Advances in Mathematics of Communications doi:10.3934/amc.2011.5.275 Volume 5, No. 2, 2011, 275–286

A Z4-LINEAR CODE OF HIGH MINIMUM LEE DERIVED FROM A HYPEROVAL

Michael Kiermaier and Johannes Zwanzger Institut f¨urMathematik Universit¨atBayreuth 95440 Bayreuth, Germany

(Communicated by Patrick Sol´e)

Abstract. In this paper we present a new non-free Z4-linear code of length 29 and size 128 whose minimum Lee distance is 28. Its Gray image is a nonlinear binary code with parameters (58, 27, 28), having twice as many codewords as the biggest linear binary codes of equal length and minimum distance. The code also improves the known lower bound on the maximal size of binary block codes of length 58 and minimum distance 28. Originally the code was found by a heuristic computer search. We give a geometric construction based on a hyperoval in the projective Hjelmslev plane over Z4 which allows an easy computation of the symmetrized weight enumerator and the automorphism group. Furthermore, a generalization of this construction to all Galois rings of characteristic 4 is discussed.

1. Introduction A bit more than 40 years ago in papers of Nordstrom, Robinson, Preparata and Kerdock [16, 17, 12], some nonlinear binary codes with better minimum distance than any linear code of equal size and length were constructed. Later it was discov- ered [14,5] that they can be considered as images of linear codes over Z4 under the so-called Gray map. This led to an increased interest and a lot of investigation in linear codes over Z4, also a theory for linear codes over more general rings like finite chain rings and Frobenius rings was developed. However, apart from the Preparata series, not many examples are known where the Gray image of a Z4-linear code provably outperforms all linear binary codes of the same length and size. ˆ Recently, a new example of such a Z4-linear code C was found by a heuristic ap- proach similar to [22]. A subsequent analysis of the code revealed a clear geometric structure. The Lee weight enumerator of Cˆ is 28 32 36 44 LeeCˆ = 1 + 105X + 7X + 14X + X . So the Gray image of Cˆ has the parameters (58, 27, 28), while the optimal minimum distance for a linear binary [58, 7]-code is 27 [19]. Furthermore, Cˆ surpasses the hitherto largest known nonlinear binary block code of length 58 and minimum distance 28 [20, 13] by 4 codewords.

2000 Mathematics Subject Classification: Primary: 94B05; Secondary: 51C05, 51E21. Key words and phrases: , Gray image, minimum distance, Hjelmslev geometry, hyperoval. The first author is supported by DFG grant WA 1666/4-1.

275 c 2011 AIMS-SDU 276 Michael Kiermaier and Johannes Zwanzger

This paper is organized as follows: In Section2, we give a brief introduction to the theory of Z4-linear codes. A more detailed overview can be found in [11, Chapter 12]. In Section3, the required basics about the projective Hjelmslev plane over Z4 are presented. For a deeper discussion of projective Hjelmslev geometries and the connection to linear codes over finite chain rings, the reader is referred to [7,9]. In Section4, a geometric construction of Cˆ based on a hyperoval in the projective Hjelmslev plane over Z4 is given, which allows an easy computation of the symmetrized weight enumerator and the automorphism group of Cˆ. The question for a generalization of the construction is discussed in Section5.

2. Z4-linear codes n A submodule C of the Z4-module Z4 is called Z4-linear code of length n. There is a unique pair of non-negative integers (r1, r2) such that

∼ r1 r2 C = Z4 ⊕ 2Z4

r1 r2 as Z4-modules. The partition 2 1 ` 2r1 + r2 is called the shape of C. Let ∗ ∗ Z4 = {1, 3} be the unit group of Z4, Rad(Z4) = Z4 \ Z4 = {0, 2} the radical of Z4 and S = {0, 1} ⊂ Z4, which is a set of representatives of the factor ring Z4/Rad(Z4). The code C is free as a Z4-module if and only if r2 = 0. In the case r1 = 0, C n n is contained in 2Z4 , which is isomorphic to F2 as F2-modules. This shows that Z4-linear codes with r1 = 0 are just binary linear codes, and for a proper Z4-linear code we may assume r1 > 0. (r1+r2)×n A matrix Γ ∈ Z4 is called a generator matrix of C if its rows generate C as Z4-module, and exactly r1 rows contain at least one unit, while in the remaining r2 rows all entries are from Rad(Z4). Without loss of generality we may assume that this will be the case for the last r2 rows of Γ. Then

t r1 r2 C = {v Γ: v ∈ Z4 × S }

r1 r2 and for every c ∈ C there is a unique information vector v ∈ Z4 × S such that c = vtΓ.

2.1. Weight and . The Lee weight  0 7→ 0  wLee : Z4 → R, 1, 3 7→ 1 2 7→ 2 n gives rise to a weight function on Z4 by summing up the weights of all the compo- nents; as there is no danger of confusion, it will also be denoted by wLee. It is easy to check that the function n n 0 0 dLee : Z4 × Z4 → R, (c, c ) 7→ wLee(c − c ) n 0 is a metric on Z4 , the Lee metric. dLee(c, c ) is also called the Lee distance between 0 0 0 c and c and the value dmin(C) := min{dLee(c, c ): c 6= c ∈ C} the minimum Lee distance of C. The Lee weight enumerator of C is given by

X wLee(c) LeeC = X ∈ Z[X]. c∈C Like in classical coding theory over finite fields, one of the main goals is to find a Z4-linear code for given parameters n, r1 and r2 whose minimum Lee distance is as

Advances in Mathematics of Communications Volume 5, No. 2 (2011), 275–286 A Z4-linear code of high minimum Lee distance 277 large as possible. As a special case of linear codes over finite chain rings, a table of Z4-linear codes of high minimum Lee distance can be found in [21]. Furthermore, for a codeword c = (c0, c1, . . . , cn−1) we define the numbers a0 = ∗ |{i : ci = 0}|, a1 = |{i : ci = 2}| and a2 = |{i : ci ∈ Z4}|. The monomial a0 a1 a2 wsym(c) = X0 X1 X2 ∈ Z[X0,X1,X2] will be called the symmetrized weight of c. Furthermore, the symmetrized weight P enumerator of C is given by symC = c∈C wsym(c). The symmetrized weight carries more information than the Lee weight: Given the symmetrized weight s of a codeword c, the Lee weight of c can be computed by evaluating s at X0 = 0, X1 = 2 and X2 = 1. Similarly, the Lee weight enumerator can be derived from the symmetrized weight enumerator. We want to mention that the indexing of the polynomial variables of the sym- metrized weight differs from [11] as well as from [7], where yet another indexing is used. The reason for this is that on the one hand, we feel the index order should re- spect the linear order on the ideals of Z4, while the zero element should be counted by X0. 2.2. Transformation into a binary code. Obviously the function  0 7→ 00  2 1 7→ 01 γ : Z4 → F2, 2 7→ 11  3 7→ 10 2 is an isometry between the metric spaces (Z4, dLee) and (F2, dHam), where dHam is 2 the Hamming metric on F2. γ is called the Gray map on Z4. Like above, it is canonically extendible to n 2n γ : Z4 → F2 . n Hence, every block code C ⊂ Z4 can be transformed via γ into a binary code γ(C) of the same size and weight distribution and of twice the length. However, Z4-linearity of C usually does not result in F2-linearity of γ(C). n 2.3. Automorphisms. The linear isometries of the (Z4 , dLee) are ex- n n n actly the monomial transformations Z4 → Z4 . Thus each linear isometry ι : Z4 → n n Z4 can be written as (σ, v) ∈ Sn × {±1} , where σ denotes the permutation of the coordinates and for all i ∈ {0, . . . , n − 1}, the ith component of all codewords is multiplied by vi. σ will be called the permutation part of ι, and for a subgroup G of the monomial group, the permutation group consisting of all permutation parts of the elements in G will be called permutation part of G, accordingly. Given a Z4- linear code C of length n, the stabilizer group of C in the group of linear isometries is called the automorphism group Aut(C) of C. The group of all linear isometries acts on the set of all Z4-linear codes of length n. Two Z4-linear codes of length n are called isomorphic if and only if they are contained in the same orbit. The kernel of this group action is cyclic of order 2 and generated by the linear isometry (id, (−1,..., −1)t). To get a faithful group action, we look at the factor group I of the group of linear isometries modulo this kernel. The notion of the permutation part will be carried forward to elements and subgroups of I, since it does not depend on the choice of the representatives. Given a Z4-linear code C of length n, we call the stabilizer group of C with respect to the group action of I the projective automorphism group of C and denote it by PAut(C).

Advances in Mathematics of Communications Volume 5, No. 2 (2011), 275–286 278 Michael Kiermaier and Johannes Zwanzger

Either PAut(C) is isomorphic to a subgroup of Aut(C) in which case Aut(C) =∼ PAut(C) × Z2, or Aut(C) is isomorphic to a non-splitting central extension of Z2 by PAut(C). Thus, the essential information about the automorphism group is contained in PAut(C).

3. The projective Hjelmslev plane PHG(2, Z4) 3 Let P and L be the set of all free submodules of Z4 of rank one respectively two. It holds |P| = |L| = 28. The geometry consisting of the point set P, the line set L and the incidence relation given by the subset relation is called projective Hjelmslev plane over Z4 and will be denoted by PHG(2, Z4). Each line contains exactly 6 points, and a point is incident with exactly 6 lines. A line l will be identified with the set of 6 points incident with l. Unlike in classical projective planes, in PHG(2, Z4) it may happen that two distinct lines meet in more than one point: Two points p1 and p2 are called neighbors if there are two distinct lines incident with p1 and p2. The neighbor relation is an equivalence relation on P, and the equivalence classes are called neighbor classes. The set of the neighbor classes will be denoted by N . It holds |N | = 7, and each neighbor class contains exactly 4 points. A line passes through 3 neighbor classes, intersecting each of these neighbor classes in 2 points. Furthermore, a point p and a line l are called neighbors if l contains a neighbor of p. For each line l there are 12 points which are neighbors of l: 6 of them being incident with l and further 6 points not being incident with l, but contained in the 3 neighbor classes met by l. The set of the 12 points which are neighbors of a given line l will be denoted by [l]. A permutation of P mapping lines to lines is called collineation. The group of all collineations of PHG(2, Z4) is given by PGL(3, Z4) acting on P in the natural way. It acts transitively on all ordered quadruples of points in general position1. For a point set U ⊆ P the stabilizer group of U in PGL(3, Z4) is called automorphism group Aut(U) of U. Furthermore, two point sets are called isomorphic if they are contained in the same orbit of PGL(3, Z4) acting on the set of all subsets of P.

3 3.1. Coordinate vectors. For a point p ∈ P, a vector v ∈ Z4 with p = Z4v 3 is called coordinate vector of p. A vector v ∈ Z4 is a coordinate vector of some point if and only if v contains at least one unit entry. Each point has exactly two coordinate vectors which arise from each other by a multiplication by the scalar −1. By κ : P → Z4 we will denote a mapping that assigns each point p one of its coordinate vectors. Example 3.1. For a point p, κ(p) may be taken as the unique coordinate vector of p whose first unit entry equals 1.

t t 3 For vectors u = (u1, u2, u3) , v = (v1, v2, v3) ∈ Z4, we define their inner product as

hu, vi := u1v1 + u2v2 + u3v3. 3 The orthogonal module of a submodule S of Z4 is defined as ⊥ 3 S := {u ∈ Z4 : hu, vi = 0 for all v ∈ S}.

1No three points are neighbors of the same line.

Advances in Mathematics of Communications Volume 5, No. 2 (2011), 275–286 A Z4-linear code of high minimum Lee distance 279

In particular, the orthogonal of a point is a line and vice versa: L = {p⊥ : p ∈ P} and P = {l⊥ : l ∈ L}. In this way, for a line l a coordinate vector v of the point l⊥ is called coordinate vector of l. For a coordinate vector v of l it holds l = v⊥. Incidence and neighbor relations can be easily formulated using coordinate vec- ⊥ tors: Let p1 = Z4v1 and p2 = Z4v2 be two points and l = u be a line. p1 and p2 are neighbors if and only if v1 − v2 ∈ {0, 2}. p1 is incident with l if and only if hu, v1i = 0, and p1 is a neighbor of l if and only if hu, v1i ∈ {0, 2}.

3.2. The Galois ring GR(64, 4). Let f ∈ Z4[X] be a monic polynomial of degree 3, such that its image modulo 2 is irreducible in F2[X]. The ring Z4[X]/(f) is called the Galois ring of order 64 and characteristic 4 and is denoted by GR(64, 4). It can be shown that up to ring isomorphism, the definition does not depend on the exact choice of f. Each element of GR(64, 4) has a unique representative of degree at most 2. By the coefficients of this representative, each element of GR(64, 4) will be identified 3 3 ∼ with a vector in Z4. This identification gives rise to the isomorphism Z4 = GR(64, 4) as Z4-modules, providing an additional multiplicative structure on the coordinate vectors. It can be shown that the unit group GR(64, 4)∗ of order 56 has a unique subgroup T of order 7, called the set of Teichm¨uller units in GR(64, 4). If the defining polynomial f of GR(64, 4) is chosen as the Hensel lift of a primitive polynomial in F2[X] of degree 3, X + (f) is a generator of T . For the multiplicative structure of a general Galois ring, see [18, Thm. 9].

3 2 Example 3.2. We choose f = X +2X +X +3 ∈ Z4[X] as the defining polynomial 3 of GR(64, 4). f is the Hensel lift of the primitive polynomial X + X + 1 ∈ F2[X]. By identifying each element of GR(64, 4) with its unique representative of degree at most 2, we get the set of Teichm¨ullerunits T = {Xi : i ∈ {0,..., 6}} = {1,X,X2, 2X2 + 3X + 1, 3X2 + 3X + 2,X2 + 3X + 3,X2 + 2X + 1}

3.3. Hyperovals. A nonempty point set O ⊂ P is called a hyperoval in PHG(2, Z4) if each line l ∈ L intersects O either in 0 or in 2 points, and each neighbor class of PHG(2, Z4) contains at most one point of O. Straightforward combinatorial considerations show the following Lemma on the structure of a hyperoval in PHG(2, Z4), see also [6]:

Lemma 3.3. (i) Let O be a hyperoval in PHG(2, Z4). Then |O| = 7, and each neighbor class of PHG(2, Z4) contains exactly one point of O. There are 7 lines in L intersecting O in 0 points and 21 lines intersecting O in 2 points. (ii) Each quadruple of points in general position can be uniquely extended to a hyperoval in PHG(2, Z4). (iii) There are 256 hyperovals in PHG(2, Z4). Furthermore, any two hyperovals are isomorphic, and the automorphism group of a hyperoval is isomorphic to the simple group GL(3, 2) of order 168.

The existence of a hyperoval in PHG(2, Z4) also follows from a theorem by Honold and Landjev [8], which carries forward to larger projective Hjelmslev planes over Galois rings of characteristic 4:

Advances in Mathematics of Communications Volume 5, No. 2 (2011), 275–286 280 Michael Kiermaier and Johannes Zwanzger

Lemma 3.4. Let T be the set of Teichm¨uller units in GR(64, 4), and O ⊂ P be the point set given by the elements of T interpreted as coordinate vectors. Then O is a hyperoval in PHG(2, Z4). Example 3.5. The Teichm¨ullerunits in Example 3.2 yield the hyperoval                 1 0 0 1 2 1 1  O = Z4 0 , Z4 1 , Z4 0 , Z4 3 , Z4 1 , Z4 1 , Z4 2  0 0 1 2 1 3 1  in PHG(2, Z4). ˆ 4. Construction of C by a hyperoval in PHG(2, Z4)

Let O be a hyperoval in PHG(2, Z4) and ( 0 if p ∈ O; µ : P → Rad(Z4), p 7→ 2 otherwise. For a point p ∈ P we define a vector κ(p) v = ∈ 3 × Rad( ). p µ(p) Z4 Z4 Furthermore, let Γ be the (3 + 1) × 28-matrix consisting of the 28 column vectors vp for p ∈ P, let Γˆ be the (3 + 1) × 29-matrix which arises from Γ by appending the t 4 column (0 0 0 2) ∈ Z4, and let ΓS be the 3×28-matrix consisting of the first 3 rows of Γ. The definition of the generator matrices Γ, Γˆ and ΓS depends on the choice of the hyperoval O, the choice of the function κ and on the order of the columns. But since all 256 hyperovals in PHG(2, Z4) are isomorphic, and the permutation of the coordinates and the multiplication by units are isomorphisms of Z4-linear codes, different choices result in isomorphic codes. Example 4.1. We choose the function κ as in Example 3.1 and the hyperoval O as in Example 3.5. Then  0022 0022 1111 1111 0022 1111 1111 0   0202 1111 0022 1133 1111 0022 1133 0  Γˆ =    1111 0202 0202 1313 1313 1313 0202 0  0222 0222 0222 2022 2202 2202 2220 2 where the points are ordered in such a way that each neighbor class appears as a group of 4 successive columns.

The Z4-linear code generated by Γ will be denoted by C, the Z4-linear code ˆ ˆ generated by Γ will be denoted by C, and the Z4-linear code generated by ΓS will 2 be denoted by CS.

Lemma 4.2. (i) C has the parameters n = 28, r1 = 3 and r2 = 1. The Lee weight enumerator of C is 26 28 32 34 42 LeeC = 1 + 49X + 56X + 7X + 14X + X and the symmetrized weight enumerator of C is

28 7 5 16 15 13 6 6 16 symC = X0 + (42X0 X1 X2 + 7X0 X1 ) + 56X0 X1 X2 12 16 3 9 16 7 21 + 7X0 X1 + 14X0 X1 X2 + X0 X1 .

2 CS is isomorphic to the simplex code Sim(3, Z4) as defined in [7, Sect. 6.1].

Advances in Mathematics of Communications Volume 5, No. 2 (2011), 275–286 A Z4-linear code of high minimum Lee distance 281

(ii) The code CS consists exactly of the codewords of C of weight 0, 28 and 32. (iii) ∼ ∼ PAut(C) = GL(3, 2) and Aut(C) = GL(3, 2) × Z2. More precisely, the projective automorphism group of C is isomorphic to its permutation part. By identifying each column vp with the point p in PHG(2, Z4), the permutation part is the collineation group of the hyperoval O.

Proof. Clearly, the matrix ΓS given by the first 3 rows of Γ has an invertible (3×3)- submatrix. Furthermore, considering the (4 × 4)-submatrix given by the columns t vp where p is an element of the neighbor class containing Z4(0 0 1) , we see that the fourth row of Γ is not contained in the Z4-span of the first three rows. So all 3 128 codewords arising as products vΓ with v ∈ Z4 × S are distinct, showing r1 = 3 and r2 = 1. Now let c = vtΓ ∈ C be a codeword with information vector vt = (xt, s)t ∈ 3 3 Z4 × S, x ∈ Z4 and s ∈ S. To compute the symmetrized weight of c, the products t v vp = hx, κ(p)i + sµ(p) of the information vector v with the single columns of Γ are interpreted geometrically for the 28 points p ∈ P. We distinguish the following cases: 3 1. x∈ / Rad(Z4) : ⊥ In this case, l = x is a line in PHG(2, Z4). From Lemma 3.3(i) we know that there are 2 · 21 = 42 coordinate vectors x such that l meets O in two points, and there are 2 · 7 = 14 coordinate vectors x such that l and O are disjoint. (a) s = 0: t Here we have v vp = hx, κ(p)i. There are 6 points in P incident with l, 6 points not incident but a neighbor of l and 16 points which are not a neighbor of l. So the 28 points in P are partitioned into 6 points p with hx, κ(p)i = 0, 6 points p with hx, κ(p)i = 2 and 16 points p with hx, κ(p)i ∈ {1, 3}. This shows that the 43 − 23 = 56 vectors x considered in this case give rise to the partial symmetrized weight enumerator

6 6 16 e1a = 56X0 X1 X2 . (b) s = 1 and |l ∩ O| = 2: We count |l ∩ O| = 2, |l ∩ (P\O)| = 4, |([l] \ l) ∩ O| = 1, |([l] \ l) ∩ (P\O)| = 5. This shows that among the 12 points p ∈ [l], we get 2 + 5 = 7 times the t value v vp = hx, κ(p)i + µ(p) = 0 and 4 + 1 = 5 times the value 2. For ∗ the remaining 16 points p ∈ P \ [l] we have hx, κ(p)i ∈ Z4 and therefore t ∗ v vp ∈ Z4 independently from the value of µ(p). So the 42 codewords in this case account for the partial weight enumerator

7 5 16 e1b = 42X0 X1 X2 . (c) s = 1 and |l ∩ O| = 0: Like in the previous case, we count |l ∩ O| = 0, |l ∩ (P\O)| = 6, |([l] \ l) ∩ O| = 3, |([l] \ l) ∩ (P\O)| = 3.

Advances in Mathematics of Communications Volume 5, No. 2 (2011), 275–286 282 Michael Kiermaier and Johannes Zwanzger

This shows that among the 12 points p ∈ [l], we get 0 + 3 = 3 times the t value v vp = hx, κ(p)i + 2µ(p) = 0 and 6 + 3 = 9 times the value 2. So the 14 codewords in this case give the partial symmetrized weight enumerator 3 9 16 e1c = 14X0 X1 X2 . 3 2. x ∈ Rad(Z4) \{0}: 0 3 0 0⊥ In this case, there is a vector x ∈ Z4 such that x = 2x . l = x is a line, and [l] does not depend on the choice of the vector x0. (a) s = 0: There are 12 points in P in [l] and 16 points not contained in [l]. So the t 28 points in P are partitioned into 12 points p with v vp = hx, κ(p)i = 0 t 2hx , κ(p)i = 0 and 16 points with v vp = 2. Thus in this case we get the partial symmetrized weight enumerator 12 16 e2a = 7X0 X1 . (b) s = 1: We count |[l] ∩ O| = 3, |[l] ∩ (P\O)| = 9, |(P\ [l]) ∩ O| = 4, |(P\ [l]) ∩ (P\O)| = 12. t 0 t So v vp = 2hx , κ(p)i + µ(p) = 2 for 9 + 4 = 13 points p, and v vp = 0 for 3 + 12 = 15 points p. Therefore, we get the partial symmetrized weight enumerator 15 13 e2b = 7X0 X1 . 3. x is the zero vector. (a) s = 0: Here c is the zero word, having symmetrized weight 28 e3a = X0 . (b) s = 1: Here c is the last line of Γ, having symmetrized weight 7 21 e3b = X0 X1 . Sorting the terms by their Lee weight, we get

symC = e3a + (e1b + e2b) + e1a + e2a + e1c + e3b 28 7 5 16 15 13 6 6 16 = X0 + (42X0 X1 X2 + 7X0 X1 ) + 56X0 X1 X2 + 12 16 3 9 16 7 21 7X0 X1 + 14X0 X1 X2 + X0 X1 , and the Lee weight enumerator of C therefore is 1 + 49X26 + 56X28 + 7X32 + 14X34 + X42. So part (i) is shown, and part (ii) follows from the discussed cases with s = 0. For part (iii) we first consider a representative ι = (id, v) of an element of PAut(CS). Because the acting group I was defined as a factor group modulo hνi, we may assume that at most 14 entries in v equal −1. Let c ∈ CS be a row of 6 6 16 ΓS. Then c is a codeword of symmetrized weight X0 X1 X2 and ι(c) − c ∈ CS consists only of zeros and at most 14 symbols 2. By part (i) and (ii), symCS = 28 6 6 16 12 16 X0 + 56X0 X1 X2 + 7X0 X1 , so ι(c) − c is the zero word and therefore vi = 1 for all coordinates i where c has a unit entry. The unit entries of the rows of ΓS cover all coordinates, implying that ι is a representative of the identity element in I. Hence,

Advances in Mathematics of Communications Volume 5, No. 2 (2011), 275–286 A Z4-linear code of high minimum Lee distance 283

PAut(CS) is isomorphic to its permutation part. Since the generator matrix of CS consists of coordinate vectors of the points of PHG(2, Z4), the permutation part of PAut(CS) is exactly the collineation group of PHG(2, Z4), which is isomorphic to GL(3, Z4). Now let ι ∈ Aut(CS) and M ∈ GL(28, Z4) be the monomial matrix that cor- responds to ι. ι induces a unique matrix A ∈ GL(3, Z4) such that AΓSM = ΓS. The two matrices induced by the two representatives of an element ¯ι ∈ PAut(CS) arise from each other by a multiplication by the scalar matrix −I3. So one of these matrices has determinant +1 and the other one −1. It is straightforward to check that by selecting the elements of Aut(CS) inducing a matrix with deter- 3 minant +1, we get a subgroup of Aut(CS) isomorphic to PAut(CS). This shows ∼ Aut(CS) = PAut(CS) × Z2. By part (ii), there is no other subcode of C isomorphic to CS. So we have PAut(C) ⊆ PAut(CS). By part (i), C contains a unique codeword c of symmetrized 7 21 weight X0 X1 which is fixed by all automorphisms of C. The zero-coordinates of c correspond to the points of the chosen hyperoval O, so by the identification with collineations, PAut(C) ⊆ Aut(O). It is easily seen that all the collineations in Aut(O) in fact correspond to projective automorphisms of C, so PAut(C) is given by Aut(O) which is isomorphic to GL(3, 2) by Lemma 3.3(iii). It follows that ∼ Aut(C) = GL(3, 2) × Z2.

Main Theorem. (i) Cˆ has the parameters n = 29, r1 = 3 and r2 = 1. The Lee weight enumerator of Cˆ is 28 32 36 44 LeeCˆ = 1 + 105X + 7X + 14X + X and the symmetrized weight enumerator of Cˆ is 29 7 6 16 15 14 13 16 3 10 16 7 22 symCˆ = X0 + (98X0 X1 X2 + 7X0 X1 ) + 7X0 X1 + 14X0 X1 X2 + X0 X1 . (ii) ˆ ∼ ˆ ∼ PAut(C) = GL(3, 2) × Z2 and Aut(C) = GL(3, 2) × Z2 × Z2. ˆ ∼ More precisely, it holds that Aut(C) = Aut(C) × Z2, where Aut(C) is the group of Lemma 4.2(iii) acting on the first 28 coordinates, and the Z2-part is generated by the multiplication of the entries in the last coordinate by −1. Proof. The code Cˆ consists of the codewords of the code C extended by a single coordinate carrying a symbol in Rad(Z4). Since C has the parameters r1 = 3 and r2 = 1 by Lemma 4.2(i), so has Cˆ. The codewords contained in the subcode CS are extended by a 0-symbol, and the codewords in C\CS are extended by a 2-symbol. By Lemma 4.2(ii), exactly the codewords of Lee weight 0, 28 and 32 are extended by a 0-symbol. Given this information, LeeCˆ and symCˆ can be easily deduced from LeeC and symC given in Lemma 4.2(i). ˆ 3 The last column of Γ is the only column in 2Z4, hence the last coordinate is fixed by the permutation part of all automorphisms of Cˆ. So the automorphism group of Cˆ splits into a direct product of the automorphisms on the first 28 coordinates and on the last coordinate, respectively. Now part (ii) follows from Lemma 4.2(iii).

3 The induced matrices of determinant +1 are exactly the elements of SL(3, Z4).

Advances in Mathematics of Communications Volume 5, No. 2 (2011), 275–286 284 Michael Kiermaier and Johannes Zwanzger

Remark 4.3. We verified part (ii) of the preceding theorem using the algorithm for the computation of the automorphism group of a Z4-linear code described in [2]. 5. Generalization According to [8] hyperovals exist in all projective Hjelmslev planes over Galois rings GR(4r, 4) of size 4r and characteristic 4 with r ≥ 1.4 Thus it is natural to ask if the construction may be generalized to such a Galois ring R = GR(4r, 4). The size of the residue field R/Rad(R) of R is q = 2r. In the following, we assume that R is equipped with the homogeneous distance  0 if a = b,  dHom : R × R → R, (a, b) 7→ q if a − b ∈ Rad(R) \{0}, q − 1 otherwise, which in the case R = Z4 specializes to the Lee distance. For the homogeneous distance there is the generalized Gray map, a distance preserving map (R, dHom) → q (Fq, dHam), which allows the construction of nonlinear codes over a q-ary alphabet from R-linear codes. For Galois rings of composition length 2, the Reed-Solomon map γ∗ in [15] is a suitable generalized Gray map. The homogeneous weight was introduced in [1] as 2 a generalization of the Lee weight to integer residue rings Zk. For the case k = p , p prime, also a generalized Gray map was given. A more general notion of the homogeneous weight and the Gray map can be found in [10,3,4]. 5.1. Direct generalization. A direct generalization is problematic due to the fact that for r > 1 we have |Rad(R) \{0}| > 1, and so for two different coordinate t t t t vectors κ1(p) and κ2(p) of a point p the vectors (κ1(p) , µ(p)) and (κ2(p) , µ(p)) need not to be projectively equivalent any more. Hence to get a definition of the code C which is unique up to isomorphism for a fixed hyperoval, the function µ has to take into account the chosen coordinate vector of p. While this is more a formal annoyance, the real problem is that it is not clear how to choose µ(κ(p)) ∈ Rad(R) \{0} for a point p not contained in a fixed hyperoval. It remains an open question if this can be done in an appropriate way, maybe by representing the elements of R3 as elements in a Galois extension Rˆ of R of degree 3 and defining the function µ as an expression involving the trace function Rˆ → R. 5.2. Starting with several copies of the simplex code. A way to circum- vent the mentioned problem is the following: The first three rows of the generator matrix Γ are defined by concatenating |Rad(R) \{0}| = q − 1 copies of the gener- ator matrix of the simplex code. If a point p is contained in a fixed hyperoval, the entries below the q − 1 coordinate vectors of p in the fourth row of Γ are all set to zero. Otherwise, each element of Rad(R) \{0} is chosen exactly once. Now the symmetrized weight enumerator of Cr can be computed similarly as in Lemma 4.2. The matrix Γˆ is defined by appending q − 1 columns (0, 0, 0, 2)t to Γ, such that in the generated code Cˆr the homogeneous weight of the minimum weight words of Cr is raised to match the second smallest homogeneous weight in Cr. The code Cˆr constructed in this way has the symmetrized weight enumerator 5 2 7 shown in Table1. Thus Cˆr is a R-linear code of length q − q + q − 1, size q and minimum homogeneous distance q6 − q5 − q3 + q2.

4 Note that Z4 = GR(4, 4).

Advances in Mathematics of Communications Volume 5, No. 2 (2011), 275–286 A Z4-linear code of high minimum Lee distance 285

Table 1. Symmetrized weight enumerator of Cˆr

#codewords X0 X1 X2 whom (q7 + 2q6 − q5 − q4 − 2q3 + q2)/2 q3 − 1 q4 − q3 − q2 + q q5 − q4 q6 − q5 − q3 + q2 (q7 − 2q6 + q5 − q4 + 2q3 − q2)/2 q3 − 2q − 1 q4 − q3 − q2 + 3q q5 − q4 q6 − q5 − q3 + 3q2 q3 − 1 q4 − q2 + q − 1 q5 − q4 0 q6 − q5 q4 − q3 − q + 1 q4 − 1 q5 − q4 − q2 + q 0 q6 − q5 − q3 + q2 1 q5 − q2 + q − 1 0 0 0 q − 1 q3 − 1 q5 − q3 − q2 + q 0 q6 − q4 − q3 + q2

Example 5.1. In the second smallest case R = GR(16, 4), we get a code Cˆ2 of length 1011, size 47 and minimum homogeneous distance 3024. Applying the generalized 7 Gray map to Cˆ2, we end up with a nonlinear (4044, 4 , 3024)-code over a quaternary alphabet having the Hamming weight enumerator 1 + 11781X3024 + 4536X3056 + 63X3072 + 3X3792.

It is hard to judge the quality of Cˆ2 since the length is too large for a direct comparison with the internet tables. However, the fact that for each r more than half of the codewords of Cˆr have minimum weight makes the generalized codes look quite promising.

Acknowledgements We would like to thank the referees for their valuable comments and sugges- tions and Ivan Landjev for the fruitful discussion at the ALCOMA10 conference in Thurnau, which inspired the generalization using several copies of the simplex code.

References

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