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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 Sn be the set of all permutations of length n. The permutation code C is just a subset of Sn. 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 ∈ Sn, their Hamming distance dH(σ, π) is the number of elements that they differ. Definition Let π = π1π2 . . . , πn, σ = σ1σ2 . . . , σn ∈ Sn. The Chebyshev distance between π and σ is dC(π, σ) = max{|πj − σj||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 dH(σ, π) = 5 and dC(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 Sn which are exactly at distance k from the identity. Clearly, |D(n, k)| = Dk n

k

• .

Theorem M(n, d) ≤ n! ⌊ d−1

2 ⌋

k=0

Dk n

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 k2 ≤ n ≤ k2 + k − 2 for some integer k ≥ 2, then n! M(n, 4) ≥ 1+ (n + 1)n(n − 1) n(n − 1) − (n − k2)((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 M(n, d) ≥ n! d−1

k=0 Dk

n

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˝

• s, 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 |F| ≤ n − 1 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. Erd˝
• s, 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 ∈ Sn are said to k-intersect if they agree on at least k points. A set I ⊂ Sn 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 ⊂ Sn 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˝
• s-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:

Ti1→j1,...,ik→jk = {σ ∈ Sn, σ(i1) = j1, . . . , σik = jk} Theorem (Ellis et al., 2011) For any fixed k and sufficiently large n, if I ⊂ Sn 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 Sn 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(Sn, 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., ∆(n, d) = |S(n, d − 1)| =

d−1

• k=1

n k

• Dk.

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.

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

Independent set and Clique

A graph is vertex-transitive if any vertex can be mapped into any

• ther 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) 881-890.

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

Main Idea

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

A result on the independent number

For m ≥ 1 and x ≥ 0, we define the function fm(x) by fm(x) = 1 (1 − t)1/m m + (x − m)tdt. Theorem (Li and Rousseau, 1996) 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 · fm(∆) ≥ N · log(∆/m) − 1 ∆ .

• Y. Li and C. C. Rousseau, On book-complete graph Ramsey numbers, J.
• Combin. Theory Ser. B 68(1) (1996), 36-44.

An improvement to the Gilbert-Varshamov bound for permutation codes 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) = 7n2 − 31n + 34.

An improvement to the Gilbert-Varshamov bound for permutation codes 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 MIS(n, d) := n! · 1 (1 − t)1/m′(n,d) m′(n, d) + [∆(n, d) − m′(n, d)] t · dt. Then M(n, d) ≥ MIS(n, d). MIS(13, 5) = 2147724 greatly improves the best known result which is M(13, 5) ≥ 878778.

• F. Gao, Y. Yang, and G. Ge, An Improvement on the GilbertCVarshamov

Bound for Permutation Codes, IEEE Tran. Inform. Theory, 59 (5), 3059-3063 (2013).

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

Our improvement

Lemma (Y. Yang et al., 2013) When n goes to infinity, m(n, d) = O(nd−3). Theorem (Y. Yang et al., 2013) When d is fixed and n goes to infinity, we have MIS(n, d) MGV (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), 3059-3063 (2013).

An improvement to the Gilbert-Varshamov bound for permutation codes Under Chebyshev distance

The permutation code under Chebyshev distance

An improvement to the Gilbert-Varshamov bound for permutation codes Under Chebyshev distance Constructions

Explicit Construction

Let n and d be given. Define C = {(π1, . . . , πn) ∈ Sn|πi ≡ 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), 2611-2617 (2010).

An improvement to the Gilbert-Varshamov bound for permutation codes 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, Cr , of size Mr as follows: for each multiset of r code words from C (π(j)

1 , . . . , π(j) n ),

j = 0, 1, . . . , r − 1, let ρj = (rπ(j)

1

− j, . . . , rπ(j)

n − j),

j = 0, 1, . . . , r − 1 and include (ρ0|ρ1| . . . |ρr−1) as a codeword in Cr. Then (ρ0|ρ1| . . . |ρr−1) ∈ Srn. Hence |Cr| = Mr.

An improvement to the Gilbert-Varshamov bound for permutation codes 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), 2611-2617(2010).

An improvement to the Gilbert-Varshamov bound for permutation codes Under Chebyshev distance Constructions

Recursive Construction II

For a permutation π = (π1, π2, . . . , πn) ∈ Sn and an integer m, 1 ≤ m ≤ n + 1 define ϕm(π) = (m, π′

1, π′ 2, . . . , π′ n) ∈ Sn+1

by π′

i = πi

if πi ≤ m π′

i = πi + 1

if πi > m

An improvement to the Gilbert-Varshamov bound for permutation codes Under Chebyshev distance Constructions

Recursive Construction II

Let C be an (n, d)-Permutation code, and let 1 ≤ s1 < s2 < . . . < st ≤ n + 1 be integers. Define C[s1, s2, . . . , st] = {ϕsj(π)|1 ≤ j ≤ t, π ∈ C}. Theorem (Kløve et al., 2010) If C is an (n, d)-permutation code of Size M and sj + d ≤ sj+1 for 1 ≤ j ≤ t − 1, then C[s1, s2, . . . , st] 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), 2611-2617(2010).

An improvement to the Gilbert-Varshamov bound for permutation codes 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 Sn 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), 2611-2617(2010).

An improvement to the Gilbert-Varshamov bound for permutation codes 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∈Sn

a1,π1 . . . an,πn. Let A(n,d) be the n × n matrix with a(n,d)

i,j

= 1 if |i − j| ≤ d and a(n,d)

i,j

= 0, otherwise. Lemma (Lehmer, 1970) V (n, d) = perA(n,d).

• 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.

An improvement to the Gilbert-Varshamov bound for permutation codes Under Chebyshev distance Lower and upper bounds

Upper bound for V (n, d)

Lemma perA ≤

n

• i=1

(ri!)1/ri, where ri 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, 2011.

• T. Klove, 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 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

=    if i > j + d or j > i + d, 2 if i + j ≤ d + 1 or i + j ≥ 2n + 1 − d, 1

• therwise.

Theorem (Kløve, 2011) perB(n,d) ≤ 22dperA(n,d).

An improvement to the Gilbert-Varshamov bound for permutation codes Under Chebyshev distance Lower and upper bounds

Example

A(6, 2) =         1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1         B(6, 2) =         2 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2        

An improvement to the Gilbert-Varshamov bound for permutation codes 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

• r column is k, then

perA ≥ n!kn/nn. Theorem (Kløve, 2011) V (n, d) ≥ n!(2d + 1)n 22dnn .

• J. H. van Lint and R. M. Wilson, A Course in Combinatorics, 2nd ed.

Cambridge, U. K.: Cambridge Univ. Press, 2011.

• T. Kløve, Lower bounds on the size of spheres of permutations under the

Chebyshev distance, Des. Codes Cryptogr., 59 183-191 (2011).

An improvement to the Gilbert-Varshamov bound for permutation codes 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−1nn dn .

• T. Klove, 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 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.

An improvement to the Gilbert-Varshamov bound for permutation codes Under Chebyshev distance Lower and upper bounds

Thank you!