$2$-arcs of maximal size in the affine and the projective Hjelmslev plane over $\mathbb{Z}_{25}$

It is shown that the maximal size of a $2$-arc in the projective Hjelmslev plane over $\mathbb{Z}_{25}$ is $21$, and the $(21,2)$-arc is unique up to isomorphism. Furthermore, all maximal $(20,2)$-arcs in the affine Hjelmslev plane over $\mathbb{Z}_{25}$ are classified up to isomorphism.


Introduction
It is well known that a Desarguesian projective plane of order q admits a 2-arc of size q + 2 if and only if q is even. These 2-arcs are called hyperovals. The biggest 2-arcs in the Desarguesian projective planes of odd order q have size q + 1 and are called ovals.
For a projective Hjelmslev plane over a chain ring R of composition length 2 and size q 2 , the situation is somewhat similar: In PHG(2, R) there exists a hyperoval -that is a 2-arc of size q 2 + q + 1 -if and only if R is a Galois ring of size q 2 with q even, see [7,6] For the case q odd and R not a Galois ring it was recently shown that the maximum size m 2 (R) of a 2-arc is q 2 , see [4].
In the remaining cases, the situation is less clear. For even q and R not a Galois ring, it is known that the maximum possible size of a 2-arc is lower bounded by q 2 + 2 [4] and upper bounded by q 2 + q [6]. The last case q odd and R a Galois ring currently is the least satisfactory one. Besides the upper bound q 2 , only the lower bound q+1 2 2 is known [4], leaving a comparatively large gap.
The current state of the lower and upper bounds on the maximal size of a 2-arc for general q is summarized in the following table: m 2 (R) R Galois ring not a Galois ring q even q 2 + q + 1 In the open cases for #R ≤ 16 the exact values were determined computationally [8,3]. The ring R = Z 25 is the smallest one where the exact value m 2 (R) was not known beforehand. In [2] the first 2-arc in PHG(2, Z 25 ) of size 20 was found. This was the biggest known 2-arc until in [9] a 2-arc of size 21 was given. The main result of this paper is: The maximum size of a 2-arc in PHG(2, Z 25 ) is 21. The (21, 2)-arc is unique up to isomorphism.
The projective Hjelmslev plane over Z 25 has 775 points, so the search space for arcs of size 21 or 22 is way too big to allow a direct attack by a backtrack search. Therefore, special computational methods had to be developed.
In Section 2, we give a brief introduction to finite chain rings and coordinate Hjelmslev planes. For details, see for example [21,22] 1 and [6], respectively. For the computation, two independent approaches are used: In Section 3, in a first step the possible images of a (22,2)-arc in the factor plane are computed up to isomorphism, and in the second step the preimages are investigated. In Section 4 first the (20, 2)-arcs in the affine plane are classified and then checked for extendability to a (22,2)-arc in the projective plane. In Section 5, further computational results are summarized, and the maximum size of a 2-arc of a given maximum point class multiplicity is determined for the projective Hjelmslev planes over Z 25 and S 5 , which is the other chain ring of composition length 2 and size 25.
Computational nonexistence-and uniqueness proofs using highly nontrivial algorithms are always a delicate matter, since already a subtle glitch in the implementation may very well cause a totally wrong result. For that reason we put a lot of effort on assuring the correctness of our implementations. The maximality and uniqueness of the (21, 2)-arc was achieved independently by our two approaches. Furthermore, both approaches independently report the number of the PGL(3, R)↓-isomorphism classes of (20, 2)-arcs in AHG(2, Z 25 ) as 488.

Finite chain rings
Let R be a finite ring. 2 R is called chain ring if the lattice of the left-ideals is a chain. A chain ring is necessarily local, so there is a unique maximum ideal N = rad(R). It can be shown that N is a principal ideal. Because R is finite, the quotient ring R/N is isomorphic to a finite field F q of order q = p r with p prime. R/N is called the residue class field of R.
We will need the projection φ : R → F q , a → a mod N , which is a surjective ring homomorphism. The number of ideals of R reduced by 1 is the composition length of R, considered as a left module R R. This number will be denoted by m. The order of R is q m .
An important subclass of the finite chain rings are the Galois rings. Their definition is a slight generalization of the construction of finite fields via irreducible polynomials: Let p be prime, r and m positive integers, q = p r and f ∈ Z p m [X] be a monic polynomial of degree r such that the image of f modulo p is irreducible in F p [X]. Then the Galois ring of order q m and characteristic p m is defined as Up to isomorphism, the definition is independent of the particular choice of f . The symbols p, q, r and m are consistent with the earlier definitions: GR(q m , p m ) / rad(GR(q m , p m )) ∼ = F q and the composition length of GR(q m , p m ) is m. Furthermore, the class of the Galois rings contains the finite fields and the integer residue class rings modulo a prime power: GR(p m , p m ) ∼ = Z p m and GR(p r , p) ∼ = F p r . The finite fields are exactly the finite chain rings of composition length 1. In the following we assume that R is a finite chain ring of composition length 2. The isomorphism types of these rings are known [23,21,1]: For a fixed size q = p r of the residue field of R there are r +1 possible isomorphism types for R. One is the Galois ring GR(q 2 , p 2 ) of characteristic p 2 , and the remaining r ones are all of characteristic p and not isomorphic to a Galois ring. Among the latter r possibilities there is a single commutative ring, which is S q = F q [X]/(X 2 ).
In particular, up to isomorphism the only chain rings of size 25 and composition length 2 are Z 25 and S 5 = F 5 [X]/(X 2 ).

Affine and projective coordinate Hjelmslev planes
The projective Hjelmslev plane PHG(2, R) over a finite chain ring R is defined as follows: The point set P(PHG(2, R)) [line set L(PHG(2, R))] is the set of the free rank 1 [rank 2] right submodules of the right R-module R 3 , and the incidence relation is given by set inclusion. For a point x a vector v ∈ R 3 with x = vR is called coordinate vector of x. x has #R × = q(q − 1) coordinate vectors. There is a unique coordinate vector of x whose first unit entry is 1.
The affine Hjelmslev plane AHG(2, R) over a finite chain ring R is defined as follows: The point set P(AHG(2, R)) is the set of the vectors in R 2 , and the line set L(AHG(2, R)) is the set of all cosets of free rank 1 submodules of the right R-module R 2 . The incidence relation is given by set membership.
The projective and affine Hjelmslev planes share a lot of structure which will be described simultaneously: Let (H, G) be either (PHG(2, R), PG(2, F q )) or (AHG(2, R) , AG(2, F q )). For a consistent description, in the following we identify a line L of PHG(2, R) with the set of points incident with L, so that the incidence relation of both PHG(2, R) and AHG(2, R) is given by set membership.
In contrast to classical planes, in Hjelmslev planes it may happen that two distinct lines meet each other in more than a single point. More precisely, there is more than one line passing through points x and y of H if and only if φ(x) = φ(y), where the mapping φ is extended from R to P(H). Each preimage φ −1 (z) with z ∈ P(G) is called point class. For x ∈ P(H), the point class φ −1 (φ(x)) containing x is denoted by [x]. Similarly, two lines L 1 and L 2 of H intersect in more than one point if and only if φ(L 1 ) = φ(L 2 ). The preimages φ −1 (l) with l ∈ L(G) are called line classes. For L ∈ L(H), the line class φ −1 (φ(L)) containing L is denoted by [L]. So by our definition, a line class [L] is a set of points.
Similarly to the classical case, the projective and the affine Hjelmslev plane over R are tightly related: By the mapping P(AHG(2, R)) → P(PHG(2, R)), v → (1, v)R, the affine Hjelmslev plane is embedded in the projective Hjelmslev plane. We will refer to this embedding as the standard embedding. The points in PHG(2, R) not contained in the image of AHG(2, R) form a line class. This line class is called line class at infinity and given by all points vR such that the first component of v is not a unit. On the other hand, removing a line class [L] from PHG(2, R) yields a plane isomorphic to AHG(2, R), and [L] is its line class at infinity.
There is also an axiomatic definition of projective and affine Hjelmslev planes which does not rely on an underlying chain ring [12,18]. This definition is more general than the one given above: Not every axiomatically defined Hjelmslev plane can be coordinatized over a finite chain ring. For that reason, the Hjelmslev planes PHG(2, R) and AHG(2, R) are also called coordinate Hjelmslev planes. The axiomatic definition of an affine Hjelmslev plane involves a parallelism, which is an equivalence relation on the set of lines satisfying Euclid's parallel axiom. In contrast to classical affine planes, in general the parallelism of an affine Hjelmslev plane is not uniquely determined by its underlying incidence structure. For a coordinate affine Hjelmslev plane AHG(2, R) embedded in PHG(2, R), each line L contained in the line class at infinity induces a parallelism in the following way: Two lines in L(AHG(2, R)) are called parallel if and only if they pass through the same point on L. Using the standard embedding, we will call the parallelism induced by the line (1, 0, 0) ⊥ the standard parallelism.

Induced subgeometries
Because φ : P(H) → P(G) maps lines to lines, the geometry of the point classes and the line classes with incidence given by the subset relation is isomorphic to the plane G. This plane is called factor plane of H.
The restriction of the geometry H to a single point class [x] is isomorphic to the affine plane AG(2, F q ). It will be denoted by Π [x] . Each line in one of these affine geometries is called line segment when considered as a subset of P(H).

For each point class [x] incident with [L] we define
Then the geometry Π [L] consisting of the point set P and the line set L with the incidence relation defined in the obvious way 3 is isomorphic to the projective plane PG(2, F q ).

Collineations
A collineation of a point-line geometry is a bijection on the point set mapping lines to lines. As usual, two multisets of points are called isomorphic if they are contained in the same orbit under the action of the collineation group. In this article we do not require that a collineation of an affine Hjelmslev plane preserves its parallelism. The reason for this is that the group of collineations will be used to reduce the search space of the maximal arc problem, which does not depend on the parallelism.
The group ΓL(3, R) consists of all elements (A, σ) with A ∈ GL(3, R) and σ ∈ Aut(R). The multiplication in ΓL(3, R) is given by is defined as the factor group ΓL(3, R)/N and acts faithfully on P(H). By the fundamental theorem of projective Hjelmslev geometry [14], PΓL(3, R) is the group of collineations of PHG(2, R). In the following it is assumed that AHG(2, R) is embedded into PHG(2, R) by the standard embedding. It is clear that all σ ∈ PΓL(3, R) fixing AHG(2, R) as a set give rise to a collineation of AHG(2, R), which will be denoted by σ↓. The stabilizer of AHG(2, R) in PΓL(3, R) will be denoted by PΓL(3, R) ∞ , and the set of induced collineations of AHG(2, R) will be denoted by PΓL(3, R)↓. The symbols PGL(3, R) ∞ and PGL(3, R)↓ are defined analogously. It can be checked that the restriction ↓ : By the standard embedding each element of AHG(2, R) has the form ρ↓ preserves the standard parallelism if and only if c = 0. Hence the mappings in PΓL(3, R)↓ preserving the standard parallelism are given by the group AΓL(2, R) consisting of all mappings

Arcs
For any geometry and any n ∈ N, a multiset of points k of size n is called (n, w)-arc, if no w + 1 elements of k are collinear 4 . We denote by m w (R) the maximum size of a w-arc in the projective Hjelmslev plane PHG(2, R). There is an online table [10] for known lower bounds on m w (R).
We want to mention that the literature is inconsistent about allowing an arc to be a proper multiset. Our definition is motivated by the connection to coding theory [5] where the possibly repeated columns of a generator matrix are interpreted as coordinate vectors in projective Hjelmslev geometries. This article is about 2-arcs, where the different possible definitions do not matter: The only proper multiset which is a 2-arc consists of a single point of multiplicity 2.
For an arc K and a point set X ⊆ P, the size #(X ∩ K) is called multiplicity of X. In this way, multiplicities are declared for point classes and line classes. A point class of multiplicity u will simply be called u-class. Furthermore, the maximum value of u such that there is a u-class will be denoted by u(K). It is the maximum point class multiplicity of K.

Computation via the images in the factor plane
We define the map Φ : N P(PHG(2,R)) → N P(PG(2,R)) as the extension of φ to multisets. For a multiset of points K in PHG(2, R), the image Φ(K) reduces K to the distribution of the points in K into the point classes of PHG(2, R).
The strategy for the algorithm is to generate all possible images Φ(K) up to PG(2, F q )-isomorphism in a first step. In a second step, the preimages of these images are generated. Concerning the algorithmic complexity, the second step is much harder than the first one.
This approach suggests itself for several reasons: Without the intermediate step, the computations must be done under the full collineation group PΓL(3, R). But to compute the possible preimages of a given point class distribution k in the second step, the acting group is reduced to the preimage in PΓL(3, R) of the stabilizer of k by the Homomorphism Principle [17] and the fact that the mapping Φ naturally extends to a homomorphism of the group actions, mapping the action of PΓL(3, R) on P(PHG(2, R) to the action of PΓL(3, F q ) on P(PG(2, F q )), Furthermore, the next lemma (compare [6, Theorem 3.5]) summarizes severe restrictions on the possible point class distributions, which allow a drastic pruning of the search tree between the two steps: Lemma 3.1 Let H be the projective Hjelmslev plane over R and assume that its factor plane G is of odd order q. Let K be a 2-arc in H.    In Theorem 5.1, this will be done explicitly for the chain rings Z 25 and S 5 .    = #L 2 + 3#L 3 . Solving this equation system for (#L 2 , #L 3 ) yields 2#L 2 = #P 2 (11 − #P 2 ) and 3#L 3 = #P 2 (#P 2 − 6). So 3 | #P 2 and therefore #P 2 ∈ {6, 9}.

Creating the point class distributions
In the case #P 2 = 9 it holds #P 1 = 4 and #L 2 = #L 3 = 9. Let  The automatic construction of the point class distribution is done by a backtracking search, checking the conditions of Lemma 3.1 in each step. Because only one representative of each isomorphism class is needed, we apply orderly generation [24]. Whenever a complete point class distribution k is found, all 2-arcs in its preimage in PHG(2, Z 25 ) are created as described below. If in this search it turns out that even a smaller 5 point class distribution k ′ < k does not admit an (n, 2)-arc in its preimage, it is clear that there does not exist an (n, 2)-arc K with Φ(K) = k. Thus k ′ , or even stronger all isomorphic copies of k ′ , are a forbidden substructure for the point class distributions of an (n, 2)-arc. This restriction on the point class distributions is not covered by Lemma 3.1 and can be used additionally for the ongoing construction of point class distributions.

Lifting point class distributions to 2-arcs in PHG(2, Z 25 )
After the generation of a possible point class distribution, the preimages in the Hjelmslev plane are built. Starting from K = ∅ we iteratively extend K point-wise, with respect to the point class distribution k and the 2-arc property. To make our computation more efficient we construct the possible point sets only up to isomorphism. For the isomorphism test we use a combination of orderly generation [24] and the ladder game, based on the homomorphism principle [16,25,17].
The orderly generation algorithm needs an assignment of the predicate canonical to exactly one representative in each orbit. The two-step approach also works for the computation of a canonical representative: Having such a predicate in the factor plane PG(2, Z 5 ) together with the homomorphism of group actions Φ, by the Homomorphism Principle we are able to arrange the definition of canonicity in N P(PHG(2,Z 25 )) in such a way that the image of a canonical point set under the mapping Φ is always canonical. Furthermore by the Homomorphism Principle the search space for the canonical candidate can be reduced: For a point set K in the preimage of a canonical point class distribution k, the canonical form of K must be in the orbit of K under the Φ-preimage of the stabilizer of k in PΓL (3, Z 5 ). This group usually is much smaller than the full group PΓL(3, Z 25 ).
These methods were implemented in C++ and executed on a single CPU of type Intel Xeon E5520. Running the program for (22, 2)-arcs in PHG(2, Z 25 ) took 8.5 hours and gave no result. Running the program for (21, 2)-arcs took 13.5 hours and returned exactly one (21, 2)-arc, the arc already found in [9]. This proves Theorem 1.1.

Canonization in AHG(2, Z 25 )
First we outline how to efficiently determine a canonical representative in AHG(2, Z 25 ) of a point set K which contains a point triple in general position (no 2 points in the same point class, no 3 points in the same line class). 6 By the map mapping (x, y) → x + 25y, computed in the integers, we can compare points. Sets of points are compared by the lexicographic ordering. The smallest point set isomorphic to K is called canonical form of K. Because In order to efficiently canonize K with respect to AGL(2, Z 25 ) we loop over all triples k 1 , k 2 , k 3 ∈ K in general position, set b = −k 1 , and uniquely , where we use the notation from subsection 2.4. 7 Thus we obtain a canonizer which needs at most #K 3 operations. We can easily extend this to a canonizer for PGL(3, Z 25 )↓ by looping over all 25 possibilities for c and determining A ′ If we have an arc K which is canonical with respect to PGL(3, Z 25 )↓ we can also revert these steps to obtain a complete list of 25 arcs which are isomorphic to K and canonical with respect to AGL(2, Z 25 ).

Classification of arcs in AHG(2, Z 25 ) by orderly generation and integer linear programming
The classification of arcs in AHG(2, Z 25 ) is explained at the example of the (20, 2)-arcs. We use an orderly generation approach (see [24]), where we utilize integer linear programming to prune the search tree. Each (20, 2)-arc K in AHG(2, Z 25 ) corresponds to a solution of the binary linear program (BLP) i∈P(AHG(2,Z 25 )) x i ≥ 20 i∈L x i ≤ 2 ∀L ∈ L(AHG(2, Z 25 )) x i ∈ {0, 1} ∀i ∈ P(AHG(2, Z 25 )), via K = {i ∈ P(AHG(2, Z 25 )) | x i = 1}. Due to symmetry and the large integrality gap, this BLP can not be solved directly using customary ILP solvers like ILOG CPLEX. To deal with the large automorphism group, one possibility would be to apply techniques e. g. from [19]. We chose another possibility: By fixing some x i to one, i. e. by prescribing some points of K, the biggest part of the symmetries is broken and we can try to solve the resulting BLP in reasonable time. Of course for the search to be complete, all the non-isomorphic possibilities of prescribing some x i must be dealt with separately.
We can utilize this approach to deduce that for all point classes if we prescribe a set K ′ of points then we only search for arcs K, where K is lexicographically minimal with respect to AGL(2, Z 25 ), i. e. where K ≤ ψ(K) for all automorphisms ψ ∈ AGL(2, Z 25 ), and where the smallest #K ′ elements of K are equal to K ′ . To this end we can formulate some necessary linear constraints: If for given K ′ there are points j ∈ K ′ , i ∈ P(AHG(2, Z 25 )) and an automorphism ψ such that ψ {i} ∪ K ′ \{j} < K ′ then we can add the constraint x i = 0 to our BLP. If there are points j 1 = j 2 ∈ K ′ , i 1 = i 2 ∈ P(AHG(2, Z 25 )) and an automorphism ψ such that ψ {i 1 , i 2 } ∪ K ′ \{j 1 , j 2 } < K ′ then we can add the constraint to the BLP. Using these constraints we were able to drastically reduce the number of lexicographically smallest sets K ′ = {k 1 , . . . , k 5 } which possibly can be extended to (20, 2)-arcs in AHG(2, Z 25 ) using the feasibility of the BLP. For each such partial arc K ′ we can easily determine a set K ′′ of points such that we have K ∩ K ′′ = ∅ for all lexicographically minimal (20, 2)-arcs K with K ′ ⊂ K. (This set consists of the i ∈ P(AHG(2, Z 25 )) where we would add x i = 0 to the BLP.) After these preparative calculations we have classified the (20, 2)-arcs in AHG(2, Z 25 ) using a branch&bound approach as follows. We start with one of the K ′ , where the points from the corresponding sets K ′′ are forbidden, as described above, and extend the partial arcs point by point by branching on the point classes [x]. (See [13] for the question where to branch.) I. e. for a given point class [x] we loop over all possibilities to extend the current partial arc with a point from [x] or to take none of the points from [x].
In the bounding step we perform two checks. If the cardinality of the partial arc K is either at most 8 or 20 we check whether there is an automorphism ψ ∈ AGL(2, Z 25 ) such that the smallest five elements of ψ(K) are lexicographically smaller than the initial K ′ , see Subsection 4.1 for the details. If such a ψ exists, we can prune the search tree (isomorphism-pruning). The maximum size of a 2-arc in AHG(2, R) is 20, this was determined computationally in [15]. Now we can give the number of such (20, 2) arcs up to different automorphism groups: In Table 1 we give the number of non-isomorphic (20, 2)-arcs in AHG(2, Z 25 ) per size of its stabilizer Aut 1 in AGL(2, Z 25 ) (collineations of AHG(2, Z 25 ) preserving the standard parallelism), Aut 2 in PGL(3, Z 25 )↓ (all collineations of AHG(2, Z 25 )) and Aut 3 in PGL(3, Z 25 ) (all collineations of PHG(2, Z 25 ), via the standard embedding). We verified that none of these arcs can be extended to a (22, 2)-arc in PHG(2, Z 25 ).
The uniqueness of K necessarily means that that all affine subsets of size 19 of the unique (21, 2)-arc k in PHG(2, Z 25 ) are isomorphic. Again this is consistent with the analysis of k given in [9]: k has maximum point class multiplicity u(k) = 1, and in the factor plane the 0-classes form a projective triangle ∆ extended by the unique fixed point under the action of the stabilizer of ∆ on the factor plane. Therefore, there are 3 possibilities for the choice of a line class at infinity, namely the 3 edges of the projective triangle. Under the stabilizer of k which has order 3, these three line classes are in the same orbit.

Number of non-isomorphic (n, 2)-arcs for large n
We used the same methods to compute further numbers of isomorphism types of 2-arcs. Wherever possible, both computational approaches were used to assure the correctness of the result. Table 3 shows the number of PGL(3, Z 25 )-isomorphism types of (n, 2)arcs in PHG(2, Z 25 ). The column s i lists the number of (n, 2)-arcs whose stabilizer has size i, and the column Σ lists the numbers of all (n, 2)-arcs. In the column "time", the running time of the algorithm described in Section 3 is given.
Remarkably, the stabilizers of large 2-arcs in PHG(2, Z 25 ) are relatively small. For comparison we mention that in PHG(2, S 5 ) there exists a (25, 2)arc whose stabilizer has order 300. This implies that using the method of prescribed automorphisms like in [11], a 2-arc of size at least 18 in PHG(2, Z 25 ) is much harder to find than a (25, 2)-arc in PHG(2, S 5 ).

The maximum size of a 2-arc of given maximum point class multiplicity
Using a combination of the computational results and geometric reasoning, we are ready to determine the maximum size of a 2-arc K in PHG(2, Z 25 ) and PHG(2, S 5 ) of a given maximum point class multiplicity u(K): Proof. Let P u be the set of all u-classes and x be a u-class.
For a line class [L], the non-increasing sequence of the multiplicities of the point classes incident with [L] is called type of [L]. The type of [L] is a partition of the multiplicity of [L] into 6 summands. In the following, we assume the usual partial order on the set of partitions.
Furthermore, types greater or equal to (3, 3, 1, 0, 0, 0) are impossible: Assume that there is a line class [L] of type (3, 3, 1, 0, 0, 0). Let x be the point of K corresponding to the entry 1 in the type. All lines of Π [L] incident with x contain at most 2 points of the induced point set k, and the line incident with x and p ∞ intersects k only in x. This gives #k ≤ 6, a contradiction.
For u = 6, it is clear that the points of K in [x] must form an oval, and no further point can be added to K. Since the type (3, 2, 2, 0, 0, 0) is not possible, at most 2 of the point classes [x i ] can be 2-classes. In total, K ≤ 2 · 3 + 2 · 2 + 2 · 1 = 12. In the case #P 3 = 1, by the restrictions on the types the line classes incident with [x] have multiplicity at most 6, so again #K ≤ 3 + 3 · (6 − 3) = 12.