Mutually Pseudo-orthogonal Latin squares, difference matrices and LDPC codes

Transcription

1 Mutually Pseudo-orthogonal Latin squares, difference matrices and LDPC codes Asha Rao (Joint Work with Diane Donovan, Joanne Hall and Fatih Demirkale) 8th Australia New Zealand Mathematics Convention Melbourne Australia 10 December, 2014

2 Outline Introduction c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

3 Outline Introduction Subclass of Difference Covering Arrays c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

4 Outline Introduction Subclass of Difference Covering Arrays Constructing Pseudo-orthogonal Latin Squares c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

5 Outline Introduction Subclass of Difference Covering Arrays Constructing Pseudo-orthogonal Latin Squares LDPC Codes c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

6 Introduction Outline Introduction Subclass of Difference Covering Arrays Constructing Pseudo-orthogonal Latin Squares LDPC Codes c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

7 Introduction Difference Matrices A fundamental tool for constructing many combinatorial objects Have diverse applications, such as design of animal studies Coding Theory - Authentication Codes, LDPC codes c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

8 Introduction Generalisations of Difference Matrices Why Generalise? - because of existence constraints Holey Difference Matrices Difference Covering Arrays Difference Packing Arrays c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

9 Introduction Difference Matrix - Definition A Difference Matrix over an abelian group (G, +) of order n is an n k matrix Q = [q(i, j)] with entries from G such that for all pairs of columns 0 j, j k 1, j j j,j = {q(i, j) q(i, j ) 0 i n 1} contains every element of G equally often, say λ times. Notation: DM(n, k; λ) c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

10 Introduction Existence of Difference matrices DM(n, k; λ) does not exist if k > λn DM(n, 3; λ) does not exist if λ is odd and n is even and the Sylow 2-subgroup of G is cyclic in particular, if λ is odd, n is even and G = Z n or if λ is odd, and n 2 mod 4 c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

11 Introduction Generalisation: Difference Covering Array Given an abelian group (G, +) of order n A Difference Covering Array DCA(k, η; n) is an η k matrix Q = [q(i, j)] with entries from G such that, for all pairs of columns 0 j, j k 1, j j, the difference set j,j = {q(i, j) q(i, j ) 0 i η 1} contains every element of G at least once. Difference Packing Array DPA(k, η; n) if the difference set contains every element of G at most once. c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

12 Introduction Some other twists! If (G, +) is the cyclic group, then the difference covering (packing) array is said to be cyclic. A cyclic DCA(k, n + 1; n), Q = [q(i, j)] (0 i n, 0 j k 1), satisfies the reflection property if, for all columns j, 0 j k 2, and all rows i {0, 1,... (n 1)/2}, q(i, j) + q(n 1 i, j) = n 1. We may assume that the last row and last column of a DCA(k, η; n) contain all zeros. c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

13 Subclass of Difference Covering Arrays Outline Introduction Subclass of Difference Covering Arrays Constructing Pseudo-orthogonal Latin Squares LDPC Codes c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

14 Subclass of Difference Covering Arrays DCA with particular Properties Why? Because these properties will build connections to pseudo-orthogonal Latin squares Want DCA(k, n + 1; n), Q = [q(i, j)], (0 i n, 0 j k 1) satisfying the properties: P1. the entry 0 G occurs at least twice in each column of Q, and P2. for all pairs of distinct columns j and j, j k 1 j, j,j = {q(i, j) q(i, j ) 0 i n 1} = G \ {0}, if G is the cyclic group over Z n, then these conditions imply that for all distinct columns j and j, j k 1 j, j,j = {0, 1, 2,..., n/2, n/2,..., n 1} with repetition retained. c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

15 Constructing Pseudo-orthogonal Latin Squares Outline Introduction Subclass of Difference Covering Arrays Constructing Pseudo-orthogonal Latin Squares LDPC Codes c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

16 Constructing Pseudo-orthogonal Latin Squares Latin Squares A Latin square of order n is an n n array in which each of the symbols of Z n occurs once in every row and once in every column. Two Latin squares A = [a(i, j)] and B = [b(i, j)], of order n, are said to be orthogonal if O = {(a(i, j), b(i, j)) 0 i, j n 1} = Z n Z n. Difference matrices can be used to construct sets of mutually orthogonal Latin squares (MOLS). There exist significant existence questions for sets of MOLS. There is no pair of MOLS(6), however it is not known if there exists a set of three MOLS(10), or four MOLS(22) etc. c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

17 Constructing Pseudo-orthogonal Latin Squares Pseudo-orthogonal Latin Squares Raghavarao, Shrikhande and Shrikhande [2002] vary the mutually orthogonal condition Each symbol of one Latin square is paired with every symbol in the other once, except except for one symbol with which it is paired twice and And one with which it is not paired at all But, are Pseudo-orthogonal Latin Squares useful? Applications to multi-factor crossover designs in animal husbandry for example. c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

18 Constructing Pseudo-orthogonal Latin Squares Near Mutually Orthogonal Latin Squares Here we are interested in a special class of Pseudo-orthogonal Latin squares The set O does not contain the pair (a, a), for any a Z n. Question: Can we use difference techniques to construct mutually pseudo-orthogonal Latin squares? Yes, first done by Raghavarao, Shrikhande and Shrikhande in 2002, Theorem: If there exists a cyclic DCA(t + 1, 2p + 1; 2p), Q = [q (i, j)], which satisfies Property P1 and P2, then there exists a set of t pseudo-orthogonal Latin squares of order 2p. General Constructions by Li and van Rees in c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

19 Constructing Pseudo-orthogonal Latin Squares A cyclic DCA with reflection property Theorem Let f to be a natural number and m = 4k + 2 where k is a non-negative integer, such that gcd(f, 4m) = 2, gcd(f 1, 4m) = 1, and f 2 + f 2 2m mod 4m. Then a DCA(4, 4m + 1; 4m) with reflection property exists. Corollary For k such that k 1 mod 3, we can always construct a DCA(4, 4m + 1; 4m) as described above. c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

20 Constructing Pseudo-orthogonal Latin Squares The Construction of a cyclic DCA with reflection property Intervals I 1 I 2 I 3 I 4 q(a, 0) = a q(a, 1) = b(a) q(a, 2) = c(a) [0, m 1] [m, 2m 1] [2m, 3m 1] [3m, 4m 1] (a + 1)f 1 (a + 1)f 1 af af (a+1)(2m f + 2) 1 a(2m f + 2) m (a+1)(2m f + 2) + m 1 a(2m f + 2) b(a) a (a + 1)(f 1) (a + 1)(f 1) a(f 1) a(f 1) c(a) a (a+1)(2m f + 1) a(2m f + 1) m (a+1)(2m f + 1) + m a(2m f + 1) c(a) (a + 1)(2m a(2m 2f +2) a(2m 2f +2)+ a(2m 2f + 2) b(a) 2f + 2) f m + 1 3m f + 1 c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

21 LDPC Codes Outline Introduction Subclass of Difference Covering Arrays Constructing Pseudo-orthogonal Latin Squares LDPC Codes c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

22 LDPC Codes LDPC Codes - What are they? Linear error-correcting codes obtained from sparse bipartite graphs. First invented by Robert Gallagher in the 19060s. Highly sought after for high-speed communication and data storage The parity-check matrix is sparse. c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

23 LDPC Codes Example Here is the parity check matrix of the [6,3] LDPC code. H = c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

24 LDPC Codes Constructing LDPC Codes Gallagher s proposed construction: randomly assign 1s to positions in H subject to constraints Difficult to randomly construct good high-rate LDPC codes of short and moderate length Why? Parity-check matrices are very dense compared to low-rate codes. Need structured LDPC codes such as finite geometry LDPC codes c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

25 LDPC Codes QC-LDPC Codes Quasi-cyclic LDPC codes are well suited for hardware implementation Vasic and Milenkovic (2004) propose construction from cyclic difference families. These too have restricted lengths c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

26 LDPC Codes Example of CDF CDF(13,3,1) over Z 13 The Blocks B 1 = {0, 1, 4} and B 2 = {0, 2, 7} are the base block of a (13, 3, 1) CDF. D (1) = D (2) = c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

27 LDPC Codes Example of QC-LDPC Take the action of G = Z v on the base blocks to get BIBD(v, c, 1). The orbits of the base blocks of the CDF(13,3,1) give BIBD(13,3,1) The incidence matrix of the BIBD is the parity-check matrix H of the LDPC code. H has 13 rows and 26 columns and is of the form H = [H 1, H 2 ] Each column has weight 3. c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

28 LDPC Codes Furture Work Extending the construction to get LDPC codes from DCA(t + 1, 2p + 1, ; 2p). Would the properties of the LDPC codes thus obtained be better? c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26

29 LDPC Codes Any Questions c Asha Rao (ANZMC) MNOLS, DM, LDPC 10 December, / 26