An improvement to the Gilbert-Varshamov bound for permutation codes

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1 An improvement to the Gilbert-Varshamov bound for permutation codes Yiting Yang Department of Mathematics Tongji University Joint work with Fei Gao and Gennian Ge May 11, 2013

2 Outline Outline 1 Introduction to permutation codes 2 Under Hamming distance Upper bounds Lower bounds Our improvement 3 Under Chebyshev distance Constructions Lower and upper bounds

3 Introduction to permutation codes Permutation codes Definition Let S n be the set of all permutations of length n. The permutation code C is just a subset of S n. The length of C is n and each permutation in C is called a codeword. Applications: Powerline communication and Flash memories P. Frankl, M. Deza, On the maximum number of permuations with givern maximal or minimal distance, J. Combin. Theory Ser. A 22 (3) (1977), N. Pavlidou, A. J. H. Vinck, J. Yazdani and B. Honary, Powerline communications: State of the art and future trends, IEEE Communications Magazine, (2003), A. Jiang, R. Mateescu, M. Schwartz, and J. Bruck, Rank modulation for flash memories, in Proc. IEEE Int. Symp. Information Theory, 2008,

4 Introduction to permutation codes Hamming and Chebyshev metrics Definition For two distinct permutations σ, π S n, their Hamming distance d H (σ, π) is the number of elements that they differ. Definition Let π = π 1 π 2..., π n, σ = σ 1 σ 2..., σ n S n. The Chebyshev distance between π and σ is d C (π, σ) = max{ π j σ j 1 j n}. P. Diaconis, Group Representations in probability and Statistics, Hayward, CA: Inst. Math. Statist., T. Kløve, T. Lin, S. Tsai, and W. Tzeng, Permutation Arrays Under the Chebyshev Distance, IEEE Tran. Inform. Theory, 56(6), (2010).

5 Introduction to permutation codes Permutation code of minimum distance d Example Let σ = and π = Then d H (σ, π) = 5 and d C (σ, π) = 2. We say a permutation code C has minimum Hamming distance d if the Hamming distance of any pair of distinct permutations in C is at least d. Similarly, C is called a permutation code with minimum Chebyshev distance d if the Chebyshev distance of any pair of distinct permutations in C is at least d. They are both called a (n, d)-permutation code.

6 Introduction to permutation codes M(n, d) and P (n, d) The maximum number of codewords in a permutation code with minimum Hamming distance d is denoted by M(n, d). The maximum number of codewords in a permutation code with minimum Chebyshev distance d is denoted by P (n, d). Problems: Construct large permutation codes with some fixed minimum Hamming or Chebyshev distance. Find M(n, d) and P (n, d), or give some good lower or upper bounds of them.

7 Under Hamming distance The permutation code under Hamming distance

8 Under Hamming distance Constructions Clique search Greedy algorithm Automorphisms Direct constructions from permutation polynomials Recursive construction W. Chu, C. J. Colbourn, and P. Dukes, Constructions for Permutation Codes in Powerline Commnications, Des. Codes Cryptogr. 32 (2004), D. H. Smith and R. Montemanin, A new table of permutation codes, Des. Codes Cryptogr. 63 (2)(2012),

9 Under Hamming distance Basic results on M(n, d) 1 M(n, 2) = n!; 2 M(n, 3) = n!/2; 3 M(n, n) = n; 4 M(n, d) nm(n 1, d). P. Dukes and N. Sawchuck, bounds on permutation codes of distance four, J. Algebraic Combin. 31(1) (2010),

10 Under Hamming distance Upper bounds Sphere-packing bound Definition Let D(n, k) (k = 0, 1,..., n) denote the set of all permutations in S n which are exactly at distance k from the identity. Clearly, D(n, k) = D k ( n k). Theorem M(n, d) n! d 1 2 ( k=0 D n ). k k P. Dukes and N. Sawchuck, bounds on permutation codes of distance four, J. Algebraic Combin. 31(1) (2010),

11 Under Hamming distance Upper bounds The upper bound for M(n, 4) Theorem (Frankl and Deza, 1977) M(n, 4) (n 1)!. Theorem (Dukes and Sawchuck, 2010) If k 2 n k 2 + k 2 for some integer k 2, then n! M(n, 4) 1+ (n + 1)n(n 1) n(n 1) (n k 2 )((k + 1) 2 n)((k + 2)(k 1) n). P. Frankl, M. Deza, On the maximum number of permuations with givern maximal or minimal distance, J. Combin. Theory Ser. A 22 (3) (1977)

12 Under Hamming distance Lower bounds Gilbert-Varshamov bound Theorem M(n, d) n! d 1 k=0 D n ). k( k P. Dukes and N. Sawchuck, bounds on permutation codes of distance four, J. Algebraic Combin. 31(1) (2010),

13 Under Hamming distance Our improvement Motivation

14 Under Hamming distance Our improvement EKR Theorem Theorem (Erdős, Ko and Rado,1961) Let k be positive integers with n > 2k. If F is a intersecting family of k-subsets of {1, 2,..., n}, then ( ) n 1 F. k 1 Moreover, F = ( n 1 k 1) if and only if F is the collection of all k-subsets that contain a fixed i {1,..., n}. P. Erdős, C. Ko, R. Rado,Intersection theorems for systenms of finite sets, Quart. J. Math. Oxford Ser. 12(2) (1961)

15 Under Hamming distance Our improvement Intersecting families of permutations Definition Two permutations σ, τ S n are said to k-intersect if they agree on at least k points. A set I S n is k-intersecting if any σ, τ I k-intersect. Conjecture (Frankl and Deza, 1977) For n sufficiently large, the size of the maximum set of permutations of an n-set that are k-intersecting is (n k)!. P. Frankl, M. Deza, On the maximum number of permuations with givern maximal or minimal distance, J. Combin. Theory Ser. A 22 (3) (1977)

16 Under Hamming distance Our improvement The proof for k = 1 Theorem (Cameron and Ku,2003, Godsil and Meagher,2009) Let n 2. If F S n is an intersecting family of permutations. Then F (n 1)!, with equality holds if and only if F is a coset of a stabilizer of a point. P. J. Cameron and C. Y. Ku, Intersection families of the permutations, European J. Combin. 24 (2003) C. Godsil and K. Meagher, A new proof of the Erdős-Ko-Rado Theorem for intersecting families of permutations, European J. Combin. 30 (2009)

17 Under Hamming distance Our improvement Proof of Frankl and Deza s conjecture k-coset: T i1 j 1,...,i k j k = {σ S n, σ(i 1 ) = j 1,..., σi k = j k } Theorem (Ellis et al., 2011) For any fixed k and sufficiently large n, if I S n is k-intersecting then I (n k)!, with equality if and only if I is a k-coset. D. Ellis, E. Friedgut and H. Pilpel, Intersecting Families of Permutations, J. Amer. Math. Soc. 24 (3) (2011)

18 Under Hamming distance Our improvement Relation between intersecting families and permutation codes Observation: A t-intersecting family of S n is a permutation code with maximum Hamming distance at most n t. Theorem Let C be a permutation code with maximum Hamming distance n t. Then C (n t)!.

19 Under Hamming distance Our improvement Graph theory model Definition A subgraph of a graph is called a clique if any two of its vertices are adjacent. An independent set is a subgraph in which no two vertices are adjacent. We define a Cayley graph Γ(n, d) := Γ(S n, S(n, d 1)), where S(n, d 1) is the set of all the permutations with more than n d fixed points.

20 Under Hamming distance Our improvement Graph theory model By the definition, Γ(n, d) is a regular graph of degree which equals the size of the generating set, i.e., d 1 ( ) n (n, d) = S(n, d 1) = D k. k k=1 The codewords of an (n, d) permutation code are vertices of an independent set in Γ(n, d). Conversely, any independent set in Γ(n, d) is an (n, d)-permutation code.

21 Under Hamming distance Our improvement Independent set and Clique A graph is vertex-transitive if any vertex can be mapped into any other by a graph automorphism. Theorem (Cameron and Ku, 2003) Let C be a clique and A an independent set in a vertex-transitive graph on n vertices. Then C A n. Theorem M(n, d) n! (d 1)!. P. J. Cameron and C. Y. Ku, Intersection families of the permutations, European J. Combin. 24 (2003)

22 Under Hamming distance Our improvement Main Idea

23 Under Hamming distance Our improvement A result on the independent number For m 1 and x 0, we define the function f m (x) by f m (x) = 1 Theorem (Li and Rousseau, 1996) 0 (1 t) 1/m m + (x m)t dt. Let m 1 be an integer, and let G be a graph of order N with average degree. If any subgraph induced by a neighborhood has maximum degree less than m, then α(g) N f m ( ) N log( /m) 1. Y. Li and C. C. Rousseau, On book-complete graph Ramsey numbers, J. Combin. Theory Ser. B 68(1) (1996),

24 Under Hamming distance Our improvement Our improvement for small d I We use G(n, d) to denote the subgraph induced by the neighborhood of identity in Γ(n, d). Then G(n, d) has vertex set V (G(n, d)) = S(n, d 1) = d 1 k=1 D(n, k). We denote the maximum degree in G(n, d) by m(n, d). Lemma (Y. Yang et al., 2013) For any positive integer n 7, we have m(n, 2) = 0, m(n, 3) = 0, m(n, 4) = 4n 8, m(n, 5) = 7n 2 31n + 34.

25 Under Hamming distance Our improvement Our improvement for small d II Theorem (Y. Yang et al., 2013) Let m (n, d) = m(n, d) + 1, and 1 (1 t) 1/m (n,d) M IS (n, d) := n! 0 m (n, d) + [ (n, d) m (n, d)] t dt. Then M(n, d) M IS (n, d). M IS (13, 5) = greatly improves the best known result which is M(13, 5) F. Gao, Y. Yang, and G. Ge, An Improvement on the GilbertCVarshamov Bound for Permutation Codes, IEEE Tran. Inform. Theory, 59 (5), (2013).

26 Under Hamming distance Our improvement Our improvement Lemma (Y. Yang et al., 2013) When n goes to infinity, m(n, d) = O(n d 3 ). Theorem (Y. Yang et al., 2013) When d is fixed and n goes to infinity, we have M IS (n, d) M GV (n, d) = Ω(log(n)). F. Gao, Y. Yang, and G. Ge, An Improvement on the GilbertCVarshamov Bound for Permutation Codes, IEEE Tran. Inform. Theory, 59 (5), (2013).

27 Under Chebyshev distance The permutation code under Chebyshev distance

28 Under Chebyshev distance Constructions Explicit Construction Let n and d be given. Define C = {(π 1,..., π n ) S n π i i(mod d)for all i [n]}. Theorem (Kløve et al., 2010) If n = ad + b, where 0 b < d, then C is an (n, d)-permutation code and C = ((a + 1)!) b (a!) d b. T. Klove, T. Lin, S. Tsai, and W. Tzeng, Permutation Arrays Under the Chebyshev Distance, IEEE Tran. Inform. Theory, 56 (6), (2010).

29 Under Chebyshev distance Constructions Recursive Construction I Let C be an (n, d)-permutation code of size M, and let r 2 be an integer. We define an (rn, rd)-permutation code, C r, of size M r as follows: for each multiset of r code words from C let (π (j) 1,..., π(j) n ), j = 0, 1,..., r 1, ρ j = (rπ (j) 1 j,..., rπ (j) n j), j = 0, 1,..., r 1 and include (ρ 0 ρ 1... ρ r 1 ) as a codeword in C r. Then (ρ 0 ρ 1... ρ r 1 ) S rn. Hence C r = M r.

30 Under Chebyshev distance Constructions Recursive Construction I Theorem (Kløve et al., 2010) If n > d and r 2, then P (rn, rd) P (n, d) r. T. Klove, T. Lin, S. Tsai, and W. Tzeng, Permutation Arrays Under the Chebyshev Distance, IEEE Tran. Inform. Theory, 56 (6), (2010).

31 Under Chebyshev distance Constructions Recursive Construction II For a permutation π = (π 1, π 2,..., π n ) S n and an integer m, 1 m n + 1 define ϕ m (π) = (m, π 1, π 2,..., π n) S n+1 by π i = π i π i = π i + 1 if π i m if π i > m

32 Under Chebyshev distance Constructions Recursive Construction II Let C be an (n, d)-permutation code, and let 1 s 1 < s 2 <... < s t n + 1 be integers. Define C[s 1, s 2,..., s t ] = {ϕ sj (π) 1 j t, π C}. Theorem (Kløve et al., 2010) If C is an (n, d)-permutation code of Size M and s j + d s j+1 for 1 j t 1, then C[s 1, s 2,..., s t ] is an (n + 1, d)-permutation code of size tm. Corollary (Kløve et al., 2010) If n > d 1, then P (n + 1, d) ( n d + 1)P (n, d). T. Klove, T. Lin, S. Tsai, and W. Tzeng, Permutation Arrays Under the Chebyshev Distance, IEEE Tran. Inform. Theory, 56 (6), (2010).

33 Under Chebyshev distance Lower and upper bounds Sphere-packing and GV bounds on P (n, d) Let V (n, d) denote the number of permutations in S n within Chebyshev distance d of the identity permutation. Theorem n! V (n, d 1) P (n, d) n! V (n, (d 1)/2 ). T. Klove, T. Lin, S. Tsai, and W. Tzeng, Permutation Arrays Under the Chebyshev Distance, IEEE Tran. Inform. Theory, 56 (6), (2010).

34 Under Chebyshev distance Lower and upper bounds Permanent and V (n, d) Definition Let A be a n n matrix. Then the permanent of A is defined by pera = π S n a 1,π1... a n,πn. Let A (n,d) be the n n matrix with a (n,d) i,j = 0, otherwise. a (n,d) i,j Lemma (Lehmer, 1970) V (n, d) = pera (n,d). = 1 if i j d and D. H. Lehmer, Permutations with strongly restricted displacements,in Combinatorial Theory and its applications II, P. Erdos, A. Renyi, and V. T. Sos, Eds. Amsterdam, The Netherlands: North Holland, 1970.

35 Under Chebyshev distance Lower and upper bounds Upper bound for V (n, d) Lemma n pera (r i!) 1/r i, i=1 where r i is the number of ones in row i. Theorem (Kløve et al., 2010) V (n, d) [(2d + 1)!] n/(2d+1). J. H. van Lint and R. M. Wilson, A Course in Combinatorics, 2nd ed. Cambridge, U. K.: Cambridge Univ. Press, T. Klove, T. Lin, S. Tsai, and W. Tzeng, Permutation Arrays Under the Chebyshev Distance, IEEE Tran. Inform. Theory, 56 (6), (2010).

36 Under Chebyshev distance Lower and upper bounds Construction of B (n,d) Define the matrix B (n,d) as follows: b (n,d) i,j = Theorem (Kløve, 2011) 0 if i > j + d or j > i + d, 2 if i + j d + 1 or i + j 2n + 1 d, 1 otherwise. perb (n,d) 2 2d pera (n,d).

37 Under Chebyshev distance Lower and upper bounds Example A(6, 2) = B(6, 2) =

38 Under Chebyshev distance Lower and upper bounds Lower bound for V (n, d) Theorem If A is an n n matrix where the sum of the elements in any row or column is k, then pera n!k n /n n. Theorem (Kløve, 2011) V (n, d) n!(2d + 1)n 2 2d n n. J. H. van Lint and R. M. Wilson, A Course in Combinatorics, 2nd ed. Cambridge, U. K.: Cambridge Univ. Press, T. Kløve, Lower bounds on the size of spheres of permutations under the Chebyshev distance, Des. Codes Cryptogr., (2011).

39 Under Chebyshev distance Lower and upper bounds Bounds for P (n, d) Theorem (Kløve et al., 2010) n! [(2d 1)!] n/(2d 1) P (n, d) 2d 1 n n d n. T. Klove, T. Lin, S. Tsai, and W. Tzeng, Permutation Arrays Under the Chebyshev Distance, IEEE Tran. Inform. Theory, 56 (6), (2010).

40 Under Chebyshev distance Lower and upper bounds Further work Improve the lower and upper bound for M(n, d). Give more explicit constructions for the permutation codes under Chebyshev distance. Give a more accurate evaluation for V (n, d). Improve the lower and upper bound for P (n, d). Apply the graph model to other codes.

41 Under Chebyshev distance Lower and upper bounds Thank you!

THE ERDŐS-KO-RADO THEOREM FOR INTERSECTING FAMILIES OF PERMUTATIONS

THE ERDŐS-KO-RADO THEOREM FOR INTERSECTING FAMILIES OF PERMUTATIONS A Thesis Submitted to the Faculty of Graduate Studies and Research In Partial Fulfillment of the Requirements for the Degree of Master