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An improvement to the Gilbert-Varshamov bound for permutation codes 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

An improvement to the Gilbert-Varshamov bound for permutation codes 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

An improvement to the Gilbert-Varshamov bound for permutation codes 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), 352-360. N. Pavlidou, A. J. H. Vinck, J. Yazdani and B. Honary, Powerline communications: State of the art and future trends, IEEE Communications Magazine, (2003), 34-40. A. Jiang, R. Mateescu, M. Schwartz, and J. Bruck, Rank modulation for flash memories, in Proc. IEEE Int. Symp. Information Theory, 2008, 1736-1740.

An improvement to the Gilbert-Varshamov bound for permutation codes 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., 1988. T. Kløve, T. Lin, S. Tsai, and W. Tzeng, Permutation Arrays Under the Chebyshev Distance, IEEE Tran. Inform. Theory, 56 (6), 2611-2617 (2010).

An improvement to the Gilbert-Varshamov bound for permutation codes Introduction to permutation codes Permutation code of minimum distance d Example Let σ = 23451 and π = 12543 . 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.

An improvement to the Gilbert-Varshamov bound for permutation codes 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.

An improvement to the Gilbert-Varshamov bound for permutation codes Under Hamming distance The permutation code under Hamming distance

An improvement to the Gilbert-Varshamov bound for permutation codes 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), 51-64. D. H. Smith and R. Montemanin, A new table of permutation codes, Des. Codes Cryptogr. 63 (2)(2012), 241-253.

An improvement to the Gilbert-Varshamov bound for permutation codes 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), 143-158.

An improvement to the Gilbert-Varshamov bound for permutation codes 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. � n � Clearly, | D ( n, k ) | = D k . k Theorem n ! M ( n, d ) ≤ � . � ⌊ d − 1 2 ⌋ � n D k k =0 k P. Dukes and N. Sawchuck, bounds on permutation codes of distance four, J. Algebraic Combin. 31 (1) (2010), 143-158.

An improvement to the Gilbert-Varshamov bound for permutation codes 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 ! ( n + 1) n ( n − 1) M ( n, 4) ≥ 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) 352-360.

An improvement to the Gilbert-Varshamov bound for permutation codes Under Hamming distance Lower bounds Gilbert-Varshamov bound Theorem n ! M ( n, d ) ≥ � . � d − 1 � n k =0 D k k P. Dukes and N. Sawchuck, bounds on permutation codes of distance four, J. Algebraic Combin. 31 (1) (2010), 143-158.

An improvement to the Gilbert-Varshamov bound for permutation codes Under Hamming distance Our improvement Motivation

An improvement to the Gilbert-Varshamov bound for permutation codes Under Hamming distance Our improvement EKR Theorem Theorem (Erd˝ os, Ko and Rado,1961) Let k be positive integers with n > 2 k . If F is a intersecting family of k -subsets of { 1 , 2 , . . . , n } , then � n − 1 � |F| ≤ . k − 1 � n − 1 � Moreover, |F| = if and only if F is the collection of all k − 1 k -subsets that contain a fixed i ∈ { 1 , . . . , n } . P. Erd˝ os, C. Ko, R. Rado, Intersection theorems for systenms of finite sets , Quart. J. Math. Oxford Ser. 12(2) (1961) 313-320.

An improvement to the Gilbert-Varshamov bound for permutation codes 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) 352-360.

An improvement to the Gilbert-Varshamov bound for permutation codes 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) 881-890. C. Godsil and K. Meagher, A new proof of the Erd˝ os-Ko-Rado Theorem for intersecting families of permutations, European J. Combin. 30 (2009) 404-414.

An improvement to the Gilbert-Varshamov bound for permutation codes Under Hamming distance Our improvement Proof of Frankl and Deza’s conjecture • k-coset: T i 1 → 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) 649-682.

An improvement to the Gilbert-Varshamov bound for permutation codes 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 )! .

An improvement to the Gilbert-Varshamov bound for permutation codes 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.

An improvement to the Gilbert-Varshamov bound for permutation codes 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.