New Upper Bounds for the Size of Permutation Codes via Linear Programming

An (n, d)-permutation code of size s is a subset C of Sn with s elements such that the Hamming distance dH between any two distinct elements of C is at least equal to d. In this paper, we give new upper bounds for the maximal size μ(n, d) of an (n, d)-permutation code of degree n with 11 6 n 6 14. In order to obtain these bounds, we use the structure of association scheme of the permutation group Sn and the irreducible characters of Sn. The upper bounds for μ(n, d) are determined solving an optimization problem with linear inequalities. 1 Permutation arrays and permutation codes An (n, d)-permutation code of distance d, size s and degree n is a non-empty subset C of the symmetric group Sn acting on the set {1, . . . , n} such that the Hamming distance between any two distinct elements of C is at least equal to d. The Hamming distance between two permutations φ, ψ ∈ Sn is defined as dH(φ, ψ) = |{i ∈ {1, . . . , n} : φ(i) 6= ψ(i)}|. The weight of a permutation φ ∈ Sn if the number of non fixed points of φ. The s× n array A associated to a (n, d)-permutation code C = {φ1, . . . , φs} of size s by Aij = φi(j) has the following properties: every symbol 1 to n occurs exactly in one cell of any row and any two rows disagree in at least d columns. Such an array is called a permutation array (PA) of distance d, size s and degree n. Permutation codes have first been proposed by Ian Blake in 1974 as error-correcting codes for powerline communications [3]. This application motivates the study of the largest possible size that a permutation code can have. Upper bounds for the maximal size μ(n, d) of a permutation code with fixed parameters n and d have been studied by the electronic journal of combinatorics 17 (2010), #R135 1 many authors, see e.g. Deza and Frankl [10], Cameron [6], and more intensively since Chu, Colbourn and Dukes [8], Tarnanen [15], and Han Vinck [2, 16]. An (n, d)− permutation code C of weight w is an (n, d)− permutation code such that all permutations have weight w. The maximal size of such a permutation code is denoted by μ(n, d, w). An (n, d)-permutation code C of size s is maximal if C is not contained in an (n, d)permutation code of larger size s > s. Note that an (n, d)-permutation code reaching the maximal size μ(n, d) is necessarily maximal while the converse is not true. The most basic upper bounds on μ(n, d) appears in Deza and Frankl [10]: Theorem 1. For n > 3 and d 6 n, μ(n, d) 6 n μ(n− 1, d) and therefore μ(n, d) 6 n! (d− 1)! In this paper, we will establish new bounds for μ(n, d) for small values of the parameters n and d. In [15], H. Tarnanen uses the conjugacy scheme of the group Sn in order to obtain new upper bounds for the size of a permutation code. We use this method to obtain new upper bounds for μ(n, d). 2 Isometries A distance D on Sn is called left-invariant (resp. right-invariant) if D(φ, ψ) = D(αφ, αψ) (resp. D(φ, ψ) = D(φα, ψα) ) for all α, φ, ψ ∈ Sn. A distance that is both leftand rightinvariant is said to be bi-invariant. For any bi-invariant distance, the left multiplications lα : φ 7→ αφ and the right multiplications rα : φ 7→ φα −1 are isometries. As noticed by Deza and Huang [11], any bi-invariant distance is invertible: D(φ, ψ) = D(φ, ψ), or equivalently, the inversion i, mapping each permutation onto its inverse, is an isometry. Let R (resp. L) denote the group of all right (resp. left-) multiplications and I denote the group generated by the inversion i. We will say that the distance D distinguishes the transpositions if there exists a constant c such thatD(φ, ψ) = c⇔ φψ is a transposition. In 1960, Farahat characterized the isometry group Iso(n) of the metric space (Sn, dH) [12]. Since the Hamming distance is bi-invariant and distinguishes the transpositions, the following result appears in [4] and generalizes the characterisation given by Farahat: Theorem 2. Let D be a bi-invariant distance distinguishing the transpositions on Sn (n > 3), then the group IsoD of isometries of (Sn, D) is (L×R)⋊I, isomorphic to Sn ≀2. Every isometry t ∈ Iso(n) can be uniquely written as lαrβi k with k = 0 or 1, α, β ∈ Sn. The action of a left multiplication lα on a given code corresponds to the permutation under α of the symbols appearing in the PA associated to the code, and the action of a rightmultiplication rβ is equivalent to the permutation under β of the columns of the PA. In other words, classifying permutation codes up to isometry is equivalent to classifying PA’s the electronic journal of combinatorics 17 (2010), #R135 2 up to permutation of their rows, their columns, their symbols and up to the inversion. It immediately follows from this theorem that the autormorphism group of the conjugacy scheme of Sn is precisely the isometry group of the metric space (Sn, dH). 3 Linear programming bound A symmetric association scheme with m classes is a finite set X with m + 1 relations R0, R1, . . . Rm on X such that: • {R0, R1, . . . Rm} is a partition of X ×X • R0 = {(x, x)|x ∈ X} • If (x, y) ∈ Ri, then (y, x) ∈ Ri for all x, y ∈ X and for all i = 0, . . . , m • For each pair (x, y) ∈ Rk , the number p k ij of elements z ∈ X such that (x, z) ∈ Ri and (y, z) ∈ Rj only depends on i, j and k The numbers pkij are called intersection numbers of the association scheme. Let n denote the size of the set X and ni := p 0 ii i = 0, . . . , m. The intersection matrices L0, . . . , Lm are defined by: (Li)jk = p k ij . The relations Ri can be described by their adjacency matrix Ai: The adjacency matrix Ai of the relation Ri is the n× n-matrix such that: (Ai)xy = { 1 if (x, y) ∈ Ri 0 otherwise In terms of adjacency matrices the conditions defining the association scheme become: • m