14, , 13, 15, , 13, 14, , 15, 16, , , 14, 15, , 16, , 17, , , 17, 18, , , Page 45 of 56

46 PDPs PDFs Theorem 6. If there exists a (1t + 3, 4, 1)-PDP with difference leave L = {1}, then there exists a (1(mt + s) + 1, 4, 1)-PDF for any m and s listed below: m s m s m s m s m s 13 7, 8, , 9, 10, , 11, , 1, 13, , 13, 14, , 15, 16, , 16, , Theorem 6.3 If there exists a (1t + 5, 4, 1)-PDP with difference leave L = [1, ], then there exists a (1(mt + s) + 1, 4, 1)-PDF for any m and s listed below: m s m s m s m s m s m s 5 5, 6 7 7, , 13, 15, , 16, , 18, Page 46 of 56

47 PDPs PDFs Theorem 6.4 If there exists a (1t + 7, 4, 1)-PDP with difference leave L = [1, 3], then there exists a (1(mt + s) + 1, 4, 1)-PDF for any m and s listed below: m s m s 5 7, 8 7 4, 9, 10, 11, 1 Theorem 6.5 If there exists a (1t + 9, 4, 1)-PDP with difference leave L = [1, 4], then there exists a (1(mt + s) + 1, 4, 1)-PDF for any m and s listed below: m s m s 5 8, 9, 10, , 13, 14, 15 Page 47 of 56

48 PDFs PDPs Theorem 6.6 If there exists a (1t + 1, 4, 1)-PDF, then we have the following assertions. (1) There exists a (1(mt + s) + 3, 4, 1)-PDP with difference leave L = {1} for any m and s listed below: m s m s m s m s m s , 11 () If there exists an (m, 4, 1)-PDP with difference leave {1}, then there also exists a ((1t + 1)m, 4, 1)-PDP with difference leave {1}. Theorem 6.7 If there exists a (1t + 1, 4, 1)-PDF, then we have the following assertions. (1) There exists a (1(mt + s) + 5, 4, 1)-PDP with difference leave L = [1, ] for any m and s listed below: m s m s m s m s m s , , , , , , , 14 () If there exists an (m, 4, 1)-PDP with difference leave [1, ], then there also exists a ((1t + 1)m, 4, 1)-PDP with difference leave [1, ]. Page 48 of 56

49 PDFs PDPs Theorem 6.8 Suppose that both a (1t + 1, 4, 1)-PDF and a PDM(3, m) exist. Then we have the following assertions. (1) There exists a (1(mt + s) + 7, 4, 1)-PDP with difference leave L = [1, 3] for any m and s listed below: m s m s m s m s m s m s () If there exists an (m, 4, 1)-PDP with difference leave [1, 3], then there also exists a ((1t + 1)m, 4, 1)-PDP with difference [1, 3]. Theorem 6.9 If there exists a (1t + 1, 4, 1)-PDF, then we have the following assertions. (1) There exists a (1(mt + s) + 9, 4, 1)-PDP with difference leave L = [1, 4] for any m and s listed below: m s m s m s () If there exists an (m, 4, 1)-PDP with difference leave [1, 4], then there also exists a ((1t + 1)m, 4, 1)-PDP with difference [1, 4]. Page 49 of 56

50 PDPs PDPs Theorem 6.10 If there exists a (1t+3, 4, 1)-PDP with difference leave L = {1}, then there exists a (1(mt + s) + u + 1, 4, 1)-PDP with difference leave [1, u] for any m, s and u listed below: m s u m s u m s u m s u Theorem 6.11 If there exists a (1t+5, 4, 1)-PDP with difference leave L = [1, ], then there exists a (1(mt + s) + u + 1, 4, 1)-PDP with difference leave [1, u] for any m, s and u listed below: m s u m s u m s u m s u Page 50 of 56

51 PDPs PDPs Theorem 6.1 If there exists a (1t+7, 4, 1)-PDP with difference leave L = [1, 3], then there exists a (1(mt + s) + u + 1, 4, 1)-PDP with difference leave [1, u] for any m, s and u listed below: m s u m s u m s u m s u m s u Theorem 6.13 If there exists a (1t+9, 4, 1)-PDP with difference leave L = [1, 4], then there exists a (1(mt + s) + u + 1, 4, 1)-PDP with difference leave [1, u] for any m, s and u listed below: m s u m s u Page 51 of 56 Theorem 6.14 If there exists a (1t + 11, 4, 1)-PDP with difference leave L = [1, 5], then there exists a (1(mt + s) + u + 1, 4, 1)-PDP with difference leave [1, u] for any m, s and u listed below: m s u m s u m s u

52 PDFs+IPDM(3, m; 1) PDFs Theorem 6.15 If there exist both a (1t + 1, 4, 1)-PDF and a (4t + 1, 4, 1)-PDF, then there exists a (1((m+1)t+s)+1, 4, 1)-PDF for any m and s listed below: m s m s m s m s , , , 7, , 8, , 9, , 9, 10, , 9, 10, 11, , 1, 13, , 11, 1, 13, 14, , 1, 13, 14, 15, , 15, 16, , 14, 15, , 16, 17, , , 17, 18, , 19, , Page 5 of 56

53 PDFs+IPDM(3, m; 1) PDPs Theorem 6.16 If there exist both a (1t + 1, 4, 1)-PDF and a (4t + 1, 4, 1)-PDF, then there exists a (1(t(m + 1) + s) + 3, 4, 1)-PDP with difference leave L = {1} for any m and s listed below: m s m s m s m s m s m s Theorem 6.17 If there exist both a (1t + 1, 4, 1)-PDF and a (4t + 1, 4, 1)-PDF, then there exists a (1(t(m + 1) + s) + 5, 4, 1)-PDP with difference leave L = [1, ] for any m and s listed below: m s m s m s m s m s m s , , , 13, Page 53 of 56

54 PDFs+IPDM(3, m; 1) PDPs Theorem 6.18 If there exist both a (1t + 1, 4, 1)-PDF and a (4t + 1, 4, 1)-PDF, then there exists a (1(t(m + 1) + s) + 7, 4, 1)-PDP with difference leave L = [1, 3] for any m and s listed below: m s m s m s m s m s m s Theorem 6.19 If there exist both a (1t + 1, 4, 1)-PDF and a (4t + 1, 4, 1)-PDF, then there exists a (1(t(m + 1) + s) + 9, 4, 1)-PDP with difference leave L = [1, 4] for any m and s listed below: m s m s m s m s m s m s Page 54 of 56

55 7 In this talk, by observing the equivalence between a PDM(n, m) and an ASP(n, m) and by using combinatorial structures with holes, we described various direct and recursive constructions for additive sequences of permutations and perfect difference families. As their immediate consequences, we showed that: Theorem 7.1 A (1t + 1, 4, 1)-PDF exists for any t 1000 except for t =, 3, and an ASP(3, m) exists for any odd 3 < m < 00 except for m = 9, 11 and possibly for m = 59. However, we should remark that it is still far from the complete settlement of the existence problems of perfect difference families with block size k, k = 4, 5, and additive sequences of permutations of length n, n = 3, 4. Novel ideas are expected in the further research for complete solutions for these challenging problems. Page 55 of 56

56 Thank You! Page 56 of 56