Combinatorial Designs: constructions, algorithms and new results - PowerPoint PPT Presentation

✬ ✩ Combinatorial Designs: constructions, algorithms and new results Ilias S. Kotsireas Wilfrid Laurier University ikotsire@wlu.ca ✫ ✪ I. S. Kotsireas, MSRI 1

✬ ✩ Combinatorial Design Theory Is it possible to arrange elements of a finite set into subsets so that certain properties are satisfied? Existence and non-existence results. Infinite classes. Tools & concepts from: linear algerbra, algebra, group theory, number theory, combinatorics, symbolic computation, numerical analysis. Applications to: cryptography, optical communications, wireless communications, coding theory. ✫ ✪ I. S. Kotsireas, MSRI 2

✬ ✩ • W. D. Wallis, A. P. Street, J. Seberry, Combinatorics: Room squares, sum-free sets, Hadamard matrices. Springer-Verlag, 1972. • A. V. Geramita, J. Seberry, Orthogonal designs. Quadratic forms and Hadamard matrices. Marcel Dekker Inc. 1979. • C. J. Colbourn, J. H. Dinitz, The CRC handbook of combinatorial designs. CRC Press, 1996. • V. D. Tonchev, Combinatorial configurations: designs, codes, graphs. Longman Scientific & Technical, John Wiley & Sons, Inc., 1988. • T. Beth, D. Jungnickel, H. Lenz, Design theory. Vols. I, II. Second edition. Cambridge University Press, Cambridge, 1999. • A. S. Hedayat, N. J. A. Sloane, J. Stufken Orthogonal arrays. Theory and applications. Springer-Verlag, 1999. • D. R. Stinson, Combinatorial designs, Constructions and analysis. Springer-Verlag, 2004. • C. J. Colbourn, J. H. Dinitz, Handbook of Combinatorial Designs. Second Edition, Chapman and Hall/CRC Press, 2006. • K. J. Horadam, Hadamard Matrices and Their Applications. Princeton ✫ University Press, 2006. ✪ I. S. Kotsireas, MSRI 3

✬ ✩ Weighing Matrices A weighing matrix W = W ( n, k ) of weight k , is a square n × n matrix with entries − 1 , 0 , +1 having k non-zero entries per row and column and inner product of distinct rows zero. W · W t = k I n Fact: If there is a W (2 n, k ), n odd, then k ≤ 2 n − 1 and k is the sum of two squares. Theorem: If there exist two circulant matrices A , B of order n each, satisfying A · A t + B · B t = k I n , then there exists a W (2 n, k ). ✫ ✪ I. S. Kotsireas, MSRI 4

✬ ✩   A B W (2 n, k ) =   − B t A t W (2 n, 2 n − 1) constructed from two circulants: infinite class W (2 n, 2 n − 3) constructed from two circulants: do not exist Ten Open Problems: [C. Koukouvinos, J. Seberry, JSPI (81), 1999] Do there exist W (2 · 23 , 41) , W (2 · 25 , 45) , W (2 · 27 , 49) , W (2 · 29 , 53) , W (2 · 33 , 61) , W (2 · 35 , 65) , W (2 · 39 , 73) , W (2 · 43 , 81) , W (2 · 45 , 85) , W (2 · 47 , 89) constructed from two circulants? Common feature: W (2 n, 2 n − 5), for n = 23 , . . . , 47. Odd large weights. R. Craigen, The structure of weighing matrices having large weights. ✫ ✪ Des. Codes Cryptogr. (5) 1995 I. S. Kotsireas, MSRI 5

✬ ✩ Plan of attack: Establish potential patterns for the locations of the 5 zeros in solutions. From 3 2 n ∼ 2 3 . 17 n ops, down to 2 2 n − 5 ops. Idea: Analyze the solutions sets for W (2 n, 2 n − 5) for all odd n up to n = 15. (bash/Maple meta-program, C code generation, supercomputing) First observation: (4 zeros) 0 0 0 0 ⋆ . . . ⋆ ⋆ . . . ⋆ a 1 . . . a n − 2 a n − 1 a n b 1 b 2 b 3 . . . b n ✫ ✪ I. S. Kotsireas, MSRI 6

✬ ✩ Second observation: (the remaining fifth zero) 0 0 0 0 0 a 1 ⋆ . . . ⋆ ⋆ . . . ⋆ a n − 2 b 3 ⋆ . . . ⋆ b n � �� � � �� � � �� � n − 2 terms 2 terms n − 3 2 terms n − 3 0 0 0 0 0 a 1 ⋆ . . . ⋆ a n − 2 b 3 ⋆ . . . ⋆ ⋆ . . . ⋆ b n � �� � � �� � � �� � 2 terms 2 terms n − 2 terms n − 3 n − 3 CRYSTALIZATION When we fix the 4 zeros as indicated above, then the fifth zero can only appear in exactly two possible places, in a W (2 n, 2 n − 5) solution. A proof will probably use Hall polynomials, PAF equations Implication: Infinite Class of W (2 n, 2 n − 5) ✫ ✪ I. S. Kotsireas, MSRI 7

✬ ✩ Results: W(2*23,41) solution -1 -1 -1 -1 -1 1 1 -1 1 -1 0 1 1 1 -1 -1 1 -1 -1 1 -1 0 0 0 0 -1 -1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 1 -1 W(2*25,45) solution 1 1 1 1 1 -1 -1 1 1 -1 1 0 1 -1 1 -1 -1 -1 -1 1 1 1 1 0 0 0 0 -1 -1 1 -1 -1 1 1 1 1 -1 1 1 -1 1 -1 1 -1 1 1 -1 -1 1 1 W(2*27,49) solution 1 1 1 1 1 1 -1 -1 -1 1 -1 -1 0 -1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 0 0 0 0 -1 -1 -1 1 1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 -1 W (2 · 29 , 53) is still out of reach, as it still requires 2 53 ops. ✫ ✪ I. S. Kotsireas, MSRI 8

✬ ✩ Periodic & non-periodic autocorrelation function The 2nd elementary symmetric function in n variables a 1 , . . . , a n � σ 2 = a 1 a 2 + · · · a n − 1 a n = a i a j 1 ≤ i<j ≤ n plays a pivotal role in building W (2 n, k ). PAF and NPAF concepts n − 1 � n � n − i = n ( n − 1) � σ 2 is made up of = 2 2 i =1 ✫ ✪ I. S. Kotsireas, MSRI 9

✬ ✩ (pairwise different) quadratic monomials:  . .  a 1 a 2 a 2 a 3 a 3 a 4 . a n − 1 a n     . .  . .  . . •  a 1 a 3 a 2 a 4    . . .   . . .  . . a 3 a n . •   . . • . • a 1 a n − 1 a 2 a n     .  .   • • . • a 1 a n         � �� � � �� � � �� � � �� � � �� �  n − 1 terms n − 2 terms n − 3 terms n − i terms 1 term ✫ ✪ I. S. Kotsireas, MSRI 10

✬ ✩  . . . ← N A (1)  a 1 a 2 a 2 a 3 a 3 a 4 a n − 1 a n    . .   . .  . . • ← N A (2) a 1 a 3 a 2 a 4     . . . . . . . . . • ← N A (3) a 3 a n   . . .  . . .   a 1 a n − 1 a 2 a n • . • . .     .  .  • • . • ← N A ( n − 1) a 1 a n Lemma: N A (1) + N A (2) + . . . + N A ( n − 1) = σ 2 Fact: P A ( s ) = N A ( s ) + N A ( n − s ) , s = 1 , . . . , n − 1 Lemma: P A (1) + P A (2) + . . . + P A ( n − 1) = 2 σ 2 ✫ ✪ I. S. Kotsireas, MSRI 11

✬ ✩ Fact: NPAF = 0 = ⇒ PAF = 0 The converse is not always true. Definition: Two sequences A = [ a 1 , . . . , a n ] and B = [ b 1 , . . . , b n ] are said to have zero PAF (resp. NPAF) if P A ( s ) + P B ( s ) = 0 , i = 1 , . . . , n − 1 resp. N A ( s ) + N B ( s ) = 0 , i = 1 , . . . , n − 1 . Weighing matrices come from sequences with zero PAF. Fact: If we can construct two sequences A and B with zero PAF, then we can construct W (2 · n, k ) from two circulants. ✫ ✪ I. S. Kotsireas, MSRI 12

✬ ✩ Power Spectral Density, PSD PSD Theorem [Fletcher, Gysin, Seberry, Australas. J. Combin., 23, 2001] Two sequences [ a 1 , . . . , a n ], [ b 1 , . . . , b n ] can be used to make up circulant matrices A and B that will give W (2 n, k ) weighing matrices if and only if PSD ([ a 1 , . . . , a n ] , i ) + PSD ([ b 1 , . . . , b n ] , i ) = k, ∀ i = 0 , . . . , n − 1 2 where PSD ([ a 1 , . . . , a n ] , k ) denotes the k -th element of the power spectral density sequence, i.e. the square magnitude of the k -th element of the discrete Fourier transform (DFT) sequence associated to [ a 1 , . . . , a n ]. ✫ ✪ I. S. Kotsireas, MSRI 13

✬ ✩ The DFT sequence associated to [ a 1 , . . . , a n ] is defined as n − 1 � a i +1 ω ik , k = 0 , . . . , n − 1 DFT [ a 1 ,...,a n ] = [ µ 0 , . . . , µ n − 1 ] , with µ k = i =0 � 2 π � 2 π � � 2 πi n = cos where ω = e + i sin is a primitive n -th root of unity. n n The proof is based on the Wiener-Khinchin Theorem • The PSD of a sequence is equal to the DFT of its PAF sequence n − 1 � PAF A ( j ) ω jk | µ k | 2 = j =0 • The PAF of a sequence is equal to the inverse DFT of its PSD sequence n − 1 PAF A ( j ) = 1 � | µ k | 2 ω − jk n j =0 ✫ ✪ I. S. Kotsireas, MSRI 14

✬ ✩ The Parseval Theorem provides a horizontal relationship between the elements of a sequence [ a 1 , . . . , a n ] and its DFT sequence: n n | a i | 2 = 1 � � PSD ([ a 1 , . . . , a n ] , i ) n i =1 i =1 The PSD theorem provides a vertical relationship between the elements of two sequences [ a 1 , . . . , a n ] and [ b 1 , . . . , b n ]. The PSD criterion for W (2 n, k ) states that: if for a certain sequence [ a 1 , . . . , a n ] there exists i ∈ { 1 , . . . , n − 1 2 } with the property that PSD ([ a 1 , . . . , a n ] , i ) > k , then this sequence cannot be used to construct W (2 n, k ). Important Consequence: we can now decouple the PAF equations, roughly corresponding to cutting down the complexity by ✫ ✪ half. I. S. Kotsireas, MSRI 15

✬ ✩ Algorithm: String Sorting Begin with PSD ([ b 1 , . . . , b n ] , i ) = k − PSD ([ a 1 , . . . , a n ] , i ) , ∀ i = 0 , . . . , n − 1 2 and take integer parts   k − 1 − [ PSD ([ a 1 , . . . , a n ] , i )] , is not an integer [ PSD ([ b 1 , . . . , b n ] , i )] = k − [ PSD ([ a 1 , . . . , a n ] , i )] , is an integer  A pair of vectors [ a 1 , . . . , a n ] and [ b 1 , . . . , b n ] can be encoded as the concatenation of the integer parts of the first n − 1 components of their 2 PSD vectors: [ b 1 , . . . , b n ] − → [ PSD ([ b 1 , . . . , b n ] , 1)] . . . [ a 1 , . . . , a n ] − → k − 1 − [ PSD ([ a 1 , . . . , a n ] , 1)] . . . ✫ ✪ I. S. Kotsireas, MSRI 16