On KM-arcs in Small Desarguesian Planes

In this paper we study the existence problem for KM-arcs in small Desarguesian planes. We establish a full classification of KMq,t-arcs for q 6 32, up to projective equivalence. We also construct a KM64,4-arc; as t = 4 was the only value for which the existence of a KM64,t-arc was unknown, this fully settles the existence problem for q 6 64.

1 Introduction and Preliminaries Definition 1.An incidence structure I is a triple (P, L, * I ), where P, L are sets and * I ⊆ P × L.More often than not, L is a collection of subsets of P (called lines) and * is the inclusion relation: p * I L ⇔ p ∈ L for any p ∈ P, L ∈ L. In such the elements of P are called points.Definition 2. The dual of I, denoted by I D , is the incidence structure (L, P, * I D ) where the incidence is preserved (i.e.(L, p) ∈ * I D ⇔ (p, L) ∈ * I ).The set L is now the set of points, and P the set of lines.
A large area of research is devoted to studying substructure of PG(2, q) and AG(2, q) with certain combinatorial properties.One substructure that has gotten a lot of attention in PG(2, q) is the following one.Definition 3. A hyperoval in PG(2, q) is a nonempty set S of points, such that every line is incident with 0 or 2 of points of S. One readily computes that a hyperoval has q + 2 points.
In this paper, we study a slight relaxation of this definition, to the following concept.Definition 4 ( [7]).A KM q,t -arc in PG (2, q), also known as a (q + t, t)-arc of type (0, 2, t), is a set S of q + t points in PG(2, q) for which every projective line meets S in either 0, 2 or t points.
The case t = 1 is a degenerate case where S is just any arc.Hence, t > 1 will almost always be assumed.The case t = q is fully classified; here the only example is the symmetric difference of two lines.Hence, for most purposes it is sufficient to study the the case 1 < t < q.
Definition 4 was introduced in [7] and there are several reasons why these structures are of interest.The first reason is that strong structural properties can be derived from this combinatorial definition.

Theorem 5 ([7]
). KM q,t -arcs of type (0, 2, t) with 1 < t < q can only exist if q is even.Moreover, t needs to be a divisor of q, i.e. t = 2 r with r h.

Theorem 6 ([4]
).All t-secants of a KM q,t -arc with t > 2 are concurrent in a point outside the set, which is called the nucleus.
Hence, every KM q,t -arc S with t > 2 has the following structure: • there are q/t + 1 concurrent lines, each containing t points of S; • all other lines contain 0 or 2 points of S.
It is interesting that such a strong structure follows from a combinatorial definition based on three intersection possibilities (0, 2 and t).Usually, such strong properties are only found for sets with two possible intersection numbers.One reason for this -which at the same time is our second motivation -is that they only have two possible intersection numbers when embedded in AG(2, q) D .Proposition 7. A set S in PG(2, q), PG(2, q) D or AG(2, q) is a set of type (0, 2) if and only if it consists of q + 2 points, no three collinear (i.e. it is a hyperoval).Proposition 8.A set S in AG(2, q) D is a set of type (0, 2) if and only if, when embedding AG(2, q) D in PG(2, q), S is either a hyperoval or a KM-arc with the point at infinity as its nucleus.
Proof.Let S be KM-arc.If t = 2, the statement is trivial.For t > 2, it follows from Theorem 6 that S is a hyperoval in AG(2, q) D , where the point of concurrency in the KM-arc is the point at infinity of AG(2, q) D .
For the reverse implication, let S be a hyperoval in AG(2, q) D and let p be any affine point not in S and let L be its line at infinity.Every affine line through p needs at least the electronic journal of combinatorics 24(1) (2017), #P1.51 Table 1: Number of projective equivalence classes of KM q,t -arcs in PG(2, q), q 64.one other point of S through it, meaning that the total number of points of S is q + t, where t 2 is the number of points of S on L. Since L was arbitrary, and the total number of points of S is fixed, that means the number t is independent of the choices of p and L, and every line through the point at infinity has either no points of S on it, or has exactly t points of S through it.Hence, S is a KM-arc.
While the above is largely a reformulation of Theorem 6, it does highlight better why these KM-arcs are special and have stronger structural properties than other sets with three intersection numbers.
The third reason is that they have been shown [1,12,11] to be crucial in the structure of the dual projective plane code, and of LDPC codes derived from certain partial geometries, where (non)existence of certain KM-arcs is relevant for their dimension and minimum distance, respectively.
The main challenge regarding these KM-arcs is construction and classification.We know that nontrivial examples only exist when q = 2 h and t = 2 r with 1 r h, but the converse is not known.Classification is known only up to q = 8 [7] and existence is known only by a few (families of) constructions1 , which together settle the existence problem for q 32.
• In [4], a KM 2 h ,2 r+1 was constructed when h − r|h.The same paper also provides a construction for KM 2 h ,2 r+m when h − r|h and a KM 2 h−r ,2 m exists.
• In [8], a KM 32,8 was constructed via random search as part of Limbupasiriporn's PhD thesis.
• In [6], the authors describe a clever random search to construct a KM 32,4 as a union of subsets of conics, and succeeded in finding such an example.
The reason the random searches in [6,8] are so important, is because they show that for q = 32, all proper divisors t yield KM q,t -arcs.Table 1 lists the number of known projective equivalence classes of KM q,t -arcs.Only the most recent paper changing the bounds on each number is listed.
In Section 2, we will describe our method to obtain a full classification of KM q,t -arcs for q 32, as well as the results.In Section 3 we discuss the new findings, as well as several observations and pattens.In Section 4, we use one of these patterns to perform a targeted search that has lead to the construction of a KM 64,4 -arc, solving the last remaining open case in Table 1.
2 Classifying the KM-arcs in PG(2, q) for q 32 To settle the notation, we recall the concept of associated polynomial from [7].

Theorem 9 ([7]
).A set S in PG(2, q) is a KM q,t -arc if and only if it can be written in the following form: and where f a polynomial with the following properties: • f reaches every value in F q either 0 or t times; • f is monic and has f (0) = 0; z+a acts injectively on its set of non-roots in F q ; • for w ∈ W \ {0}, w −1 + F a (z) has no roots in F q \ {a}.
We can now describe KM-arcs by their associated polynomial.Note however that multiple associated polynomials can describe equivalent KM-arcs.It is also worth remarking that the properties of an associated polynomial from 9 look very similar to those of o-polynomials of hyperovals.
Remark 10.In [7,Theorem 2], the conditions on the second bullet are optional, and in the original version from [7] also contained f (1) = 0.However, sometimes one can obtain easier polynomials when not requiring this condition.We will always list the easiest associated polynomial with f (0) = 0 and 0 ∈ W , where easiest is defined below.Notation 11.Define a function ϕ : {0, 1, . . ., q − 1} → F q that represents the lexicographical order of the elements in additive notation: When choosing an associated polynomial to represent the KM-arc, we will always pick the one with smallest degree (and in case of equal degree, the lexicographically smallest one w.r.t.ϕ) among all possible associated polynomials.
Remark 12.In the case t = 2, the definition of associated polynomial is slightly different from the classical definition for hyperovals (the difference is a term +z), because of a slightly different coordinate system.Now, we will discuss all cases that need to be considered.Some cases have already been dealt with.
Proposition 13.A KM q,q -arc is always projectively equivalent to x acts transitively on the set of KM-arcs found in [12] (as they are cosets of hyperplanes), resulting in the following proposition.
For planes of order up to 32, the remaining cases to classify are hence (q, t) ∈ {(16, 4), (32, 4), (32, 8)}.Since the classification will be computational, one needs to fix the primitive polynomials of the fields for the coordinates to make sense.For F 16 we pick X 4 + X + 1 as primitive polynomial, for F 32 we pick X 5 + X 2 + 1.
We will now outline the algorithm used for our classification result.First we fix the nucleus N = (0, 0, 1).Let L N be the set of lines through N .Consider the set P(L N ) q t +1 of all subsets of L N of size q t + 1.Now, we will partition this set into orbits under the PΓL(3, q) N , the stabilizer of N in the collineation group of PG(2, q).To make this computationally feasible, we used a breadth-first backtracking search to find a representative for each PΓL(3, q) N -orbit of subsets of P(L N ) of size 1, 2, 3, . . ., q t + 1 in that order, where each size is obtained by considering all possibilities to extend the previous size, and only keeping a unique canonical representative per orbit.
We denote by L N the set of representatives obtained, i.e. a set of sets of q t + 1 lines.Since N must be a fixed point of every collineation in the automorphism group of any KMarc, different elements S 1 , S 2 ∈ L N represent different orbits under PΓL(3, q) N , and hence a KM-arc having S 1 as its set of t-secants (with t > 2) is always projectively inequivalent to a KM-arc having S 2 as its set of t-secants.Hence, this splits the problem in disjoint subproblems.
Now, for any given such line set L ∈ L N , let S L = ∅ and pick an arbitrary line L ∈ L (computationally, it is wise to pick one with the smallest orbit size under PΓL(3, q) L , but that is not mandatory).Now consider the set T L of all PΓL L,L -inequivalent t-sets on L. For each T ∈ T L we use a backtracking-based constraint solver to find the possible placings of the remaining q points on the lines of L, such that the KM-arc properties are satisfied.In the vast majority of the cases this solver only requires milliseconds to determine that there will be no solutions for this T .When a solution S is found, it is tested explicitly for projective equivalence with every element in S L , and if no equivalent element is found, S is added to S L .
Then, at the end, is the set of all KM q,t -arcs up to projective equivalence.For q = 16, nothing surprising was found.The three projective equivalence classes of KM 16,4 -arcs found in [7] are the only existing classes of KM 16,4 -arcs, which could be expected given that no new constructions had been found despite the very small order of the plane.
• f (z) = z 8 + z 2 , its automorphism group has order 3840 and acts transitively on its points.
For q = 32, t = 8 the search found one class of translation arcs (the first one below) and two classes of non-translation arcs.All three can be constructed from [2, Theorem 4.6].
Result 20.There are exactly 3 projective equivalence classes of KM 32,8 -arcs.One representative of each class is given below.
For q = 32, t = 4 the search did find new results results.Despite t = 4 being the case that has been the hardest to find examples for (only one example known for q = 32, constructed by random search, and no examples known for q = 64), we found no less than eight projective equivalence classes of KM 32,4 -arcs, and none of them are translation KM-arcs.

Observations
Remark 22.It is remarkable that while (for the same value of q) examples for smaller t are remarkably harder to construct, the number of examples is definitely not smaller.The examples indeed get less symmetrical, but their number increases.
Remark 23.In [12], the author observes that all t-secants in his example stem from (affine) linear sets, and expresses belief that this could hold in general.In the coordinates used in Section 2, that means if we partition the arc in q t + 1 sets of size t based on which t-secant they belong to, and replace each point by its last coordinate (since the first two coordinates are identical for all points in the partition class), each class is now a coset of an additive subgroup of F q .We verified this property and now confirm that this holds for all KM q,t -arcs with q 32.Unfortunately, we did not manage to find a similar pattern in the coordinates of the secant lines.A criterion (even conjectured) that would eliminate a large portion of the possible sets of t-secants, or a pattern that all of the sets of t-secants fulfill, would be extremely helpful in constructing larger new KM-arcs.
Remark 24.In [2], the authors observe the property that the KM q,q/4 -arcs they studied, had the following property.Label the t-secants of S as L 0 , L 1 , L 2 , L 3 , L 4 .Then for any such labeling, the electronic journal of combinatorics 24(1) (2017), #P1.51 are well defined (the equality holds), and either in which case they say L 0 has Property II, or in which case they say L 0 has Property I. Our search found the following.
• For the second and third class in (32, 8) and for the second class in (16, 4), all t-secants have Property II.
• For the third class in (16, 4), all t-secants have property I.
Moreover, we discovered that a stronger result holds.Remark 22 states that the nonnucleus points on each line have a natural correspondence to the additive group (F q , +) by the last coordinate of the points.In this group, it turns out that for all KM 16,4 -arcs and all KM 32,8 -arcs, the subsets corresponding to • S 1 , S 2 , S 3 (for t-secants L of type I) • S 1 ∩ S 2 , S 1 ∩ S 3 , S 2 ∩ S 3 (for t-secants L of type II) are exactly the three cosets of the additive subgroup corresponding to S ∩ L 0 .Food for thought.
Remark 25.When we consider the subgroup of the automorphism group which stabilizes every single t-secant, all KM q,t -arcs with q 32 yield an elementary abelian 2-group, i.e.
The action of this group within each t-secant is closely linked to the linearity property above, but is not a requirement for it: even for the KM 32,4 -arcs with automorphism group order 2 the linearity property still holds, despite the absence of a subgroup of order 4 to act on the points within each t-secant.
Remark 26.Let f (z)− q−1 i=0 a i z i be the associated polynomial of a KM-arc.Let ν(i) be the number of 1s in the base 2 representation of the integer i. Denote by Then, [7, Proposition 6.3] states that a KM-arc with associated polynomial f is a translation KM-arc if and only if f = f 1 (i.e. it is a linearized polynomial).Non-translation KM-arcs with t = q/4, as well as some KM 32,4 arcs with relatively large automorphism group, have f = f 1 + f 2 .Hence, the limited data we have suggests that it may be interesting to study KM-arcs with f = f 1 + f 2 , as we may be able to find such a KM q,t -arc for any q, t with t a proper divisor of q = 2 h .Remark 27.The first example for (16, 4) and (32, 4) share a peculiar property.The set of values obtained by the associated polynomial f is respectively {ϕ(i)|i ∈ {0, . . ., 3}} and {ϕ(i)|i ∈ {0, . . ., 7}}.This is the key observation for the construction in Section 4.
4 Construction of a KM 64,4 -arc Since we have no clear requirements on which line sets L can yield KM-arcs (or equivalently, what sets can be the image of the associated polynomial of a KM-arc), and since iterating over all possible 65 17 line sets of size 17 through the nucleus is not feasible on today's computer hardware (even up to isomorphism), a full classification is not possible for q = 64 with our techniques.However, in this section we will present a partial search, solving the last open case in Table 1.
We applied the following simplifications of the search, to allow the search to finish within a reasonable time.We only look for KM-arcs which satisfy the linearity constraint.This reduces the search space within each lines from q t to roughly q log 2 (t)+1 sets.Moreover, we will not look at all projectively inequivalent sets of secants, but we will only look at one specific set L of lines through the nucleus, which we select based on the following observation.
Since there are only 4 such line sets up to projective equivalence in PG(2, 64), it is computationally feasible to try them all, and one of these sets effectively lead to a new KM-arc.We obtained the following result.
Result 31.The polynomial f (z) = z 48 + z 34 + z 20 + z 16 + z 6 + z 4 + z is an associated polynomial of a KM 64,4 -arc, whose automorphism group has order 192 and partitions its points in orbits of sizes 32, 32, 4.This result, combined with the results from Section 3, yields the updated results in Table 2.
Its α-powers consist of two arithmetic progressions (one with common difference 9 and one with common difference 21); and one geometric progression (with common ratio 2).