The Dual Construction for Arcs in Projective Hjelmslev Spaces

Advances in Mathematics of Communications doi:10.3934/amc.2011.5.11 Volume 5, No. 1, 2011, 11–21

THE DUAL CONSTRUCTION FOR ARCS IN PROJECTIVE HJELMSLEV SPACES

Thomas Honold Zhejiang Provincial Key Laboratory of Information Network Technology and Department of Information and Electronic Engineering Zhejiang University, 38 Zheda Road, Hangzhou, Zhejiang 310027, China

Ivan Landjev New Bulgarian University 18 Montevideo str., Soﬁa 1618, Bulgaria and Institute of Mathematics and Informatics 8 Acad. G. Bonchev str., Soﬁa 1113, Bulgaria

(Communicated by Marcus Greferath)

Abstract. In this paper, we present a duality construction for multiarcs in projective Hjelmslev geometries over chain rings of nilpotency index 2. We compute the parameters of the resulting arcs and discuss some examples.

1. Introduction Duality constructions for arcs without multiple points in projective geometries over finite fields have been known for a long time (cf. [5]). These have been generalized by Dodunekov and Simonis [4] to multiarcs using a duality construction of Brouwer-van Eupen [1] for linear codes over finite fields. It turns out that similar constructions work for multiarcs in projective Hjelmslev geometries over finite chain rings. However, the situation here is more complicated. One major difference from the classical case is that dual constructions yield arcs in the dual geometry, which in the case of a non-commutative chain ring may not be isomorphic to the original geometry. The more complex structure of projective Hjelmslev geometries results also in more complicated formulas for the parameters of the dual arc. In this paper, we generalize the duality construction for multiarcs to projective Hjelmslev geometries over finite chain rings of length (nilpotency index) 2 and derive formulas for the parameters and the spectrum of the resulting dual arcs. It is possible (in principle) to prove analogous results for projective Hjelmslev geometries over finite chain rings of arbitrary length. The reason for not doing this in the present paper is twofold: on one hand, the formulas obtained are expected to be very complicated; on the other hand, almost no examples of nice arcs are known in Hjelmslev geometries over chain rings of length larger than 2, which would illustrate the construction. The paper is structured as follows. In Section 2, we introduce some basic notions and results on finite chain rings and projective geometries over finite chain rings. In Section 3, we describe a duality construction for multiarcs in projective Hjelmslev

2000 Mathematics Subject Classiﬁcation: 51C05, 51E21, 51E22, 94B05. Key words and phrases: Finite chain ring, projective Hjelmslev space, duality construction, dual space, Galois ring, multiarc.

11 c 2011 AIMS-SDU 12 Thomas Honold and Ivan Landjev geometries over rings of nilpotency index two. We prove our main theorem, which gives the type of an arbitrary hyperplane in the dual geometry. This enables us to compute the parameters and the spectrum of the resulting arc. In Section 4, we discuss some examples of applying the dual construction that yield interesting arcs.

2. Preliminaries Throughout the paper, R will denote a chain ring with nilpotency index 2 and ∼ r residue field R/ Rad R = Fq, where q = p is a prime power. All such rings have cardinality q2. For fixed q, there exist r + 1 isomorphism classes of such rings [3, 15, 16]: (1) the Galois ring Gq = Zp2 [X]/(h), where h ∈ Zp2 [X] is monic of degree r and irreducible over Zp; these rings are commutative and have characteristic 2 (i) 2 p ; (2) r truncated (skew) polynomial rings Tq = Fq[X; σi]/(X ), σi ∈ Aut(Fq), pi (0) σi : α → α , of characteristic p. Apart from Tq these rings are non-commutative. (0) For the sake of convenience, we set Sq = Tq . Let MR be a free right R-module of rank rk M = k ≥ 3. Denote by P (resp. L) the sets of free rank 1 (resp. rank 2) submodules of MR. The (right) (k − 1)- dimensional projective Hjelmslev geometry over R, denoted by PHG(MR), is defined as the incidence structure (P, L, ⊆) with point set P, line set L and incidence rela- k tion ⊆ (set inclusion). The structure PHG(MR) is isomorphic to PHG(RR) (and in 0 particular well-defined), since every module isomorphism φ : MR → MR between free R-modules induces an isomorphism between the corresponding incidence struc- 0 tures PHG(MR) and PHG(MR). Thus there is no loss of generality in restricting k ourselves to the geometries PHG(RR). k As usual lines of PHG(RR) will be identified with subsets of P. Two points x and y are neighbours (notation x _^ y) if there exist at least two lines incident with both of them; two lines K and L are neighbours (notation K _^ L) if for every point x ∈ K there exists a point y ∈ L with x _^ y, and vice versa. A non-empty point set T ⊆ P is called a Hjelmslev subspace of PHG(MR) if for every two distinct points x, y ∈ T there exists a line L ∈ L with {x, y} ⊆ L ⊆ T . Every Hjelmslev subspace is equal to the set of points contained in some free submodule U ≤ MR and thus forms a projective Hjelmslev geometry of its own with dimension rk U − 1. Hjelmslev subspaces of codimension 1 are called hyperplanes. The neighbour relation can be extended to Hjelmslev subspaces of higher dimension in the following way. We say that the Hjelmslev subspace S1 is a neighbour to S2 (notation S1 _^ S2) if every point from S1 has a neighbour on S2. Of course, the relation _^ is not necessarily symmetric under this definition. On Hjelmslev subspaces of the same dimension neighbourhood is an equivalence relation. The class of points that are neighbours of a given point x ∈ P (resp., given hyperplane H) is denoted by [x] (resp., [H]). Left projective Hjelmslev geometries PHG(RM) are defined in a similar fashion. k ∼ k op Notice that PHG(RR ) = PHG(SS), where S = R is the opposite ring of R. In what follows, we confine ourselves to right projective Hjelmslev geometries. In order to save space, we refer for the standard facts about the structure of the geometries k PHG(RR) to [7,8,9, 10, 13, 14]. k The coordinate Hjelmslev geometries PHG(RR) exhibit many similarities to the classical projective geometries PG(k − 1, q). For the R-module MR, denote by

Advances in Mathematics of Communications Volume 5, No. 1 (2011), 11–21 Dual construction for arcs 13

S(MR) (resp. F(MR)) the set of all submodules (resp. the set of all free submodules) of MR. Furthermore, set

Fs(MR) = {U ∈ F(MR) | rk U = s}. Consider the map S( Rk) → S(Rk ) π : R R U → U ⊥ ⊥ k where U = {y ∈ RR | xy = 0 for every x ∈ U}. Here xy denotes the usual inner product xy = x1y1 +...+xkyk of the vectors x = (x1, . . . , xk) and y = (y1, . . . , yk). ⊥ k The map π is a bijection with inverse V → V := {x ∈ RR | xy = 0 for every y ∈ k ⊥ k V }. One can show that U ≤ RR is free of rank s if and only if U ≤ RR is free k k of rank k − s, so that π induces bijections Fs(RR ) → Fk−s(RR) for all 0 ≤ s ≤ k. This in turn induces a bijection between the Hjelmslev subspaces of dimension t k k in PHG(RR ) and those of dimension k − 2 − t in PHG(RR) for t = 0, . . . , k − 2. k k Thus hyperplanes in PHG(RR) can be viewed as points in PHG(RR ), Hjelmslev k k subspaces of codimension 2 in PHG(RR) can be viewed as lines in PHG(RR ), etc. If the ring R is commutative, there is no need to distinguish between the left and the right geometry and π defines a polarity in the usual sense. The geometry k k PHG(RR ) will be called the dual of PHG(RR), and vice versa. Recall that R was assumed to be a chain ring of length 2 and with residue field r of order q. As mentioned earlier, R is either the Galois ring Gq, q = p , p prime, or 2 else a factor ring of a skew-polynomial ring Fq[X; σ]/(X ), σ ∈ Aut Fq. It is easily 2 op ∼ −1 2 checked that if R = Fq[X; σ]/(X ) then R = Fq[X; σ ]/(X ). Since the right geometry over R is isomorphic to the left geometry over Rop of the same dimension, k we can conclude that in this case the dual geometry of PHG(RR) is isomorphic to k −1 2 PHG(SS), where S = Fq[X; σ ]/(X ). Let Π = (P, L, ⊆) be a projective Hjelmslev geometry. Let us denote by H the set of all hyperplanes in Π. A multiset in Π is defined as a mapping K : P → N0 from the points of Π to the nonnegative integers. The mapping K is extended in a natural way to subsets of P by X K(Q) = K(x), Q ⊆ P. x∈Q A multiset K in Π is called an (n, w)-arc (resp. (n, w)-blocking multiset) if |K| = n and K(H) ≤ w (resp. K(H) ≥ w) for every hyperplane H ∈ H. For a hyperplane H and a multiset K in Π, we define the type of H (with respect to K) as the triple a(H) = a0(H), a1(H), a2(H) , where X X X a0(H) = K(x), a1(H) = K(x), a2(H) = K(x). x:x6∈[H] x:x∈[H]\H x:x∈H

In other words, a0(H) is the sum of the multiplicities of the points that are not neighbours to H, a1(H) is the sum of the multiplicities of the points that are neighbours to H, but do not lie on H, and, ﬁnally, a2(H) is the sum of the multiplicities of the points that lie on H. Let a = (a0, a1, a2) ∈ N0 × N0 × N0. Deﬁne

Aa = |{H | H ∈ H,H has type a}|.

The sequence {Aa | a ∈ N0 × N0 × N0} is called the spectrum of K. In what follows, we denote by W (K), or just by W , should the multiset be clear from the context, the set of all triples a ∈ N0 × N0 × N0 with Aa > 0. Note that the set W (K) is analogous to the weight set of a multiset of points in PG(k − 1, q) as deﬁned in [4].

Advances in Mathematics of Communications Volume 5, No. 1 (2011), 11–21 14 Thomas Honold and Ivan Landjev

Below we give examples for some multiarcs which will be used as illustration of the results.

3 2 2 Examples. (i) (cf. [11]) Hyperovals in PHG(RR), where R = GR(q , 2 ) with q = 2r. These have q2 + q + 1 points, one in each neighbour class, and meet every line in either 0 or 2 points. Hence W = (q2, q − 1, 2), (q2, q + 1, 0) .A particular example is the so-called Teichm¨ullerset, which is obtained from the 3 × 3 ∼ subgroup of order q − 1 of S in the representation PHG(RR) = PHG(SR) with S = GR(q6, 22).1 (i) 2 (ii) The Baer subplane for planes over rings Tq . This has q + q + 1 points, one in each neighbour class, and meets every line in either 1 or q + 1 points (and thus forms a (q2 + q + 1, 1)-blocking set). Here W = (q2, 0, q + 1), (q2, q, 1) . (iii) (cf. [2]) A (q(q2 + q + 1), 2q)-arc consisting of q2 + q + 1 line segments, one segment in each neighbour class of points, and such that no two segments are collinear. Here W = (q3, q2, q), (q3, q2 − q, 2q) . (iv) (cf. [10]) In the geometries over all chain rings with 16 elements and length 2, there exist (18, 2)-arcs with the following structure: There are nine neighbour classes of points with two points and twelve empty classes. Further the nonempty classes form an Hermitian curve in the factor plane. The set of line types is W = {(12, 4, 2), (12, 6, 0), (16, 0, 2), (16, 2, 0)}. For the two rings of characteristic 2, these arcs are (n, 2)-arcs of the largest possible size n. 3 (v) The Teichm¨ullerset (see Example (i)) in PHG(Z9) forms a (13, 3)-arc with one point in each neighbour class and W = (9, 4, 0), (9, 3, 1), (9, 2, 2), (9, 1, 3) . 3 (vi) In PHG(Z9) there exist (39, 5)-arcs [12]. Arcs with these parameters have three points in each neighbour class forming a triangle, and they meet every line in either 2 or 5 points [6]. Hence W = (27, 10, 2), (27, 7, 5) in this case. Moreover, as shown in [6], these are (n, 5)-arcs of the largest possible size in 3 PHG(Z9). 3 (vii) In PHG(RR), R = GR(16, 4), there exist (84, 6)-arcs having four points in each neighbour class forming a quadrangle with three pairs of parallel sides, and meeting every line in either 2 or 6 points. These arcs have W = (64, 18, 2), (64, 14, 6) .

3. The dual construction k Let Π = PHG(RR) and let K be an (n, w)-arc in Π. Let τ : W → N0 be an arbitrary function and deﬁne

H → ; (1) Kτ : N0 H → τ(a(H)).

The map Kτ can be extended to subspaces ∆ of arbitrary dimension by Kτ (∆) = P τ τ H⊇∆ K (H). The arc K is called the τ-dual of K. The parameters of a τ-dual

1This subgroup intersects R× in a subgroup of order q − 1 and therefore accounts for only 3 q −1 2 q−1 = q + q + 1 distinct points.

Advances in Mathematics of Communications Volume 5, No. 1 (2011), 11–21 Dual construction for arcs 15 arc are (n0, w0), where 0 X n = τ(a)Aa, a∈W X w0 = max Kτ (x) = max Kτ (H). x∈P x∈P H:x∈H Note that the τ-dual arc of K is not contained in the same, but in the dual Hjelmslev k geometry PHG(RR ). In ring geometries we are interested not only in the intersection numbers of the arcs in question, but also in the possible types of the hyperplanes in Π. It is diﬃcult to derive formulas for the spectrum of dual arcs for arbitrary functions τ. That is why we consider only a particular class of choices for τ. Let H be a hyperplane of type a = (a0, a1, a2). We consider H as a point of multiplicity

τ(a) = αa0 + βa1 + γa2

= α(n − a1 − a2) + βa1 + γa2

= αn + (β − α)a1 + (γ − α)a2 ∗ k in the dual geometry Π = PHG(RR ). Thus it is reasonable to consider mappings of the form

(2) τ(a) = α + βa1 + γa2, where α, β and γ are chosen in such a way that the numbers τ(a) are nonnegative integers for all a ∈ W . We call such mappings linear (for obvious reasons). If τ is linear with coeﬃcients as in (2), then the τ-dual of K is given by (3) Kτ (H) = α + βK([H]) + (γ − β)K(H). For our main result, we need a counting lemma. k Lemma 1. Let K be an (n, w)-arc in Π = PHG(RR), where R is a chain ring with 2 ∼ |R| = q , R/ Rad R = Fq. Then the following identities hold: P k−2 qk−1−1 (a) H∈H K(H) = nq · q−1 ; P k−3 qk−2−1 k−3 k−2 2(k−2) (b) H:x∈H K(H) = nq · q−1 + q (q − 1)K([x]) + q K(x); P k−1 qk−1−1 (c) H∈H K([H]) = nq · q−1 ; P k−2 qk−2−1 2(k−2) (d) H:x∈H K([H]) = nq · q−1 + q K([x]). Proof. (a) This equality follows by the fact that each point from Π is contained in qk−2(qk−1 − 1)/(q − 1) hyperplanes. (b) The multiplicity of each point outside [x] is counted qk−3(qk−2 − 1)/(q − 1) times, the multiplicity of each point y ∈ [x], y 6= x, is counted qk−2(qk−2 −1)/(q−1) times and, ﬁnally x is contained in qk−2(qk−1 − 1)/(q − 1) hyperplanes and hence its multiplicity is counted this number of times. This implies

X qk−2 − 1 qk−2 − 1 qk−2 − 1 K(H) = nqk−3 · + qk−2 · − qk−3 · K([x]) q − 1 q − 1 q − 1 H:x∈H qk−1 − 1 qk−2 − 1 + qk−2 · − qk−2 · K(x), q − 1 q − 1 whence we get the equality.

Advances in Mathematics of Communications Volume 5, No. 1 (2011), 11–21 16 Thomas Honold and Ivan Landjev

(c) The multiplicities of the points in the neighbour class [H] are counted qk−1 S times, once for each hyperplane in it. The union H [H] of all neighbour classes qk−1−1 covers the whole space exactly q−1 times. This implies the desired equality. (d) The number of hyperplanes H0 ∈ [H] with x ∈ H0 is qk−2 if x ∈ [H] and zero otherwise. Hence X X K([H]) = qk−2 K([H]) H:x∈H [H]:x∈[H] h qk−2 − 1 qk−1 − 1i = qk−2 n − K([x]) · + K([x]) · q − 1 q − 1 qk−2 − 1 = qk−2n · + q2(k−2)K([x]) q − 1 as asserted.

k Theorem 2. Let K be an (n, w)-arc in PHG(RR), where R is a chain ring with 2 ∼ |R| = q , R/ Rad R = Fq. Let α, β, γ ∈ Q be such that α + βa1 + γa2 ∈ N0 for all a = (a0, a1, a2) ∈ W . For any hyperplane H of type a = (a0, a1, a2), let τ K (H) = τ(a(H)) = α + βa1 + γa2. Then the type of an arbitrary hyperplane x∗ = xR ∈ P in the dual geometry is b = (b0, b1, b2), where 2k−2 2k−4 2k−4 b0 = αq + βnq (q − 1) + γnq − βq2k−4(q − 1) + γq2k−4 K([x]),

k−2 k−1 k−3 k−2 k−3 k−2 b1 = αq (q − 1) + βnq (q − 1)(q − 1) + γnq (q − 1) + βqk−3(qk − 2qk−1 + qk−2 − 1) + γqk−3(qk−1 − qk−2 + 1) K([x]) −(γ − β)q2k−4K(x), qk−1 − 1 qk−2 − 1 b = αqk−2 · + βnqk−3(qk−2 − 1) + γnqk−3 · 2 q − 1 q − 1 + βqk−3(qk−1 − qk−2 + 1) + γqk−3(qk−2 − 1) K([x]) +(γ − β)q2k−4K(x).

Proof. Note that the size of the dual arc is b0 + b1 + b2. Now by Lemma1(a)(c), we get X τ b0 + b1 + b2 = K (H) H∈H X = α + β · K([H]) + (γ − β) · K(H) H∈H X X X = α + β K([H]) + (γ − β) K(H) H∈H H∈H H∈H qk − 1 qk−1 − 1 qk−1 − 1 = αqk−1 + βnqk−1 + (γ − β)nqk−2 q − 1 q − 1 q − 1 qk − 1 qk−1 − 1 = αqk−1 + βnqk−2(qk−1 − 1) + γnqk−2 . q − 1 q − 1

Advances in Mathematics of Communications Volume 5, No. 1 (2011), 11–21 Dual construction for arcs 17

The number of points in the dual plane incident with the hyperplane xR is

X τ b2 = K (H) H:x∈H X = α + β · K([H]) + (γ − β) · K(H) H:x∈H X X X = α + β K([H]) + (γ − β) K(H) H:x∈H H:x∈H H:x∈H qk−1 − 1 qk−2 − 1 = αqk−2 + βnqk−2 + βq2(k−2)K([x]) q − 1 q − 1 qk−2 − 1 + (γ − β) nqk−3 + qk−3(qk−2 − 1)K([x]) + q2(k−2)K(x) q − 1 qk−1 − 1 qk−2 − 1 = αqk−2 + βnqk−3(qk−2 − 1) + γnqk−3 q − 1 q − 1 + βq2(k−2) + (γ − β)qk−3(qk−2 − 1) K([x]) + (γ − β)q2(k−2)K(x) as claimed. Finally, the number of points in the dual geometry that are neighbours to a ﬁxed hyperplane xR is b1 + b2. Hence

X τ b1 + b2 = K (H) H:x∈[H] 1 X X = α + β · K([H]) + (γ − β) · K(H) qk−2 y∈[x] H3y  1 X qk−1 − 1 X qk−2 − 1 = α qk−2 + β nqk−2 + q2(k−2)K([x]) qk−2  q − 1 q − 1 y∈[x] y∈[x]  X qk−2 − 1 +(γ − β) nqk−3 + qk−3(qk−2 − 1)K([x]) + q2(k−2)K(y) q − 1  y∈[x] qk−1 − 1 qk−2 − 1 = αqk−1 + βnqk−2(qk−2 − 1) + γnqk−2 q − 1 q − 1 + βq2k−3 + (γ − β)q2(k−2) K([x]).

Solving for b0, b1, b2, we get the desired result.

In the planar case, the above formulas are less complicated.

3 Corollary 3. Let K be an (n, w)-arc in PHG(RR), where R is a chain ring with 2 ∼ |R| = q , R/ Rad R = Fq. Let α, β, γ ∈ Q be such that α + βa1 + γa2 ∈ N0 for all a = (a0, a1, a2) ∈ W . For any line L of type a = (a0, a1, a2), let

τ K (L) = τ(a(L)) = α + βa1 + γa2.

Advances in Mathematics of Communications Volume 5, No. 1 (2011), 11–21 18 Thomas Honold and Ivan Landjev

∗ Then the type of an arbitrary line x ∈ P in the dual plane is b = (b0, b1, b2), where 4 2 2 2 b0 = αq + βnq (q − 1) + γnq − β(q − 1) + γ q K([x]), 2 2 b1 = αq(q − 1) + βn(q − 1) + γn(q − 1) + β(q3 − 2q2 + q − 1) + γ(q2 − q + 1)K([x]) − (γ − β)q2K(x),

b2 = αq(q + 1) + βn(q − 1) + γn + β(q2 − q + 1) + γ(q − 1)K([x]) + (γ − β)q2K(x).

k Corollary 4. Let K be an (n, w)-arc in PHG(RR) = (P, L, ⊆), where R is a chain 2 ∼ ring with |R| = q , R/ Rad R = Fq. Let τ(a) be linear with coeﬃcients as in (2). Then the number of types of hyperplanes in W (Kτ ) is equal to the number of pairs K([x]), K(x), where x runs through P.

4. Examples In this section, we illustrate the dual construction by applying it to the examples from Section2. The numbering of the examples is the same as before.

Examples. (i) Since |W | = 2, every mapping τ : W → N0 is linear. We take τ :(q2, q + 1, 0) → 1, (q2, q − 1, 2) → 0, so that Kτ consists of the 0-lines of the hyperoval K, taken with multiplicity 1. The mapping τ is realized by the choice of coeﬃcients 1 q − 1 α = 0, β = , γ = − . q + 1 2(q + 1) We have n = q2 + q + 1 and K([x]) = 1 for all x. Using Corollary3, we get the possible types of lines in the dual plane (it is isomorphic to the original one, since hyperovals exist over Galois rings only and these are commutative) as (b0, b1, b2), where q4 − q3 b = , 0 2 q3 − q2 − q 1 b = + q2K(x), 1 2 2 q2 1 b = − q2K(x). 2 2 2 So the dual arc is a ((q4 − q)/2, q2/2)-arc with two intersection numbers 0 and q2/2. Moreover, every neighbour class of points contains exactly (q2 − q)/2 points. It can easily be checked that the dual arcs are optimal, i.e. 4 3 q − q m 2 (R ) = , q /2 R 2 r 3 where R = Gq with q = 2 . Here mw(RR) denotes the largest possible size 3 n of an (n, w)-arc in PHG(RR). It should be noted that in case of q = 2 the hyperoval is self-dual. In case of q = 4, one obtains a (126, 8)-arc which was ﬁrst announced in [12].

Advances in Mathematics of Communications Volume 5, No. 1 (2011), 11–21 Dual construction for arcs 19

2 2 τ (ii) Here we take τ : W → N0,(q , q, 1) → 0, (q , 0, q + 1) → 1, so that K consists of the q + 1-lines of the Baer subplane K (taken with multiplicity 1). The map τ is realized by the choice of coeﬃcients 1 1 α = 0, β = − , γ = . q(q + 1) q + 1 It has the same parameters as the original point set, i.e. is a Baer subplane of the dual Hjelmslev plane. We say that the Baer subplane is self-dual. (iii) The (q(q2 + q + 1), 2q)-arc K in this example turns also out to be self-dual. We have n = q(q2 + q + 1), line types (q3, q2 − εq, q + εq), where ε = 0 or 1, and 3 2 K([x]) = q for all classes of points [x]. If we take τ : W → N0 as (q , q , q) → 0, (q3, q2 − q, 2q) → 1 or, equivalently, as the linear map with coeﬃcients 1 1 α = 0, β = − , γ = , q(q + 1) q + 1 the dual arc Kτ consists of the 2q-lines of K (taken with multiplicity 1). By Corollary3 we have 3 2 b0 = q , b1 = q − qK(x), b2 = q + qK(x), which implies that the dual arc Kτ has the same parameters. This class of arcs can be generalized to projective Hjelmslev geometries k PHG(RR) of arbitrary dimension. The parameters of the generalized arcs are k−2 k k−2 k−2 (q (q −1)/(q −1), q (q +q −2)/(q −1)) with lines of type (a0, a1, a2), where 2k−3 a0 = q , qk−1 − 1 a = − 1 (qk−2 − qk−3) + (1 − ε)qk−2, 1 q − 1 qk−1 − 1 a = qk−3 − 1 + εqk−2, 2 q − 1 ε = 0, 1, and where K([x]) = qk−2 for every neighbour class of points [x]. These arcs are again self-dual with qk−2 − 1 q − 1 α = 0, β = − , γ = . qk−2(qk−1 − 1) qk−1 − 1 (iv) Here we apply the construction to an arc, in which not all the neighbour classes of points have the same cardinality. We set 1 1 1 α = , β = , γ = − . 2 4 4 Thus lines of type (12, 6, 0) become points in the dual plane of multiplicity 2; lines of type (12, 4, 2) or (16, 2, 0) become points of multiplicity 1 and lines of type (16, 0, 2) become points of multiplicity 0.2 The spectrum of the (18, 2)-arc is

a = (a0, a1, a2) (16, 2, 0) (16, 0, 2) (12, 6, 0) (12, 4, 2) Aa 108 36 48 144

2Note that the dual plane is isomorphic to the original plane for all three rings GR(16, 4), 2 2 F4[X]/(X ), and F4[X; σ1]/(X ) considered here.

Advances in Mathematics of Communications Volume 5, No. 1 (2011), 11–21 20 Thomas Honold and Ivan Landjev

Hence we obtain that the resulting dual arc has parameters (348, 24). By Theorem2 we can obtain the spectrum of the dual arc: (K([x]), K(x)) (0, 0) (2, 0) (2, 1) b = (b0, b1, b2) (272, 57, 19) (256, 68, 24) (256, 76, 16) Ab 192 126 18 Theorem2 also applies to the arc in the dual plane formed by the 0-lines, τ a2 which are of two diﬀerent types. This arc is equal to K for τ(a) = 1 − 2 , i.e. 1 α = 1, β = 0, γ = − 2 . It is an (156, 11)-arc with spectrum (K([x]), K(x)) (0, 0) (2, 0) (2, 1) b = (b0, b1, b2) (112, 33, 11) (128, 20, 8) (128, 28, 0) Ab 192 126 18 (v) The spectrum of the (13, 3)-arc K is

a = (a0, a1, a2) (9, 4, 0) (9, 3, 1) (9, 2, 2) (9, 1, 3) Aa 26 39 39 13 1 τ We take τ(a) = a1 − 9 a0, so that K consists of the 0-, 1-, and 2-lines of K taken with multiplicities 3, 2, and 1, respectively. By Theorem2 the dual arc Kτ has the following spectrum: (K([x]), K(x)) (1, 0) (1, 1) b = (b0, b1, b2) (135, 39, 21) (135, 48, 12) Ab 104 13 3 It is thus a non-projective (195, 21)-arc in PHG(Z9) with constant point neighbour class multiplicity 15 and only two diﬀerent line types (intersection numbers). The arcs in Examples (vi), (vii) are also self-dual. Dual arcs with the same parameters as the original arcs are obtained by taking in both cases all 2-lines of the original arc with multiplicity 1.

Acknowledgements The authors wish to thank the reviewers for their careful reading of the manu- script and for several corrections. The research of the first author was supported by the Open Project of Zhejiang Provincial Key Laboratory of Information Network Technology and by the National Natural Science Foundation of China under Grant No. 60872063. The research of the second author was supported by the Scientific Research Fund of Sofia University under Contract No 192/22.04.2010.

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[5] J. W. P. Hirschfeld, “Projective Geometries over Finite Fields,” 2nd edition, Oxford University Press, 1998. [6] T. Honold, M. Kiermaier and I. Landjev, New arcs of maximal size in projective Hjelmslev planes of order nine, Comptes Rendus l’Acad. Bulgare Sci., 63 (2010), 171–180. [7] T. Honold and I. Landjev, Projective Hjelmslev geometries, in “Proceedings of the Interna- tional Workshop on Optimal Codes,” Sozopol, Bulgaria, (1998), 97–115. [8] T. Honold and I. Landjev, Linearly representable codes over chain rings, Abhandlungen math. Seminar Univ. Hamburg, 69 (1999), 187–203. [9] T. Honold and I. Landjev, Linear codes over ﬁnite chain rings, Electr. J. Combin., 7 (2000), 22. [10] T. Honold and I. Landjev, On arcs in projective Hjelmslev planes, Discrete Math., 231 (2001), 265–278. [11] T. Honold and I. Landjev, On maximal arcs in projective Hjelmslev planes, Finite Fields Appl., 11 (2005), 292–304. [12] M. Kiermaier and A. Kohnert, New arcs in projective Hjelmslev planes over Galois rings, in “Optimal Codes and Related Topics,” White Lagoon, Bulgaria, (2007), 112–119. [13] A. Kreuzer, Fundamental theorem of projective Hjelmslev spaces, Mitteilungen Math. Gesellschaft Hamburg, 12 (1991), 809–817. [14] I. Landjev and T. Honold, Arcs in projective Hjelmslev planes, Discrete Math. Appl., 11 (2001), 53–70. [15] A. A. Nechaev, Finite principal ideal rings, Sbornik. Math., 20 (1973), 364–382. [16] R. Raghavendran, Finite associative rings, Compositio Math., 21 (1969), 195–229. Received February 2010; revised September 2010. E-mail address: [email protected] E-mail address: [email protected] [email protected]

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