Complete transitivity of the Nordstrom-Robinson codes

arXiv:1205.3878v2 [math.CO] 18 May 2015 NEW REGULARITY PROPERTIES OF THE NORDSTROM-ROBINSON CODES NEIL I. GILLESPIE AND CHERYL E. PRAEGER Abstract. In his doctoral thesis, Snover proved that any binary (m, 256, δ) code is equivalent to the Nordstrom-Robinson code or the punctured Nordstrom-Robinson code for (m, δ) = (16, 6) or (15, 5) respectively. We prove that these codes are also characterized as completely regular binary codes with (m, δ) = (16, 6) or (15, 5) , and moreover, that they are completely transitive. Also, it is known that completely transitive codes are necessarily completely regular, but whether the converse holds has up to now been an open question. We answer this by proving that certain completely regular codes are not completely transitive, namely, the (Punctured) Preparata codes other than the (Punctured) NordstromRobinson code. 1. Introduction In [28], Hammons et al. proved that certain interesting non-linear codes can be efficiently described as the image under the Grey map of Z4 -linear codes (see Section 5 for appropriate definitions). Their result has led to a significant research effort into Z4 -linear codes; for various classifications and constructions, see, for example, [6, 9, 10, 14, 21, 20, 33, 38, 42]; for interesting applications to steganography, see [4, 29]; for connections to unimodular lattices, respectively to semifield planes, see [5], and [31, 35]; for a database of Z4 -linear codes, see [2], and references within. In their paper, Hammons et al. also gave an explanation to one of the outstanding problems in coding theory, that the weight enumerators of the non-linear Kerdock codes and the Preparata codes satisfy the MacWilliams identities. The first member of both of these families is the well known Nordstrom-Robinson code N , which is a non-linear (16, 256, 6) binary code with several interesting properties. It is optimal, in the sense that it is the largest possible binary code of length 16 with minimum distance 6 , and it is twice as large as any linear binary code with the same length and minimum distance. Moreover, Snover [40] proved that any binary (16, 256, 6) code is equivalent to the Nordstrom-Robinson code. Analogous properties also hold for the Punctured Nordstrom-Robinson code, a non-linear (15, 256, 5) code. In this paper, we prove that the Nordstrom-Robinson codes have other exceptional properties. First we prove that the codes are completely transitive, and hence completely regular (see Definition 2.1). Then we Date: draft typeset May 19, 2015 2000 Mathematics Subject Classification: 20B25, 94B05, 05C25. Key words and phrases: Nordstrom-Robinson codes, completely transitive codes, completely regular codes, automorphism groups, Preparata codes, Octacode. This research was supported by the Australian Research Council Federation Fellowship FF0776186 of the second author. 1 2 NEIL I. GILLESPIE AND CHERYL E. PRAEGER show that binary completely regular codes with the same length and minimum distance parameters are equivalent to the Nordstrom-Robinson codes. Theorem 1.1. Any binary completely regular code of length m with minimum distance δ is equivalent to the Nordstrom-Robinson code, respectively the Punctured Nordstrom-Robinson code, if (m, δ) = (16, 6) or (15, 5). Moreover, such a code is completely transitive. It is known that completely transitive codes are necessarily completely regular [26]. A consequence of Theorem 1.1 is that the converse holds for binary codes with (m, δ) = (16, 6) or (15, 5). This is similar to a result in [24] in which the authors proved that a binary completely regular code with (m, δ) = (12, 6) or (11, 5) is unique up to equivalence, and that such codes are completely transitive. We demonstrate that the converse does not hold for any other code in an infinite family containing these two codes. As mentioned above, the Nordstrom-Robinson code of length 16 is the first member of a family of completely regular codes called the Preparata codes (see [34, Section 7.4.3] for a nice definition of the Preparata codes). It turns out that no other Preparata code is completely transitive, and similarly, no other Punctured Preparata code apart from the Punctured Nordstrom-Robinson code is completely transitive. Theorem 1.2. The (Punctured) Nordstrom-Robinson code is the only member of the (Punctured) Preparata codes that is completely transitive. In particular, other than the (Punctured) NordstromRobinson code, the (Punctured) Preparata codes are completely regular but not completely transitive. As far as the authors are aware, these are the first examples of completely regular codes shown not to be completely transitive. In Section 2, we introduce the necessary definitions and preliminary results. Then in Section 3 we prove that the Nordstrom-Robinson code and the Punctured Nordstrom-Robinson code are completely transitive. We prove Theorem 1.1 in Section 4. In the final section we prove Theorem 1.2. We also give a discussion on Z4 -linear codes. In particular, we describe the Octacode, which is a Z4 -representation of the Nordstrom-Robinson code. It is natural to ask whether the complete transitivity of the NordstromRobinson code be determined from the Z4 -linear structure of the Octacode. To answer this, we introduce the notion of (X, Z4 )-completely transitive codes (see Definition 5.5), and prove a result which suggests that the binary representation is the correct setting to prove the complete transitivity of the NordstromRobinson code. 2. Definitions and Preliminaries Let Γ be an undirected connected graph with vertex set V (Γ), and let d(α, β) denote the length of the shortest path, or distance, in Γ between α, β ∈ V (Γ). Let Γs (α) be the set of vertices in V (Γ) that are at distance s from α . A distance regular graph is a regular graph with the property that for any two vertices α, β , the number |Γi (α) ∩ Γj (β)| depends only on i, j and k = d(α, β). As in [19], we define a code C in a distance regular graph Γ to be a subset of V (Γ). The minimum distance, δ , of C is the smallest distance between distinct codewords of C . For any vertex γ ∈ Γ, we NEW REGULARITY PROPERTIES OF THE NORDSTROM-ROBINSON CODES 3 define the distance of γ from C to be d(γ, C) = min{d(γ, β) | β ∈ C}, and the covering radius of C to be ρ = max d(γ, C). γ∈V (Γ) We let Ci denote the set of vertices that are distance i from C . It follows that {C = C0 , C1 , . . . , Cρ } forms a partition of V (Γ), called the distance partition of C . The distance distribution of C is the (m + 1)-tuple a(C) = (a0 , . . . , am ) where ai = |{(α, β) ∈ C 2 : d(α, β) = i}| . |C| We observe that ai > 0 for all i and a0 = 1 . Moreover, ai = 0 for 1 6 i 6 δ − 1 and |C| = Pm i=0 ai . The automorphism group of Γ, Aut(Γ), is the group of permutations of V (Γ) that preserve adjacency. The automorphism group of C , Aut(C), is the setwise stabiliser in Aut(Γ) of C . We say two codes C and C ′ in Γ are equivalent if there exists x ∈ Aut(Γ) such that C x = C ′ . Definition 2.1. Let Γ be a distance regular graph, C be code in Γ with distance partition {C, C1 , . . . , Cρ } , and γ ∈ Ci . We say C is completely regular if |Γk (γ) ∩ C| depends only on i and k , and not on the choice of γ ∈ Ci . If there exists X 6 Aut(Γ) such that each Ci is an X -orbit, then we say C is X -completely transitive, or simply completely transitive. The following result is found in [37]. Lemma 2.2. Let C be a completely regular code in a distance regular graph Γ with distance partition {C, C1 , . . . , Cρ } . Then Cρ is a completely regular code with distance partition {Cρ , Cρ−1 , . . . , C} . 2.1. The Hamming graph. The binary Hamming graph Γ = H(m, 2) has vertex set V (Γ) = Fm 2 , the set of m-tuples with entries from the field F2 = {0, 1} , and an edge exists between two vertices if and only if they differ in precisely one entry. The Hamming distance between α, β ∈ Fm 2 is the number of entries in which the two vertices differ. Let M = {1, . . . , m} , and view M as the set of vertex entries of Γ. For α ∈ Fm 2 , the support of α is the set supp(α) = {i ∈ M : αi 6= 0} , and the weight of α is wt(α) = | supp(α)|. The automorphism group Aut(Γ) of the binary Hamming graph is semi-direct product B ⋊ L where B ∼ = S2m and L ∼ = Sm , see [8, Theorem 9.2.1]. Let g = (g1 , . . . , gm ) ∈ B, σ ∈ L and α = (α1 , . . . , αm ) ∈ V (Γ). Then gσ acts on α in the following way: (1) g g αgσ = (α11σσ−1 , . . . , αmmσσ−1 ). −1 −1 Since the base group B ∼ = S2m of Aut(Γ) acts regularly on V (Γ), we may identify B with the group of translations of Fm 2 , and Aut(Γ) with a subgroup of the affine group AGL(m, 2). More precisely B consists of the translations gβ , where αgβ = α + β for α, β ∈ Fm 2 , and if 0 is the zero vector, then Aut(Γ) = B ⋊ Aut(Γ)0 where Aut(Γ)0 (the stabiliser of 0 in Aut(Γ)) is the group of permutation matrices in GL(m, 2). For a binary code, we let Perm(C) denote the group of permutation matrices that fix C setwise. 4 NEIL I. GILLESPIE AND CHERYL E. PRAEGER Remark 2.3. In traditional coding theory, only weight preserving automorphisms of a code are considered, and so in the binary case, Perm(C) is defined as the automorphism group of a code. Consequently, established results about automorphism groups of certain codes refer to Perm(C), not Aut(C). However, if 0 ∈ C we note that Aut(C)0 is equal to Perm(C). If a code C is a subspace of Fm 2 with dimension k , we say C is a linear [m, k, δ] code. If C is not a linear code we say C is a (m, |C|, δ) code, where |C| denotes the cardinality of C . In the Hamming graph, the MacWilliams transform of the distance distribution of C , a(C), is the (m + 1)tuple a′ (C) = (a′0 , . . . , a′m ) where a′k := (2) m X ai Kk (i) i=0 with Kk (x) := k X (−1)j j=0    x m−x . j k−j It follows from [34, Lemma 5.3.3] that a′k > 0 for k ∈ {0, 1, . . . , m} . A code C in Fm 2 is antipodal if α + 1 ∈ C for all α ∈ C , where 1 = (1, . . . , 1), otherwise we say C is non-antipodal. The following result is straight forward to prove. Lemma 2.4. Let C be an antipodal code in Γ with distance distribution a(C) = (a0 , . . . , am ). Then ai = am−i for 0 6 i 6 m. Let p ∈ M = {1, . . . m} , and C be a code in Fm 2 . By deleting the same coordinate p from each m−1 codeword of C , we generate a code in F2 , which we call the punctured code of C with respect to p. We can also describe this code in the following way. Let J = {i1 , . . . , ik } ⊆ M and define the following map |J| −→ F2 πJ : Fm 2 (α1 , . . . , αm ) 7−→ (αi1 , . . . , αik ) We define the projected code of C with respect to J to be the set πJ (C) = {πJ (α) : α ∈ C} . It follows that if J = M \{p} then πJ (C) is equal to the punctured code of C with respect to p. When we project we would like to have some group information available to us. We have an induced action of Aut(Γ)J = {gσ ∈ Aut(Γ) : J σ = J} as follows: for x ∈ Aut(Γ)J , we define (3) χ(x) : |J| |J| −→ F2 F2 πJ (α) 7−→ πJ (αx ), and observe that ker χ = {(g1 , . . . , gm )σ ∈ Aut(Γ)J : j σ = j and gj = 1 for j ∈ J}. Let D = (P, B) where P is a set of points of cardinality m, and B is a set of k subsets of P called blocks. Then D is a t − (m, k, λ) design if every t-subset of P is contained in exactly λ blocks of B . We let b denote the number of blocks in a design. If D is a t-design, then it is also an j − (m, k, λj ) design for 0 6 j 6 t − 1 [12, Corollary 1.6] where     k−j m−j =λ . (4) λj t−j t−j NEW REGULARITY PROPERTIES OF THE NORDSTROM-ROBINSON CODES 5 Using this fact we can deduce that (5)     m k λj = b . j j For further concepts and definitions about t-designs see [12]. There is an analogous concept of t-designs m for subsets of vertices in Fm 2 . Let α, β be two vertices in F2 . Then we say α is covered by β if for each non-zero component αi of α it holds that αi = βi . Definition 2.5. Let D be a set of vertices of weight k in Γ. Then we say D is a 2 -ary t- (m, k, λ) design if for every vertex ν of weight t, there exist exactly λ vertices of D that cover ν . This definition coincides with the usual definition of a t-design, in the sense that the set of blocks is the set of supports of vertices in D . Therefore, when we consider 2 -ary t-designs in Γ, we simply refer to them as t-designs. The following is a consequence of a result proved by Van Tilborg [43, Theorem 2.4.7]. For a code C and a positive integer k we denote by C(k) the set of weight k codewords of C . Theorem 2.6. Let C be a completely regular code in Γ that contains the zero vertex. Then for each k with δ 6 k 6 m and C(k) 6= ∅ , it holds that C(k) forms a 2 -ary t-design with t = ⌊ δ2 ⌋. 2.2. The Nordstrom-Robinson code N . We give the description of the Nordstrom-Robinson code N due to Goethals [27]. It is a non-linear code in F16 2 constructed from the extended binary Golay code G , which is a linear code in F24 (as defined, for example, in [12, p.131]). This requires concatenation 2 24 notation. For any ᾱ ∈ F2 , we can write ᾱ as the concatenation of a vertex in F82 followed by a vertex in F16 2 . That is ᾱ = (ᾱ1 , . . . , ᾱ24 ) = uα, where u = (ᾱ1 , . . . , ᾱ8 ) ∈ F82 , α = (ᾱ9 , . . . , ᾱ24 ) ∈ F16 2 . ᾱ + β̄ = uα + vβ = (u + v)(α + β). We note that for β̄ = vβ ∈ F24 2 , Let G be the [24, 12, 8] extended binary Golay code, chosen so that γ̄ = (18 , 016 ) ∈ G . It is known that Perm(G) ∼ = M24 [36, Ch. 20], and hence Aut(G) = TG ⋊ Perm(G) where TG is the group of translations generated by G [26]. Furthermore, by [16, p.96] H := Perm(G)γ̄ ∼ = AGL(4, 2) ∼ = 24 : A8 . Let J ∗ = {1, . . . , 8} and J = M \J ∗ . Then H has an induced action on J ∗ that is permutationally isomorphic to A8 , and also a faithful action on J . We define the following subcode of G : C = {ᾱ ∈ G : supp(ᾱ) ∩ J ∗ = ∅}. Clearly C is a linear subcode of G , and it follows that H 6 Perm(C). For 1 6 i 6 7 , let ᾱi be a codeword in G with supp(ᾱi ) ∩ J ∗ = {i, 8} (such codewords exist in G , see [36, p.73]), and let C i be the coset ᾱi + C . It follows that C i consists of all the codewords ᾱ ∈ G such that supp(ᾱ) ∩ J ∗ = {i, 8} . Let u0 be the zero vertex in F82 and for i = 1, . . . , 7 , let ui ∈ F82 such that supp(ui ) = {i, 8} . For i = 0, 1, . . . , 7 it holds that πJ ∗ (ᾱ) = ui for all ᾱ ∈ C i , where C 0 = C . Definition 2.7. Let A = ∪7i=0 C i , where C 0 = C . The Nordstrom-Robinson code N is defined to be the projection code of A onto J . That is, N = πJ (A). 6 NEIL I. GILLESPIE AND CHERYL E. PRAEGER Let R be the subcode of N equal to the projection code of C onto J , and for i = 1, . . . , 7 let Ri be S the projection code of C i onto J , so N = 7i=0 Ri where R = R0 . The code R is the Reed Muller code R(1, 4), which is a linear [16, 5, 8] code [36, p.74], and it follows, for each i = 1, . . . , 7 , that Ri is a coset of R . Berlekamp proved that N is a binary (16, 256, 6) code, and that Perm(N ) = 24 : A7 acting 3 transitively on 16 points [3]. Therefore, by our definition of the automorphism group of a code, Aut(N )0 = 24 : A7 , where 0 is the zero codeword in N . Furthermore, N is an even code, by which we mean that every codeword in N has even weight [36, p.74], and the covering radius of N is ρ = 4 [1, Cor. 5.2]. 3. Complete transitivity of the Nordstrom-Robinson codes Let Γ = H(16, 2), and recall that Aut(N ) is the stabiliser of N in Aut(Γ). The following homomorphism defines an action of Aut(N ) on M = {1, . . . , 16} . (6) µ : Aut(N ) −→ S16 gσ 7−→ σ Lemma 3.1. Let K be the kernel of the map µ in (6) and let R be the Reed Muller code contained in N . Then K = TR , the group of translations of F16 2 generated by R . Proof. Let g = (h1 , . . . , h16 ) ∈ K . Then there exists β ∈ F16 2 such that we can identify g with the translation gβ of F16 . Consequently, because g ∈ Aut(N ) and 0 ∈ N , it follows that β = 0 + β = 0gβ ∈ N . β 2 Suppose first that β ∈ R , and let α ∈ N . Then β̄ = u0 β ∈ C and there exists i ∈ {0, . . . , 7} such that ᾱ = ui α ∈ C i . As G is a linear code, we deduce that ν̄ := ui α + u0 β = (ui + u0 )(α + β) = ui (α + β) ∈ G . If i 6= 0 then supp(ν̄) ∩ J ∗ = {i, 8} , and if i = 0 then supp(ν̄) ∩ J ∗ = ∅ . Thus ν̄ = ui (α + β) ∈ C i , and so αgβ = α + β ∈ N . Consequently, TR 6 K . Now suppose that β ∈ N \ R. Then there exists j ∈ {1, . . . , 7} such that β̄ = uj β ∈ C j . Let α ∈ Ri where i ∈ / {0, j} , so ᾱ = ui α ∈ C i . Since gβ ∈ Aut(N ) it follows that α + β ∈ N , and so there exists k ∈ {0, . . . , 7} such that uk (α + β) ∈ C k . Furthermore, as G is linear, we have that ᾱ + β̄ = (ui + uj )(α + β) ∈ G . It follows from the definitions that ui + uj is a weight 2 vertex in F82 with non-zero entries in positions i and j . Consequently  2 if k ∈ {0, i, j} d(uk (α + β), (ui + uj )(α + β)) = 4 otherwise which contradicts the fact that G has minimum distance 8 . Hence K = TR .  Theorem 3.2. N is Aut(N )-completely transitive. Proof. We first prove that for each β ∈ N , there exists an x ∈ Aut(N ) such that β x = 0, and hence Aut(N ) acts transitively on N . Let β ∈ N . If β ∈ R then, by Lemma 3.1, gβ ∈ Aut(N ), and it follows NEW REGULARITY PROPERTIES OF THE NORDSTROM-ROBINSON CODES 7 that βgβ = β + β = 0. Now suppose that β ∈ N \ R. Then there exists a unique i ∈ {1, . . . , 7} such that β̄ = ui β ∈ C i = ᾱi + C . Let g be the translation of F24 2 generated by ᾱi , let σ ∈ H such that σ i = 8 , and let x = gσ ∈ Aut(G). We claim that χ(x) ∈ Aut(N ), where χ is as in (3). x Since σ ∈ Perm(C) it follows that (C i )x = (ᾱi + ᾱi + C)σ = C σ = C . In particular, β̄ ∈ C . Furthermore, C x = (ᾱi + C)σ = ᾱσi + C . Now, because supp(ᾱσi ) ∩ J ∗ = {iσ , 8σ } = {8, 8σ } and σ stabilises J ∗ , it follows that ᾱσi ∈ ᾱ8σ + C and so C x = ᾱ8σ + C = C k , where k = 8σ . Now, for j 6= i or 0 , consider C j = ᾱj + C . Then (ᾱj + C)x = (ᾱj + ᾱi + C)σ = (ᾱj + ᾱi )σ + C , because σ ∈ Perm(C). It follows that supp(ᾱj + ᾱi )∩J ∗ = {j, i} , and so supp((ᾱj + ᾱi )σ )∩J ∗ = {j σ , iσ } = {j σ , 8} . Consequently (ᾱj + ᾱi )σ ∈ ᾱj σ + C , and so (C j )x = ᾱj σ + C = C ℓ , where ℓ = j σ . Hence x fixes setwise A. Because N = πJ (A) we deduce that χ(x) ∈ Aut(N ). x x Since β̄ ∈ C , there exists η ∈ R such that πJ (β̄ ) = η . By Lemma 3.1, gη ∈ K , and so y = χ(x)gη ∈ Aut(N ), and we have, by (3), x β y = πJ (β̄)χ(x)gη = πJ (β̄ )gη = η gη = η + η = 0. Consequently, Aut(N ) acts transitively on N . Recall that N has covering radius ρ = 4 , and so {N , N 1 , N 2 , N 3 , N 4 } is the distance partition of N . Recall also that Aut(N )0 ∼ = 24 : A7 acting 3 -transitively on 16 points. Since N has minimum distance 6 , it follows that Γi (0) ⊆ N i for i = 1, 2, 3 . Therefore Aut(N )0 acts transitively on Γi (0) = Γi (0) ∩ N i , and so by [24, Lemma 2.2], Aut(N ) acts transitively on N i for i = 1, 2, 3 . Let ν ∈ Γ4 (0). Then ν is the neighbour of a weight 3 vertex, which, as stated above, is an element of N 3 . Therefore ν ∈ N 2 ∪ N 3 ∪ N 4 . Suppose ν ∈ N 3 . Then there exists α ∈ N such that d(ν, α) = 3 . As ν has weight 4 , this implies that α has odd weight, which contradicts the fact that every codeword of N has even weight. Thus ν ∈ N 2 ∪ N 4 . Let ν be an element of N 4 . Then there exists α ∈ N such that d(ν, α) = 4 . As Aut(N ) acts transitively on N , there exists x ∈ Aut(N ) such that αx = 0. In particular d(0, ν x ) = 4 , and because Aut(N ) preserves the distance partition of N , it follows that Γ4 (0) ∩ N 4 6= ∅ . Also, let β be any codeword of weight 6 , and let ν ∗ ∈ Γ4 (0) be such that supp(ν ∗ ) ⊆ supp(β). Then d(ν ∗ , β) = 2 , and so Γ4 (0) ∩ N 2 6= ∅ . Consequently, because Aut(N )0 fixes setwise Γ4 (0) and preserves the distance partition of N , Aut(N )0 has at least 2 orbits on Γ4 (0). Moreover, we see in [17, Table XI] that Perm(N ) = Aut(N )0 has exactly two orbits on Γ4 (0). Thus Aut(N )0 acts transitively on Γ4 (0) ∩ N 4 , and so, by [24, Lemma 2.2], Aut(N ) acts transitively on N 4 .  Corollary 3.3. Aut(N )/K ∼ = 24 : A8 . Proof. Since Aut(N )0 ∩ K = 1 , where K is the kernel of the map µ in (6), it follows Aut(N )0 ∼ = 24 : A7 , and so µ(Aut(N )) is a 3 -transitive = µ(Aut(N )0 ). Recall that Aut(N )0 ∼ group of S16 containing 24 : A7 . By the classification of finite 2 -transitive groups [11], it follows µ(Aut(N )) ∼ = 24 : A7 , 24 : A8 , A16 or S16 . Because Aut(N ) acts transitively on N , it holds | Aut(N )| = 28 × | Aut(N )0 |. Moreover, by Lemma 3.1, |K| = |TR | = 25 , so | Aut(N )/K| = |24 : Consequently, the only possibility is that µ(Aut(N )) ∼ = 24 : A8 . that subthat that A8 |.  8 NEIL I. GILLESPIE AND CHERYL E. PRAEGER 3.1. The Punctured Nordstrom-Robinson Code PN . Let Γ = H(15, 2), p ∈ M = {1, . . . 16} and J = M \{p} . Recall that the punctured Nordstrom-Robinson code with respect to p is the code generated by deleting the pth entry of each codeword of the Nordstrom-Robinson code N . It is also equal to the projected code πJ (N ). By Theorem 3.2, N is completely transitive, and so it is completely regular. Consequently, as N is an even code with covering radius equal to 4 , a result by Brouwer [7] implies that πJ (N ) is completely regular with covering radius ρ = 3 . Therefore, as 0 ∈ πJ (N ), it follows that the minimum distance δ of πJ (N ) is equal to the minimum non-zero weight of codewords in πJ (N ), which is equal to either 5 or 6 as the minimum non-zero weight of codewords in N is 6 . By [36, p.74], | N (6)| = 112 , where N (6) is the set of codewords in N of weight 6 , and so Theorem 2.6 implies that N (6) forms a 3 − (16, 6, 4) design. Hence the number of codewords of weight 6 whose support contains p is 42 , and these correspond to 42 codewords in πJ (N ) of weight 5 . Thus πJ (N ) has δ = 5 . Consequently, πJ (N ) is a (15, 256, 5) code. Snover [40] proved that a (15, 256, 5) binary code is unique up to equivalence. Thus, for any p′ ∈ M , the punctured N code with respect to p′ is equivalent to πJ (N ). Therefore, without loss of generality, we can assume that p = 1 as in [3] (so J = M \{1} ), and we denote πJ (N ) by PN . By [3, Lemma 6.5], Aut(PN )0 ∼ = A7 acting 2 -transitively on 15 points. The action of Aut(PN )0 ∼ = A7 on Γ3 (0) is equivalent to its action on the 3 -element subsets of J . The permutation characters for actions of A7 on J , and on the 3 -element subsets of J , have inner product equal to 2 , see [16, p.10]. Hence A7 has exactly two orbits on 3 -element subsets of J , so Aut(PN )0 has exactly two orbits on Γ3 (0). Theorem 3.4. PN is Aut(PN )-completely transitive. Proof. Let J be as above, and recall the homomorphism χ from (3) with kernel equal to ker χ = h(g1 , . . . , g16 )i, where g1 = (0 1) and gi = 1 for i 6= 1 . Also, we note that Aut(N )J is equal to Aut(N ){1} , because J and {1} are disjoint sets. Now, since Aut(N ) ∩ B = K = TR (Lemma 3.1), it follows that Aut(N ) ∩ ker χ = 1 . Hence χ(Aut(N ){1} ) ∼ = Aut(N ){1} , and it is straight forward to show that χ(Aut(N ){1} ) 6 Aut(PN ). Also K 6 Aut(N ){1} , and Aut(N )/K ∼ = 24 : A8 , by Corollary 3.3. Thus Aut(N ){1} /K ∼ = A7 , it follows from the orbit stabiliser theorem that = A8 . As Aut(PN )0 ∼ | Aut(PN )| 6 | PN || Aut(PN )0 | = |K||A8 | = | Aut(N ){1} |. Hence, we deduce that Aut(N ){1} ∼ = Aut(PN ) and Aut(PN ) acts transitively on PN . Recall that PN has covering radius ρ = 3 , and so has distance partition {PN , PN 1 , PN 2 , PN 3 } . Since δ = 5 we have that Γi (0) ⊆ PN i for i = 1, 2 . Furthermore, because Aut(PN )0 ∼ = A7 acts 2 transitively on entries, it follows that Aut(PN )0 acts transitively on Γi (α) = Γi (α) ∩ PN i for i = 1, 2 . Thus, by [24, Lemma 2.2], Aut(PN ) acts transitively on PN i for i = 1, 2 . Now, let ν ∈ Γ3 (0). As δ = 5 , it follows that ν ∈ PN 2 ∪ PN 3 . By following a similar argument to that used in the proof of Theorem 3.2, we deduce that Γ3 (0) ∩ PN i 6= ∅ for i = 2, 3 . Thus because Aut(PN )0 fixes setwise Γ3 (0) and preserves the distance partition of PN , it follows that Aut(PN )0 has at least two orbits on Γ3 (0). As we saw above, Aut(PN )0 has exactly two orbits on Γ3 (0). Consequently Aut(PN )0 acts transitively on Γ3 (0) ∩ PN 3 . Hence, by [24, Lemma 2.2], Aut(PN ) acts transitively on PN 3 .  NEW REGULARITY PROPERTIES OF THE NORDSTROM-ROBINSON CODES 9 4. Proof of Theorem 1.1 Let Γ = H(m, 2) and C be a completely regular code in Γ with minimum distance δ for (m, δ) = (16, 6) or (15, 5). Complete regularity and minimum distance are preserved by equivalence, therefore, by replacing C with an equivalent code if necessary, we can assume that 0 ∈ C . Since C contains 0 and is completely regular, it follows that C(δ) 6= ∅ , where C(δ) is the set of codewords of weight δ . Hence, by Theorem 2.6, C(δ) forms a t − (m, δ, λ) design for t = ⌊ 2δ ⌋ and some positive integer λ. Using (4) with j = 1 in the case (16, 6) and (5) with j = 2 in the case (15, 5), we deduce that 2 divides λ. Let S be the set of α ∈ C(δ) such that {1, . . . , t} ⊂ supp(α). It follows that |S| = λ, and as C has minimum distance δ , we deduce that supp(α) ∩ supp(β) = {1, . . . , t} for all distinct pairs of codewords α, β ∈ S . Consequently, a simple counting argument gives that λ6 m−t . δ−t In both cases we deduce that λ < 5 , so λ = 2 or 4 . However, by Line 21 of [15, Table 3.37] and Line 16 of [15, Table 1.28], it follows that a t − (m, δ, λ) design does not exist in both cases for λ = 2 . Thus λ = 4. Case (m, δ) = (16, 6): In this case C(6) forms a 3 − (16, 6, 4) design, which is therefore also a j − (16, 6, λj ) design for j 6 3 , and in particular, λ = λ3 = 4 , λ2 = 14 , λ1 = 42 and λ0 = 112 . Let β ∈ C(6) and define ni = |{γ ∈ C(6) : supp(γ) ∩ supp(β) = i}|. By applying a result of Cameron and Soicher [13], the following relationship holds for j = 0, 1, 2, 3 : (7)   6   X i 6 ni . λj = j j i=0 Because C(6) is necessarily a simple design, it follows that n6 = 1 , and since δ = 6 we deduce that n4 = n5 = 0 . Putting these values into (7) gives four linear equations which solve to give n0 = 0 , n1 = 36 , n2 = 15 and n3 = 60 . Because n1 6= 0 , it follows that Γ10 (β) ∩ C 6= ∅ , and therefore, because C is completely regular, C(10) 6= ∅ . We now claim that 1 ∈ C , and consequently, that C is antipodal. Suppose to the contrary. Then 1 +Cρ = C and ρ > δ − 1 = 5 , where ρ is the covering radius of C [22, Lemma 2.2]. Moreover, because C(10) 6= ∅ , it holds that ρ 6 6 , and because δ = 6 , it follows from Lemma 2.2 that 1 +Cρ−i = Ci for i = 1, 2 . We now calculate the size of the set C3 in the distance partition of C . To do this, we count the pairs {(ν, γ) ∈ C3 × C : d(ν, β) = 3} . Counting this set in two ways gives   6 |C3 ||Γ3 (ν) ∩ C| = |C| , 3 where ν is any vertex in C3 . Fix ν ∈ Γ3 (0). It follows that that if γ ∈ Γ3 (ν) ∩ C then either γ = 0 or γ ∈ C(6), and if the later holds then γ covers ν . Therefore, because C(6) forms a 3 − (16, 6, 4) design, we deduce that |Γ3 (ν) ∩ C| = 5 , and so |C3 | = |C| × 112 . Now suppose that ρ = 5 . Then, by Lemma
2.2, |C2 | = |C3 |. However, |C2 | = |C| 16 2
which is a contradiction. Thus ρ = 6 . However, then Lemma 16 2.2 implies that |C|(2 + 2 × 16 + 2 × 16 2 + 112) = 2 , which is a contradiction. Hence 1 ∈ C and C is antipodal. Therefore, if a(C) = (a0 , . . . , am ), we deduce, by Lemma 2.4, that ai = am−i for all i . As 10 NEIL I. GILLESPIE AND CHERYL E. PRAEGER |C(6)| = 112 , it follows that a(C) = (1, 0, 0, 0, 0, 0, 112, a7, a8 , a7 , 112, 0, 0, 0, 0, 0, 1). We conclude from (2) and [34, Lemma 5.3.3] that the following constraints must hold: 240 − 12a7 − 8a8 > 0 −840 − 28a7 + 28a8 > 0 with a7 > 0 and a8 > 0 . Solving these constraints gives that a7 = 0 and a8 = 30 . Consequently C is a (16, 256, 6) binary code, and so, by Snover’s result [40], C is equivalent to the Nordstrom-Robinson code, proving the first part of Theorem 1.1. Case (m, δ) = (15, 5): Here C(5) forms a j − (15, 5, λj ) design for j 6 2 with λ = λ2 = 4 , λ1 = 14 , and λ0 = 42 . As above let β ∈ C(5) and define ni = |{γ ∈ C(5) : supp(γ) ∩ supp(β) = i}|. Using an equivalent expression to (7), we deduce that n0 = 6 , and therefore that a10 6= 0 in the distance distribution of C . Now, by following a similar argument to the one we used in the previous case, we deduce that C is in fact antipodal, and so a(C) = (1, 0, 0, 0, 0, 42, a6, a7 , a7 , a6 , 42, 0, 0, 0, 0, 1) Again, using the MacWilliams transform, we generate certain inequalities that solve to give a6 = 70 and a7 = 15 , and so C is a (15, 256, 5) binary code. Thus, by Snover’s result [40], C is equivalent to the punctured Nordstrom-Robinson code, proving the second part of Theorem 1.1. By [25, Lemma 2], complete transitivity is preserved by equivalence, and by Theorem 3.2 and Theorem 3.4, the Nordstrom-Robinson codes are completely transitive. Consequently, in both cases, C is completely transitive, proving the final statement of Theorem 1.1. 5. Nordstrom-Robinson Code as a Z4 -linear code Before we discuss the Nordstrom-Robinson code as a Z4 -linear code, we prove another uniqueness property that it, and the Punctured Nordstrom-Robinson code, possess. The Nordstrom-Robinson code N is the first member the Preparata codes [39], an infinite family of non-linear binary codes. For each odd k > 3 , the Preparata code P(k) has length 2k+1 , contains 2k+1 − 2(k + 1) codewords and has minimum distance 6 (see for example, [34, Section 7.4.3]). The code P(3) is equal to the Nordstrom-Robinson code N of length 16 . Theorem 1.2 follows from the following result. Proposition 5.1. The Preparata code P(k) and the Punctured Preparata Code P(k)∗ are completely transitive if and only if k = 3 . In particular, for k > 3 odd, P(k) and P(k)∗ are completely regular but not completely transitive. Proof. To prove that P(k) and P(k)∗ are completely transitive if and only if k = 3 , we examine Perm(P(k)) and Perm(P(k)∗ ), the group of permutation matrices that fix the respective code setwise. In [30, Theorems 3 and 4], Kantor showed that, for odd k > 3 , Perm(P(k)) acts imprimitively on entries and Perm(P(k)∗ ) has order (2k − 1)k . However, it is known that for any binary completely transitive code C of length n with minimum distance at least 5 , the group Perm(C) acts 2 -homogeneously on NEW REGULARITY PROPERTIES OF THE NORDSTROM-ROBINSON CODES 11
n entries [23, Prop. 2.5]. Therefore Perm(C) acts primitively on entries and 2 divides | Perm(C)|. By combining this result with Kantor’s results, and recalling that n = 2k+1 , we deduce that P(k) and P(k)∗ are not completely transitive for k > 3 . The forwards implication of the statement is a consequence of Theorem 1.1. It is well known that P(k) is completely regular for all odd k > 3 (see, for example, [41, Ex. 6.3]). Moreover, by construction, P(k) consists entirely of even weight codewords. Therefore a result of Brouwer [7] implies that the Punctured Preparata code P(k)∗ is completely regular for all odd k > 3 , which proves the final statement.  Remark 5.2. As in [23], a code C in a graph Γ is 2 -neighbour transitive if there exists an automorphism group X 6 Aut(Γ) such that Ci is an X -orbit for i = 0, 1, 2 . Completely transitive codes with covering radius ρ > 2 are necessarily 2 -neighbour transitive. We observe that the proof Proposition 5.1 also implies that the (Punctured) Preparata codes, other than the (Punctured) Nordstrom-Robinson code, are not 2 -neighbour transitive. The Nordstrom-Robinson code is also the first member of another infinite family of non-linear binary codes, the Kerdock codes [32]. For each odd k > 3 , the Kerdock code K(k) is a code of length 2k+1 , with K(3) equal to the Nordstrom-Robinson code N . The codes K(k) and P(k) are distance invariant, and interestingly they are formally dual, by which we mean the distance distribution of one can be obtained by taking the MacWilliams transform of the distance distribution of the other. In particular, the Nordstrom-Robinson code is formally self dual. However, as these codes are non-linear, neither is the dual code of the other. It was not until work by Hammons et al. [28] on linear codes over Z4 that an explanation for this phenomenon was discovered. To describe Hammons et al. work, we first define the Lee metric. We arrange the elements 0, 1, 2, 3 of Z4 in order around a circle, and define dL (a, b), for a, b ∈ Z4 , to be the number of steps apart they are. In particular, dL (a, b) = 2 if and only if {a, b} = {0, 2} or {1, 3} , otherwise dL (a, b) = 1 . We extend this definition to m-tuples of Z4 , that is, the Lee distance between α, β ∈ Zm 4 is dL (α, β) = m X dL (αi , βi ). i=1 We define the Grey map to be the bijection f : Z4 7−→ F22 given by (8) f (0) = 00, f (1) = 01, f (2) = 11, f (3) = 10, 2m and we extend this map to a bijection from Zm by 4 to F2 φ((α1 , . . . , αm )) = (f (α1 ), . . . , f (αm )). 2m The map φ is an isometry from Zm 4 , with the Lee metric, to F2 , with the Hamming metric [28, Thm. 1]. m A linear code C over Z4 of length m is an additive subgroup of Zm 4 . An inner product on Z4 is defined to be α · β = α1 β1 + . . . + αm βm mod 4 from which the usual notion of a dual code C ⊥ can be defined. Hammons et al. proved that the Kerdock codes and the Preparata codes length 2k+1 are the k image under φ of certain linear codes CK and CP in Zm 4 , where m = 2 . Moreover, these codes are 12 NEIL I. GILLESPIE AND CHERYL E. PRAEGER ⊥ dual codes of each other in Zm 4 , that is CK = CP . This explains why the distance distributions of the non-linear binary Kerdock and Preparata codes are related as though they are dual codes. Remark 5.3. It is more accurate to say that φ(CP ) is of ‘Preparata’ type, in that it is not equal to Pn , as Preparata originally defined it, but it does have the same distance distribution as Pn . Hammons et al. suggest that the definition they give is the correct definition for the Preparata codes. Let Γ be the graph with V (Γ) = Zm 4 and adjacency given by the Lee metric, that is, α, β ∈ V (Γ) are adjacent if and only if dL (α, β) = 1 . Since φ is a bijective isometry from Γ to H(m, 2), it follows that φ induces a bijection from the edge set of Γ to the edge set of H(m, 2). In particular, φ is a graph isomorphism. Therefore, Γ and H(2m, 2) have isomorphic automorphism groups, namely Aut(Γ) ∼ = S2 wr S2m . Moreover, a code C is completely transitive in Γ if and only if it is completely transitive in H(2m, 2). It is natural to ask if one can prove that a code in Γ is completely transitive without appealing to its binary representation. Our interpretation of this question is that the symmetries involved in the proof m should preserve the “ Zm 4 structure” of Γ. The largest subgroup of Aut(Γ) which preserves the Z4 structure is determined in the following lemma. Lemma 5.4. Let Γ be defined as above. Then the subgroup G of Aut(Γ) that preserves the Zm 4 structure in Γ is isomorphic to D8 wr Sm . Proof. Any automorphism of Γ that preserves Zm 4 structure must preserve the partition {{1, 2}, {3, 4}, . . . , {2m − 1, 2m}} in its action on the vertex entries of H(2m, 2). The largest subgroup of Aut(H(2m, 2)) that preserves this partition is S2 wr (S2 wr S8 ). Writing this as a subgroup of the wreath product acting on Zm 4 , this is equal to (S2 wr S2 ) wr S8 . Now S2 wr S2 = D8 . Therefore the group G of automorphisms of Γ that preserve the Zm 4 structure is a subgroup of D8 wr Sm . Now let H be the group generated by the permutations (0, 1, 2, 3) and (0, 2) of Z4 , so H ∼ = D8 . The m m group H wr Sm = H ⋊ Sm acts on the vertices of Z4 in its product action (similar to the action of S2 wr S2m on the vertices of the Hamming graph H(2m, 2) given in (1)). It is clear that Sm preserves adjacency in Γ. Moreover, by placing the elements of Z4 on the corners of a square, one deduces that H preserves the Lee metric on Z4 , and so H m preserves adjacency in Γ. Thus H wr S8 6 G.  This result naturally leads to the following definition. Definition 5.5. A code C ⊆ Zm 4 with distance partition {C, C1 , C2 . . . , Cρ } is (X, Z4 )-completely transitive if there exists X 6 D8 wr Sm such that each Ci is an X -orbit. Our view of symmetry of a Z4 -code C allows all symmetries of C in Aut(Γ) = S2 wr S2m when it is viewed as a binary code in the Hamming graph H(2m, 2). Namely we consider the full symmetry group to be the setwise stabiliser of C in S2 wr S2m . Since D8 wr Sm < S2 wr S2m , this group may be the same as the stabiliser of C in D8 wr Sm , or it may be larger. If it is larger then there is the potential for the larger group to act completely transitively while the group preserving the Z4 -structure does not. We NEW REGULARITY PROPERTIES OF THE NORDSTROM-ROBINSON CODES 13 show below, by considering the Nordstrom-Robinson code and its Z4 -representation the Octacode, that this indeed can happen. That is to say, the stabilizer of the Octacode in D8 wr Sm is properly contained in its stabilizer in Aut(Γ) and does not act completely transitively on the code. A generator matrix for the Octacode O is   1 3 1 2 1 0 0 0  1 0 3 1 2 1 0 0       1 0 0 3 1 2 1 0  1 0 0 0 3 1 2 1 by which we mean that O is equal to the set of all Z4 combinations of the rows of the matrix. It was already known prior to the paper by Hammons et al. that the Nordstrom-Robinson code N is the image under φ of O [18]. Proposition 5.6. The Octacode is completely transitive, but not (X, Z4 )-completely transitive. Proof. The Nordstrom-Robinson code N is a completely transitive code in H(16, 2) by Theorem 1.1. Therefore, because φ is a bijective isometry from Γ to H(16, 2), it follows that the Octacode O is a completely transitive in Γ. Now, because N has minimum distance δ = 6 and covering radius ρ = 4 , it follows that O has the same parameters. Let {O, O1 , O2 , O3 , O4 } be the distance partition of O . We observe that ν1 = (2, 0, 0, 0, 0, 0, 0, 0), ν2 = (1, 1, 0, 0, 0, 0, 0, 0) ∈ Γ2 (0), so ν1 , ν2 ∈ O2 because δ = 6 . Suppose that O is (X, Z4 )-completely transitive. Then, by Lemma 5.4, there exists x = (h1 , . . . , h8 )σ ∈ D8 wr S8 such that ν1x = ν2 . Again, because δ = 6 , one easily deduces that 0x = 0, that is, x stabilises 0. In particular, 0hi = 0 for i = 1, . . . , 8 . However, if ν1x = ν2 , then there exists i ∈ {2, . . . , 7} such that 0hi = 1 , which is a contradiction. Hence O is not (X, Z4 )-completely transitive.  Proposition 5.6 suggests that to prove the complete transitivity of the Nordstrom-Robinson code (and thus the Octacode), one should consider its binary representation. References [1] Aoki, T., Gaborit, P., Harada, M., Ozeki, M., Solé, P.: On the covering radius of Z4 -codes and their lattices. IEEE Trans. Inform. 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With a Dutch summary, Doctoral dissertation, University of Technology Eindhoven [Gillespie] Heilbronn Institute for Mathematical Research, School of Mathematics, Howard House, The University of Bristol, Bristol, BS8 1SN, United Kingdom. E-mail address: neil.gillespie@bristol.ac.uk [Praeger] Centre for the Mathematics of Symmetry and Computation School of Mathematics and Statistics The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia, 6009. Also affiliated with King Abdulaziz University, Jeddah, Saudi Arabia. E-mail address: cheryl.praeger@uwa.edu.au