Constructing Permutation Arrays from Groups

arXiv:1511.04494v1 [cs.DM] 14 Nov 2015

Constructing Permutation Arrays from Groups Sergey Bereg∗

Avi Levy†

I. Hal Sudborough∗

Abstract Let M (n, d) be the maximum size of a permutation array on n symbols with pairwise Hamming distance at least d. We use various combinatorial, algebraic, and computational methods to improve lower bounds for M (n, d). Some lower bounds are based on a new computational approach called a coset method. We also use algebraic techniques, involving computing the Hamming distances of affine semilinear groups and projective semilinear groups, and unions of cosets of AGL(1, q) and P GL(2, q) with Frobenius maps to obtain infinitely many new, improved lower bounds for M (n, d). We give new randomized algorithms for computing cosets. For example, we show that M (13, 4) is at least 60,635,520, where 41,712,480 was previously the best lower bound. Furthermore, this permutation array consists of a union of 638 cosets of a group of 95,040 permutations, so it can be recorded and verified quickly. An up-to-date table of lower bounds for M (n, d) is also given.

1

Introduction

Two permutations π and σ on n symbols have Hamming distance hd(π, σ) = d if, for exactly d distinct elements x, π(x) 6= σ(x). The Hamming distance of a permutation array A is d, denoted by hd(A), is the minimum hd(π, σ) for all permutations π 6= σ in A. Arrays of permutations on n symbols and with Hamming distance at least d between any two permutations in the array have been used for error correcting codes in communication over very noisy power line channels [13, 18]. For positive integers n and d, with d ≤ n, denote by M(n, d) the maximum size of such an array. Constructing maximum size permutation arrays is difficult and, except for ∗

Department of Computer Science, Erik Jonsson School of Engineering and Computer Science, University of Texas at Dallas. † Department of Mathematics, University of Washington.

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special cases of n and d, work has generally been limited to finding good upper and lower bounds. It is known that sharply k-transitive groups G of permutations on n symbols form a maximum size permutation array for pairwise Hamming distance d = n − k + 1 [10]. Except for the exceptional case, i.e. 4- and 5-transitive Mathieu groups [4, 7, 8], sharply k-transitive permutation groups on n symbols are only known for k = 2 and k = 3 and for n a power of a prime and one more than a power of a prime, respectively, [17]. Furthermore, it is known that sharply k-transitive groups do not exist otherwise. Combinatorial arguments [5] are known, which give upper and lower bounds for M(n, d) for all n and d. There are also computational approaches that have been used to construct good permutation arrays for small values of n. Due to the growth rate of n!, these computational approaches have generally been limited to small values of n and the use of automorphism groups to factor the set of all permutations into a smaller space and hence extend the range of computational processes [20]. It is also known that M(n, n − 1) ≥ kn, where k is the number of mutually orthogonal latin squares (MOLS) of order n [6], which means computing large collections of MOLS is related to searching for large permutation arrays. Other techniques that have been used include permutation polynomials [5], and special groups, such as the Mathieu groups M22 , M23 and M24 [4]. Let n be a positive integer and Zn = {0, 1, 2, . . . , n − 1}. Let Sn be the symmetric group on Zn . If G is a group, then Gσ = {gσ | g ∈ G} and σG = {σg | g ∈ G} are called a right coset of G and a left coset of G, respectively, with the representative σ [12]. Our results, giving infinitely many improved lower bounds are obtained by describing collections of cosets of groups. We show that groups AΓL(1, n = pk ) and P ΓL(2, n = pk ), where p is prime, obtained from the affine general group AGL(1, n) and the projective general group P GL(2, n) by adding k Frobenius en∗ domorphisms, both have pairwise Hamming distance n − pk , where k ∗ denotes the largest proper1 factor of k. For example, AΓL(1, n = 2k ), when k is prime, consists of kn(n − 1) permutations on n symbols and has pairwise Hamming distance 2k − 2, so M(n, n − 2) ≥ kn(n − 1), P ΓL(2, n = 2k ), when k is prime, consists of k(n + 1)n(n − 1) permutations on n + 1 symbols and has pairwise Hamming distance 2k − 2, so M(n + 1, n − 2) ≥ k(n + 1)n(n − 1). We also show that the groups AGL(1, n) and P GL(2, n), where n = pk , together with the coset defined by the single Frobenius endomorphism f (x) = xp , has pairwise Hamming distance n − p. Using a coset technique and computations involving random choices, we give several improved lower bounds for M(n, d), For example, we show that M(13, 5) ≥ 6, 639, 048 and M(14, 5) ≥ 58, 227, 624, which improve previous lower bounds. 1

A factor of k is proper if it is smaller than k.

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2

New Theoretical Bounds

In this section we observe that the Frobenius endomorphisms when added to the groups AGL(1, n) and P GL(2, n) create PA’s of relatively large Hamming distance. Let Fn be a field of order n. Then Fn has prime-power order n = pk . A polynomial f with coefficients in Fn is called a permutation polynomial if f is a bijective function from Fn to Fn . A permutation polynomial is a permutation of Fn . We consider initially two groups: the affine general semilinear group AΓL(1, n) and the projective general semilinear group P ΓL(2, n). These groups are described in the appendix. We state initially the theorems and later give the proofs. ∗

Theorem 1. The Hamming distance of AΓL(1, n) is n − pk , where n = pk and k ∗ denotes the largest proper factor of k. Since AΓL(1, n) contains kn(n − 1) elements, we conclude the following. Corollary 1. Let n = pk where p is a prime number and let k ∗ be the largest proper ∗ factor of k ≥ 1. Then M(n, n − pk ) ≥ kn(n − 1). Suppose n = 2k where k is prime, then M(n, n − 2) ≥ kn(n − 1). Example. M(2048, 2046) ≥ 11 · 2048 · 2047 = 46, 114, 816. The following result is also based on AΓL(1, n). Theorem 2. Let n = pk where p is a prime number and k ≥ 1. Let s be the smallest prime factor of k. Then M(n, n − p) ≥ s · q(q − 1). Examples. M(16, 14) ≥ 2 · 16 · 15 = 480. M(64, 62) ≥ 2 · 64 · 63 = 8064. M(81, 78) ≥ 2 · 81 · 80 = 12960. M(256, 254) ≥ 2 · 256 · 255 = 130560. M(512, 510) ≥ 3 · 512 · 511 = 784896. ∗

Theorem 3. The Hamming distance of P ΓL(2, n) is n − pk , where k ∗ denotes the largest proper factor of k. Since P ΓL(2, n) contains k(n + 1)n(n − 1) elements, we conclude the following. Corollary 2. Let n = pk where p is a prime number and k ≥ 1. Let k ∗ be the largest ∗ proper factor of k. Then M(n + 1, n − pk ) ≥ k(n + 1)n(n − 1). Examples. M(17, 14) ≥ 2 · 17 · 16 · 15 = 8160. M(28, 24) ≥ 3 · 28 · 27 · 26 = 58968, and M(33, 30) ≥ 5 · 33 · 32 · 31 = 163680. Specifically, if n = 2p where p is prime, then M(n + 1, n − 2) ≥ p(n + 1)n(n − 1).

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Let G ⊆ Sn be a permutation group. Let S and T be sets of permutations. Then hd(S, T ) denotes the minimal Hamming distance between distinct elements of S and T . If S or T contains just one permutation, we use it instead of the set, for example hd(s, T ) = hd(S, T ) where S = {s}. The goal in this section is to compute hd(G). First, we observe a well known property: Lemma 1. For any τ, σ, ρ ∈ Sn , hd(στ, σρ) = hd(τ, ρ). Proof. It follows from στ (x) = σρ(x) ⇐⇒ τ (x) = ρ(x). The following lemma shows that computing the Hamming distance of a group G does not require computing the Hamming distance between all the pairs of element in G. It shows that the computation time is O(|G|) rather than O(|G|2). Lemma 2. Let G be a group of permutations G ⊆ Sn with |G| > 1. The Hamming distance of G, i.e. hd(G), is equal hd(e, G \ {e}), where e is the identity permutation. Proof. Pick any two distinct permutations a, b ∈ G. Then, hd(a, b) = hd(e, a−1 b). Since a−1 b is in G and is not the identity, the result follows. Lemma 3. Let G and H be subgroups of Sn , such that G = ∪0≤i≤r ai H, for some r > 0, where a0 = e. Then, hd(G) = min({hd(ai , H) | 0 < i ≤ r}, hd(e, H \ {e})). Proof. Observe that, since G = ∪0≤i≤r ai H, then G = ∪0≤i≤r Ha−1 i . Indeed, if g ∈ G, −1 then g = ai h for some 0 ≤ i ≤ r and h ∈ H. Then g ∈ G and g −1 = h−1 a−1 ∈ i −1 Hai . So, hd(G) = min({hd(e, ai H) | 0 < i ≤ r}, hd(e, H \ {e})) = min({hd(e, Ha−1 i ) | 0 < i ≤ r}, hd(e, H \ {e})) = min({hd(ai , H) | 0 < i ≤ r}, hd(e, H \ {e})).

Using the following lemma, one can compute the Hamming distance simply by counting roots of polynomials. For any polynomial f with coefficients over a finite field Fn , let r(f ) denote the number of roots of f in Fn . If f and g are polynomials that define permutations on Fn , then the Hamming distance between the permutations f and g is equal to n minus the number of roots in the polynomial f − g, i.e. hd(f, g) = n − r(f − g). This is easily seen by observing that f (x) = g(x) is equivalent to f (x) − g(x) = 0; in other words, x is a root of the polynomial f − g. 4

Lemma 4. For any distinct polynomials f, g ∈ Fn [x] we have hd(f, g) = n − r(f − g). Theorem 4. Let G = AGL(1, n), with n = pk , where p is prime and k is a positive integer. For any i(1 ≤ i < k), let fi be the Frobenius function, fi : Fn → Fn , defined i by fi (x) = xp . Then, hd(fi , G) ≥ n − pgcd(i,k) . Proof. The elements of G = AGL(1, n) are the polynomials of the form ax + b, i where a 6= 0 and b are elements of Fn . So, by Lemma 4, hd(fi , G) ≥ n − max{r(xp + i ax + b) | a 6= 0, b ∈ Fn }. Let ta,b (x) be the polynomial xp + ax + b, ta (x) be the i i polynomial xp + ax, and t(x) be the polynomial xp − x. We show that, for all a, b, r(ta,b (x)) ≤ r(ta (x)), and, for all a, r(ta (x)) ≤ r(t(x)). First, let y be a root of ta,b (x). (If ta,b (x) has no root in Fn , then clearly r(ta,b (x)) ≤ i r(ta (x)).) For any root yi of ta,b (x), we have ta (yi − y) = (yi − y)p + a(yi − y) = i i yip − y p + ayi − ay = ta (yi ) − ta (y) = 0, where the second equation follows from the property (a+b)p = ap +bp of Frobenius endomorphisms [4]. Thus, yi −y is a root of ta . Since the mapping y1 → (y1 − y) is an injection, it follows that r(ta,b (x)) ≤ r(ta (x)), whenever ta,b (x) has at least one root. Thus, in general, r(ta,b (x)) ≤ r(ta (x)). We show, for all a, r(ta (x)) ≤ r(t(x)), by showing that, for all a, r(t◦a (x)) ≤ i i r(t◦ (x)), where t◦a (x) = ta (x)/x = xp −1 + a, and t◦ (x) = t(x)/x = xp −1 − 1. Suppose that a = 0. Then 0 is the only root of t◦a (x). Since t◦ (x) also has the root 1, the inequality is trivially true. So, assume that a 6= 0. Then 0 is not a root of t◦a (x). We may also assume that t◦a (x) has at least one root; otherwise, the inequality is trivially true. Let z be a root of t◦a (x). As zi ranges over all roots of t◦a (x), map zi to zi /z. Observe that: ◦

t

z  i

z

=

 z pi −1 i

z

i

zip −1 −a − 1 = pi −1 − 1 = − 1 = 0. −a z

So, zi /z is a root of t◦ (x). Since the map is injective, it follows that r(ta (x)) ≤ r(t(x)). i Let S be the set of all roots of t(x) = xp − x. Observe that S forms a finite field, since the set of roots are closed under the operations of addition, multiplication, and division. Thus, S is a subfield of Fn , and hence the cardinality of S divides the cardinality of Fn , which is pk . So, |S| = pj , for some j, where j | k. Now consider the i extension of t(x) = xp − x into its splitting field [9, 15]. In this field, the expanded root set forms Fpi . So, S is a subfield of Fpi , and so j | i. Thus, j divides both i and k, i.e. j = r(t(x)) ≤ pgcd(i,k). By transitivity we have, for all a 6= 0, b ∈ Fn , r(ta,b (x)) ≤ pgcd(i,k) . Thus, for all i(1 ≤ i < k), hd(fi , G) ≥ n − pgcd(i,k) . 5

Corollary 3. Let G = AΓL(1, n), with n = pk , where p is prime and k is a positive ∗ integer. Let k ∗ denote the largest proper divisor of k. Then hd(G, G) ≥ n − pk . i

Proof. Fix G = AΓL(1, n) = ∪0≤i 1, M(n + 1, n − 2) ≥ 2 · (n − 1) · n · (n + 1) and M(n, n − 2) ≥ 2 · (n − 1) · n. Proof. By Lemma 3, the Hamming distance between the coset f ◦ G and G, where G = P GL(2, n) and f (x) = xp is the Frobenius endomorphism, is hd(f, G). By Theorem 5, hd(f, G) = n − 2. So, the set S = G ∪ f ◦ G is a set of 2 · (n − 1) · n · (n + 1) permutations with hd(S) = n−2. It follows that M(n+1, n−2) ≥ 2·(n−1)·n·(n+1). Similarly, for the group G′ = AGL(1, n), the set S ′ = G′ ∪f ◦G′ is a set of 2·(n−1)·n permutations with hd(S ′ ) = n − 2. It follows that M(n, n − 2) ≥ 2 · (n − 1) · n. Examples. M(17, 14) ≥ 8, 160. M(65, 62) ≥ 524, 160. M(16, 14) ≥ 480. M(64, 62) ≥ 8, 064. Theorem 6. Let n = pk , where p is prime. P ΓL(2, n) has Hamming distance n−pk , where k ∗ denotes the largest proper factor of k. ∗

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Proof. Let G = P ΓL(2, n) and H = {g ∈ G | g(∞) = ∞}. H can be identified with AΓL(1, n) by means of the isomorphism i : AΓL(1, n) → H defined by:   g(x) if x ∈ Fn and g(x) ∈ Fn , i(g)(x) = ∞ if x = ∞,  g(x) if x ∈ Fn and g(x) = ∞.

Let g and h be distinct elements of G. If g(x) 6= h(x), for all x, then hd(g, h) = n+ 1. Otherwise, pick a point y such that, for some w, g(y) = h(y) = w. Choose the following elements of G: ( 1 if w ∈ Fn , τ1 (x) = x−w x if w = ∞. ( yx + 1 if y ∈ Fn , τ2 (x) = x x if y = ∞. Let g ′ = τ1 ◦ g ◦ τ2 and h′ = τ1 ◦ h ◦ τ2 . We observe that g ′ (∞) = ∞ and h′ (∞) = ∞. That is, (a) if both y and w are in Fn , then 1 y·∞+1 ) = τ1 (g(y)) = τ1 (w) = = ∞, g ′ (∞) = τ1 (g(τ2 (∞))) = τ1 (g( ∞ w−w y·∞+1 1 h′ (∞) = τ1 (h(τ2 (∞))) = τ1 (h( ) = τ1 (h(y)) = τ1 (w) = = ∞, ∞ w−w (b) if y = ∞ and w ∈ Fn , then 1 = ∞, g ′ (∞) = τ1 (g(τ2 (∞))) = τ1 (g(∞)) = τ1 (g(y)) = τ1 (w) = w−w 1 ′ h (∞) = τ1 (h(τ2 (∞))) = τ1 (h(∞)) = τ1 (h(y)) = τ1 (w) = w−w = ∞, (c) if y ∈ Fn and w = ∞, then g ′ (∞) = τ1 (g(τ2 (∞))) = τ1 (g( y∞+1 ) = τ1 (g(y)) = τ1 (w) = τ1 (∞) = ∞, ∞ y∞+1 ′ h (∞) = τ1 (h(τ2 (∞))) = τ1 (h( ∞ ) = τ1 (h(y)) = τ1 (w) = τ1 (∞) = ∞, and (d) if y = ∞ and w = ∞, then g ′ (∞) = τ1 (g(τ2 (∞))) = τ1 (g(∞)) = τ1 (g(y)) = τ1 (w) = τ1 (∞) = ∞, h′ (∞) = τ1 (h(τ2 (∞))) = τ1 (h(∞)) = τ1 (h(y)) = τ1 (w) = τ1 (∞) = ∞. So, it follows that g ′ and h′ are in H. By Lemma 1, hd(g ′, h′ ) = hd(τ1 ◦g◦τ2 , τ1 ◦h◦τ2 ) = hd(g, h). Moreover, the isomorphism i preserves Hamming distance, so hd(g ′, h′ ) = hd(i−1 (g ′), i−1 (h′ )). Since i−1 (g ′) and i−1 (h′ ) are in AΓL(1, n), it follows by Theorem 1 that hd(i−1 (g ′), i−1 (h′ )) ≥ ∗ ∗ n − pk . So, hd(g, h) ≥ n − pk .

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3

Computing Permutation Arrays using Cosets

Searching for permutation arrays of permutations of length n and pairwise hamming distance d can quickly run into trouble. For even relatively small values of n the number of permutations, n!, is very large and simple search techniques are ineffective. Previous search techniques for M(n, d) worked successfully for small values of n, or cleverly used automorphism groups to factor the space (of all permutations), to achieve effectiveness [5, 20]. On the other hand, the lower bound given by Gilbert-Varshimov [10, 16, 19], which states that M(n, d) is at least as large as NGV (n, d), where NGV (n, d) is given by: n! , NGV (n, d) = V (n, d − 1)
P where V (n, r) = nk=0 nk Dk and Dk is the number of derangements of length k, is proven by an implicit algorithm for searching the space of all permutations. That is, V (n, d−1) represents the number of permutations at distance less than d from a given permutation, say σ, i.e. the volume of a sphere of radius d − 1 around σ, denoted by Sd−1 (σ). If σ is a chosen permutation for a constructed PA, then other permutations within Sd−1 (σ) cannot be included, as they are at distance less than d from σ. So, if one iteratively chooses permutations σ1 , σ2 , σ3 , . . . for the constructed PA, one cannot choose permutations within Sd−1 (σ1 ), Sd−1 (σ2 ), Sd−1 (σ3 ), . . . One observes that, even if the spheres of radius d − 1 around σ1 , σ2 , σ3 , . . . are disjoint, one is guaranteed to be able to choose another permutation for the constructed PA at least as many times as the total number of permutations divided by the size of the spheres of radius d−1. In fact, one can choose the permutations in the sequence σ1 , σ2 , σ3 , . . . in an arbitrary way and obtain at least NGV (n, d) permutations. Of course, this implicit algorithm overlooks some practical implementation issues. For example, keeping track of which permutations have been chosen, and which cannot be chosen, as they are too close to chosen ones, requires an enormous table and searching the table takes enormous time. Due to the growth rate of the factorial function, the implicit algorithm is not practical. It should be noted that one can often get far more than NGV (n, d) permutations in a PA, because the assumption that the spheres are disjoint is not required. In fact, this combinatorial lower bound has recently been improved [11]. We designed a simple, practical algorithm to search the space of all permutations using randomness and taking advantage of the groups that underlie many PA’s. In fact, the algorithm looks at each step for a possibly large coset of a group, and not just a single permutation. It does not create a new PA for M(n, d) from scratch. 8

We show that looking for a coset of a group is equivalent to looking for a new permutation, but the coset search makes the search process more efficient in both space and time. For every integer n, there is a known group of permutations of length n and pairwise hamming distance d. For example, if there is no larger group, the identity permutation of length n, and all cyclic shifts of it, forms a cyclic group of permutations, denoted by C(n), and it has pairwise hamming distance n. For n a prime number, or a prime power, there are the groups AGL(1, n), and for numbers n one more than a prime, or a prime power, there are the groups P GL(2, n) [3], which have pairwise hamming distance n−1 and n−2, respectively. In fact, there are many groups that can be used as a starting point in search for a better PA. There are many advantages in a search designed to use groups. Testing if a permutation is at a desired hamming distance to every permutation in the group can be performed efficiently. Furthermore, for a given group G, if a permutation σ is found such that hd(σ, G) ≥ d, then the entire coset σG has pairwise hamming distance at least d from every permutation in G. That is, for a elements g1 , g2 in G, hd(σg1 , g2) = hd(σ, g2 g1−1 ), and g2 g1−1 ∈ G. Therefore, the entire coset can be added to the constructed PA and one can simply record σ, as the others in the coset are easily obtained from σ. As an example, we used the sharply 5-transitive Mathieu group M12 of permutations of length 12, which has pairwise hamming distance 8. It contains 95040 permutations. We can create a group, say G, of 95040 permutations of length 13, which also has pairwise hamming distance 8, by transforming each permutation σ of M12 into a permutation σ ′ such that, for every symbol s in {0, 1, . . . , 11}, σ ′(s) = σ(s), and, in addition, σ ′ (12) = 12. Then, as one example, by reducing the hamming distance to 4, our search program found 638 cosets of G and, hence, a PA for M(13, 4) with 638 · 95040 = 60, 635, 520 permutations. In a recent paper by Smith and Montemanni [20], the authors created a new PA for M(12, 4), consisting of 20, 908, 800 permutations, and then stated it ” is too large to check fully, but has been extensively checked.” Due to our use of groups and cosets, one only needs to check 638 coset representatives to verify the correctness of the computed 60, 635, 520 permutations for M(13, 4). This can be done in minutes. Again, in a recent paper by Chu, Colbourn, and Dukes [5], the authors created a PA for M(16, 9) of 58, 032 permutations in their Example 3.6 and observed that the so-called trivial lower bound of NGV (16, 9) = 97, 569 is larger. They stated ” However, it should be mentioned that the latter is not constructive, while the recursive method offers more structure for the resulting PA.” Using the group G = AGL(1, 16) of 240 permutations and pairwise hamming distance 15, our search program found 5,739 cosets, when the pairwise hamming distance is reduced to 9. So, in this way, we constructed a PA for M(16, 9) with 5739 · 240 = 1, 377, 360 permutations, more 9

than 14 times larger than NGV (16, 9), which moreover has considerable structure due to the nature of groups and cosets. Again, the correctness of the PA can be verified in minutes, as the set of coset representatives is much smaller than the PA itself. The effectiveness of our search strategy depends heavily on the probability of a randomly chosen permutation being at an appropriate distance from the permutations in the group, say G, from which it starts. Of course, the probability decreases dramatically as the distance is increased. For example, our search program found a single coset representative for pairwise hamming distance 16 when it was started with the group G = P GL(2, 16), which has pairwise hamming distance 18 and consists of 6,840 permutations. Using this, we have a PA for M(20, 16) with 2 · 6840 = 13, 680 permutations. It took hours of computation time on many computers and found the coset representative: 1, 2, 3, 4, 6, 8, 15, 5, 19, 18, 10, 7, 17, 16, 12, 20, 9, 11, 13, 14. This permutation, or any other of the coset representatives it might have found, once known, is sufficient to conclude that M(20, 16) ≥ 13, 680. However, this exercise shows the need to archive coset representatives. For this reason we have a website that records coset representatives and, hence, the PA’s to which they correspond [1]. Our search strategy is designed so that it always finds a new permutation σ that is exactly at the specified hamming distance, say d, from permutations already in the constructed PA. In that way, there is guaranteed to be considerable overlap between the spheres of radius d−1 with permutations chosen to be in the PA. That is, compared to the number NGV (n, d), which is obtained under the assumption that the spheres are disjoint, the number of permutations found by our search strategy is much larger. In fact, we observe that the coset representatives our program finds are often at distance exactly d from not just one, but many previously chosen permutations, so the amount of overlap is large. Our search program does not terminate until we intercede, i.e. when we decide it is unlikely to find any additional permutations. This decision is usually made when the program has not found anything for quite awhile. And, of course, there could be more coset representatives to find, but it may be that their number is small and the program has trouble finding them. Also, the random choice of coset representatives could affect the number found. That is, the same program, run again, could find more (or less) permutations, because of the randomness of choices. This is another reason we feel it is important to archive our results in order to substantiate our improved lower bounds. Our search strategy, using groups and cosets, has many advantages: (1) computing the pairwise hamming distances of permutations in the constructed PA’s is more 10

efficient, as it can be done by coset representatives and the group alone, (2) there is less time in searching, as each permutation found gives an entire new coset and not just a single new permutation, (3) less space is required to record the permutations in the constructed PA, as the coset representatives and the group are sufficient, and (4) testing the hamming distance of a new permutation to the existing set of permutations in the PA, can be accomplished by testing the hamming distance of permutations with the group and this can be done efficiently. Let G be a group of permutations on Zn = {0, 1, . . . , n − 1} with pairwise Hamming distance (denoted by hd) d. Choose d′ < d and look for a permutation π on Zn that has at least d′ from all elements of G. Then, every permutation in the left coset πG has hd at least d′ from all elements of G. Thus, one can start with G, find a permutation π at hd d′ from G and obtain G ∪ πG as an initial M(n, d′ ) set. Moreover, the process can be iterated. Given a M(n, d′ ) set of the form A = G ∪ π1 G ∪ π2 G · · · ∪ πk G, if one finds a permutation, say πk+1 , which has hd at least d′ from all permutations in A, then A ∪ πk+1 G is a larger M(n, d′ ) set. Note that permutation πk+1 has hd at least d′ from all permutations in A if πi−1 πk+1 has hd at least d′ from every permutation in G, where π0 denotes the identity permutation. Furthermore, the M(n, d) set constructed can be uniquely described simply from the coset representatives, i.e. {π1 , π2 , . . . , πk , πk+1 } and G, and hence has a more compact description than a listing of every permutation in the set. We call this process the coset method of constructing M(n, d′ ) sets. The technique is quite robust. For example, we have computed a set for M(13, 5) with 6, 639, 048 permutations and a set for M(14, 5) with 58, 227, 624 permutations which are considerably larger than the recently published lower bounds [11, 20]. Tables 1 and 2 given below give some of our improved lower bounds obtained by the coset method.

3.1

Efficient Testing of a New Permutation

Suppose that there are k cosets A = {πi G | i = 1, 2, . . . , k}. The critical step of our randomized construction is the computation of distance hd(π, A) where π is a random permutation. The definition of Hamming distance suggests the computation of hd(π, πi · g) for all representatives πi and all g ∈ G. Then the running time is O(nk|G|). We show that, if G is cyclic or has the cyclic subroup, the algorithm for testing π can be improved. Let Cn denote the cyclic group, i.e. Cn = {gj | gj = (j, j + 1, . . . , n − 1, 0, 1, . . . , j − 1), j ∈ Zn }.

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Lemma 6. The Hamming distance from a permutation π ∈ Sn to Cn can be computed in O(n) time. Proof. The Hamming distance hd(π, A) can be computed as hd(π, Cn ) = min0≤j≤n−1 {hd(π, gj )}. Straightforward computation of hd(π, gj ) takes O(n) time. We show that it can be computed in O(1) amortized time. Let D[0..n − 1] be an array and we store a tentative distance of hd(π, gj ) in D[j]. The algorithm has 2 steps. 1. (1) We initialize D[j] = n for all j ∈ Zn . 2. (2) For each m ∈ Zn , subtract one from D[j] where j ≡ (π(m) − m) mod n. Clearly, the running time is O(n). We show that the algorithm is correct. If D[j] decreases for some m, then the m-th element of π is j + m(mod n). Thus, π and gj have matching elements in m-th position. In the end, D[j] is equal to n − n′ where n′ is the number of matching elements of π and gj . The claim follows since hd(π, gj ) = n − n′ . Lemma 7. If G is cyclic then, for any permutation π ∈ Sn , the Hamming distance hd(π, A) can be computed in O(kn) time. Proof. The Hamming distance hd(π, A) can be computed as hd(π, A) = hd(π, ∪ πi Cn ) = min {hd(π, πi Cn }. 1≤i≤k

1≤i≤k

It suffices to show that di = hd(π, πi Cn ) can be computed in O(n) time. Since di = hd(πi−1 π, Cn ), we first compute σ = πi−1 π and then hd(σ, Cn ) using the algorithm from Lemma 6. Theorem 7. If Cn is a subgroup of G then the Hamming distance hd(π, A) can be computed in O(k|G|) time for any permutation π ∈ Sn . Proof. Let G/Cn = {σCn | σ ∈ G} be the set of left cosets of Cn in G. Let σ0 , σ1 , . . . , σm be representatives of these cosets. The Hamming distance hd(π, A) can be computed as [ [ hd(π, A) = hd(π, πi G) = hd(π, πi σt Cn ) = min {hd(π, πi σt Cn )}. 1≤i≤k

1≤i≤k 0≤t≤m

1≤i≤k 0≤t≤m

Since hd(π, πi σt Cn ) = hd(α, Cn ) where α = (πi σt )−1 π, it can be computed in O(n) time using the algorithm from Lemma 6. The total time is kmn = k|G|. 12

4

New tables and conclusions

We give in Tables 1 and 2 an updated partial list of lower bounds for M(n, d), with n ≥ 9 and d ≥ 4. We also created a webpage [1] that allows one to obtain the PA’s, or coded versions of them, for verification. Not all of our new results appear in the table, many are described by theorems in Section 2. Others are given here: We use the following notation in the table to describe how the results are obtained: a - A bound derived from M(n, d − 1) ≥ M(n, d). Examples: M(22, 13), M(23, 13), M(23, 19), M(25, 20). b - a bound derived from M(n + 1, d) ≥ M(n, d). Examples: M(21, 12), M(21, 13). d - A bound derived from M(n−1, d) ≥ M(n, d)/n. For example, we have M(15, 12) ≥ 2520, because M(16, 12) ≥ 40320. t - The Gilbert-Varshamov lower bound [10, 16, 19]. G - A lower bound based on known permutation groups. This includes the Mathieu groups M11 , M12 , M22 , M23 , M24 . m - A bound derived from mutually orthogonal squares [6]. p - A bound for the case where n is a prime or is a prime power or one more than a prime power, namely AGL(1, n) or AGL(2, n), which are sharply 2-transitive and sharply 3-transitive, respectively. u - A result obtained by partitioning and extending, which is contained in [2]. The technique allows one to obtain, in certain cases, M(n + 1, d + 1) ≥ M(n, d). For example, it is used to show: (1) M(10, 7) ≥ 1492, since M(9, 6) ≥ 1512. (2) M(18, 15) ≥ 8160, since M(17, 14) ≥ 8160. (3) M(18, 14) ≥ 12240 and M(19, 15) ≥ 12240, since M(17, 13) ≥ 12240. (4) M(22, 21) ≥ 104, since M(21, 20) ≥ 105. (5) M(20, 19) ≥ 92, since M(19, 18) ≥ 342. (6) M(21, 17) ≥ 13680 and M(22, 18) ≥ 13680, since M(20, 16) ≥ 13680. C - A result obtained by contraction, which is an improvement of the M(n − 1, d − 3) ≥ M(n, d) result, described in [21]. We show that, under certain conditions, one gets M(n−1, d−2) ≥ M(n, d), M(n−2, d−4) ≥ M(n, d), M(n−3, d−6) ≥ M(n, d), . . . For example, one gets M(23, 14 ≥ 2.44·108 from the Mathieu group 13

M24 , which has the Hamming distance 16 and contains more than 2.44 · 108 permutations. Similarly, one gets M(23, 20) ≥ 12144 from M(24, 22) ≥ 12144. In several cases we give a reference to the paper originally given the result. We note that M(9, 6) ≥ 1512, M(16, 14) ≥ 480, M(17, 14) ≥ 8160 follow from Corollaries 1 and 2. We also have M(27, 22) ≥ 23919, M(28, 24) ≥ 58968, M(29, 22) ≥ 235872, M(28, 21) ≥ 1316952, M(29, 26) ≥ 24360, M(29, 25) ≥ 39312, M(30, 26) ≥ 58968, M(30, 24) ≥ 170, 520, M(30, 23) ≥ 1291080, and many others, by the techniques described. We are confident that these techniques and results can be improved and challenge readers to continue this effort.

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d=4 d=5 d=6 d=7 d=8 d=9 d=10 d=11 d=12 d=13 d=14 d=15 d=16 1512 [20] 18144 [20] 3024 [20] 1512 p 504 p 72 9 u 1492 [20] 150480 [20] 18720 [20] 8640 a 720 G 720 [14] 49 10 [20] 1742400 [20] 205920 [20] 95040 a 7920 G 7920 [5] 154 p 110 11 2 190080 [20] 20908800 [20] 2376000 [20] 190080 a 95040 G 95040 a 1320 p 1320 m 60 12 638 110 19 4 60635520 10454400 1805760 380160 [5] 41712480 b 2376000 t 271908 a 95040 b 95040 [20] 4810 b 1320 [20] 195 p 156 13 636 114 20 4 10 60445440 10834560 1900800 380160 21840 [5] 550368000 [11] 22767826 t 890338 t 97547 b 95040 a 6552 [5] 6552 a 2184 p 2184 m 56 14 618 163 20 83 15 3 - 58734720 15491520 1900800 181272 32760 6552 [11] 4.01E9 [11] 263832788 t 8991655 t 888533 t 97572 t 12014 a 6076 6076 d 2520 [20] 243 m 60 15 744 169 5739 687 2 - 70709760 16061760 1377360 164880 480 [11] 1.268E11 [11] 3.317E9 - t 8972298 t 888755 t 97579 a 40320 a 40320 [20] 40320 [20] 1266 [20] 269 p 240 16 791 2298 303 46 3 2 - 75176640 9375840 1236240 187680 12240 8160 [11] 7.93E11 - t 8974885 t 888727 t 97569 t 12014 [5] 83504 a 4080 a 4080 p 4080 p 272

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d=11 d=12 d=13 d=14 d=15 d=16 d=17 d=18 d=19 d=20 d=21 d=22 d=23 d=24 251 40 5 2 1228896 195840 24480 12240 u 8160 t 97569 t 83504 a 4896 a 4896 a 4896 p 4896 90 18 3572 486 1221624 166212 u 12240 t 888729 t 97569 a 65322 [21] 65322 b 4896 a 342 p 342 19 1571 215 26 3 2 - 10745640 1470600 177840 20520 13680 u 92 t 8974608 t 888729 t 97569 12014 65322 a 6840 a 6840 p 6840 m 80 20 174 23 5 2 10 - 10745640 1470600 1190160 157320 34200 u 13680 210 t 9.95E7 t 8974609 t 888729 t 97569 b 65322 a 6840 a 6840 b 6840 a 105 m 105 21 47233 1250 - 10200960 10200960 1039126 27500 u 13680 1100 1012 u 104 t 1.20E9 2.44E8 t 8974609 t 888729 a 443520 443520 a 6840 b 6840 - [5] 220 m 66 22 2.44E08 2.44E08 12144 12144 t 1.57E10 t 1.20E9 - a 10200960 10200960 a 291456 [5] 291456 a 506 p 506 23 - t 1.57E10 t 1.20E9 a 2.44E8 a 2.44E8 2.44E8 t 97569 b 291456 a 12144 b 12144 a 12144 p 12144 m 168 24 - 15600 15600 - b 2.44E8 - b 12144 a 600 p 600 102 2 - 1591200 - 31200 - t 12144 - p 15600

References [1] Table of cosests. http://www.utdallas.edu/~ sxb027100/cosets. [2] S. Bereg, L. Morales, and I. H. Sudborough. Extending permutation arrays for hamming distances. Submitted for publication, September 2015. [3] I. F. Blake. Permutation codes for discrete channels. IEEE Trans. Inform. Theory, 20(1):138–140, 1974. [4] P. Cameron. Permutation Groups. London Mathematical Society Student Texts. Cambridge University Press, 1999. [5] W. Chu, C. J. Colbourn, and P. Dukes. Constructions for permutation codes in powerline communications. Des. Codes Cryptogr., 32(1-3):51–64, 2004. [6] C. Colbourn, T. Klove, and A. Ling. Permutation arrays for powerline communication and mutually orthogonal latin squares. IEEE Transactions on Information Theory, 50(6):1289–1291, 2004. [7] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, , and R. A. Wilson. Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, 1985. [8] J. Dixon and B. Mortimer. Permutation Groups. New York: Springer-Verlag, 1996. [9] D. S. Dummit and R. M. Foote. Abstract Algebra (2nd ed.). John Wiley & Sons, 1999. [10] P. Frankl and M. Deza. On the maximum number of permutations with given maximal or minimal distance. Journal of Combinatorial Theory, Series A, 22(3):352 – 360, 1977. [11] F. Gao, Y. Yang, and G. Ge. An improvement on the Gilbert-Varshamov bound for permutation codes. IEEE Transactions on Information Theory, 59(5):3059– 3063, 2013. [12] I. N. Herstein. Topics in Algebra. John Wiley & Sons, 2nd edition edition, 1975. [13] S. Huczynska. Powerline communication and the 36 officers problem. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 364(1849):3199–3214, 2006. 17

[14] I. Janiszczak and R. Staszewski. An improved bound for permutation arrays of length 10. Technical Report 4, Institute for Experimental Mathematics, University Duisburg-Essen, 2008. [15] R. Lidl and H. Niederreiter. Finite Fields. Cambridge University Press, 2nd edition, 1997. [16] F. J. MacWilliams and N. J. A. Sloane. The Theory of Error-Correcting Codes. Amsterdam: North-Holland/Elsevier, 1977. [17] D. Passman. Sharp transitivity. In Permutation Groups. New York: Benjamin, Inc., 1968. [18] N. Pavlidou, A. H. Vinck, J. Yazdani, and B. Honary. Power lines communications: State of the art and future trends. IEEE Communications Magazine, pages 34–40, 2003. [19] V. S. Pless and W. C. H. (Editors). Handbook of Coding Theory. Amsterdam: North-Holland/Elsevier, 1998. [20] D. H. Smith and R. Montemanni. A new table of permutation codes. Des. Codes Cryptogr., 63(2):241–253, 2012. [21] L. Yang, K. Chen, and L. Yuan. New constructions of permutation arrays. CoRR, abs/0801.3987, 2008.

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Appendix Finite Fields. Given n = pk , a power of a prime, call the unique field with n elements Fn , i.e. Fn = {0, 1, g, g 2, · · · , g n−2} where g is any generator of Fn . Groups. AGL(1, n) is the group of linear polynomials AGL(1, n) = {ax + b|a, b ∈ Fn , a 6= 0} where the group operation is function composition. This group is sharply 2−transitive and has n(n − 1) elements. P GL(2, n) is the projective general linear group acting on Fn ∪ {∞}   ax + b a, b, c, d ∈ Fn , ad 6= bc P GL(2, n) = cx + d

These are called fractional linear functions. The group P GL(2, n) is sharply 3−transitive and yields an optimal M(n + 1, n − 1) with (n + 1)n(n − 1) elements. AΓL(1, n) is the group of semilinear polynomials. Starting from AGL(1, n), append the Frobenius automorphism xp and take the group closure, yielding: i

AΓL(1, n) = {axp + b|a, b ∈ Fn , a 6= 0, 0 ≤ i < k} This group has kn(n − 1) elements. P ΓL(2, n) is the projective group of semilinear polynomials. In analogy with AΓL, start from P GL(2, n) then append xp and take the group closure. ( ) i axp + b P ΓL(2, n) = a, b, c, d ∈ Fn , ad 6= bc, 0 ≤ i < k cxpi + d This group has k(n + 1)n(n − 1) elements.

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