Maximal arcs, codes, and new links between projective planes of order 16

In this paper we consider binary linear codes spanned by incidence matrices of Steiner 2-designs associated with maximal arcs in projective planes of even order, and their dual codes. Upper and lower bounds on the 2-rank of the incidence matrices are derived. A lower bound on the minimum distance of the dual codes is proved, and it is shown that the bound is achieved if and only if the related maximal arc contains a hyperoval of the plane. The binary linear codes of length 52 spanned by the incidence matrices of 2-(52, 4, 1) designs associated with previously known and some newly found maximal arcs of degree 4 in projective planes of order 16 are analyzed and classified up to equivalence. The classification shows that some designs associated with maximal arcs in nonisomorphic planes generate equivalent codes. This phenomenon establishes new links between several of the known planes. A conjecture concerning the codes of maximal arcs in PG(2, 2m) is formulated. Mathematics Subject Classifications: 05B05,51E20,94B05


Introduction
We assume familiarity with the basic facts and notions from design theory, finite geometries, and coding theory [3,5,10,13,23].
A 2-(v, k, λ) design (or shortly, a 2-design) is a pair D={X, B} of a set X of v points and a collection B of subsets of X of size k called blocks, such that every two points appear together in exactly λ blocks. Every point of a 2-(v, k, λ) design is contained in r = λ(v − 1)/(k − 1) blocks, and the total number of blocks is b = v(v − 1)λ/k(k − 1).
The incidence matrix of a design D is a (0, 1)-matrix A = (a ij ) with rows labeled by the blocks, columns labeled by the points, where a i,j = 1 if the ith block contains the jth point, and a i,j = 0 otherwise. If p is a prime, the p-rank of a design D is the rank of its incidence matrix over a finite field of characteristic p. Two designs are isomorphic if there is a bijection between their point sets that maps every block of the first design to a block of the second design. An automorphism of a design is any isomorphism of the design to itself. The set of all automorphisms of D form the automorphism group Aut(D) of D.
The dual design D ⊥ of a design D has as points the blocks of D, and as blocks the points of D. A 2-(v, k, λ) design is symmetric if b = v, or equivalently, r = k. The dual design D ⊥ of a symmetric 2-(v, k, λ) design D is a symmetric design with the same parameters as D. A symmetric design D is self-dual if D and D ⊥ are isomorphic.
A design with λ = 1 is called a Steiner design. An affine plane of order n (n 2), is a Steiner 2-(n 2 , n, 1) design. A projective plane of order n is a symmetric Steiner 2-(n 2 + n + 1, n + 1, 1) design with n 2. The classical (or Desarguesian) plane P G(2, p t ) of order n = p t , where p is prime and t 1, has as points the 1-dimensional subspaces of the 3-dimensional vector space V 3 over the finite field of order p t , and as blocks (or lines), the 2-subspaces of V 3 .
A parallel class of D is a set of v/k = n pairwise disjoint blocks, and a resolution of D is a partition of the collection of blocks B into r = (v − 1)/(k − 1) = sk + 1 disjoint parallel classes. A design is resolvable if it admits a resolution. Any 2-((sk − s + 1)k, k, 1) design with s = 1 is an affine plane of order k, and admits exactly one resolution. If s > 1, a resolvable 2-((sk − s + 1)k, k, 1) design may admit more than one resolution.
Let P be a projective plane of order q, and let m and k be positive integers such that k q + 1 and k m q 2 + q + 1. An (m, k)-arc (or an arc of size m and degree k) is a set A of m points such that every line of P contains at most k points from A. An arc of degree 2 is also called an oval. Let x be a point in an (m, k)-arc A, let L 1 , . . . , L q+1 be the lines through x, and let n i = |A ∩ L i |, 1 i q + 1. Then Since n i k, it follows that m qk + k − q, and the equality m = qk + k − q holds if and only if every line of P is either disjoint from A or meets A in exactly k points. An (m, k)-arc is called maximal if m = qk + k − q. A hyperoval is a maximal arc of degree 2. Let A be a maximal (qk + k − q, k)-arc with k q, thus m = qk + k − q q 2 , and let y be a point outside A. Let n be the number of lines that meet A in k points. We have Thus, k divides q, and q = sk for some integer s 1 1 . If q = sk, the size qk + k − q of a maximal arc of degree k can be written in terms of s and k as (sk − s + 1)k.
The set of lines that are disjoint from a maximal ((sk −s+1)k, k)-arc A in a projective plane P of order q = sk, form a maximal ((sk − k + 1)s, s)-arc A ⊥ in the dual plane P ⊥ , called the dual arc of A.
If k > 1, the non-empty intersections of a maximal ((sk − s + 1)k, k)-arc A with lines of a projective plane P of order q = sk are the blocks of a resolvable 2-((sk − s + 1)k, k, 1) design D. Similarly, if s > 1, the dual ((sk − k + 1)s, s)-arc A ⊥ in the dual plane P ⊥ is the point set of a resolvable 2-((sk − k + 1)s, s, 1) design D ⊥ . We will refer to D (resp. D ⊥ ) as a design embeddable in P (resp. P ⊥ ) as a maximal arc.
Two maximal arcs A , A in a projective plane P are equivalent if there is a collineation of P that maps A to A . Designs associated with equivalent arcs are necessarily isomorphic, while the converse is not true in general.
Let D be a resolvable Steiner 2-(v, k, 1) design. Two resolutions R 1 , R 2 of D, where are called compatible [36], if they share one parallel class, P j , and The following theorem gives an upper bound on the number of pairwise compatible resolutions of a resolvable 2-((sk − s + 1)k, k, 1) design, and characterizes the designs for which this upper bound is achieved.  holds if and only if there exists a projective plane P of order q = sk such that D is embeddable in P as a maximal ((sk − s + 1)k, k)-arc.
In Section 2, we consider binary linear codes spanned by the rows of incidence matrices of Steiner 2-designs associated with maximal arcs in projective planes of even order, and their dual codes. Upper and lower bounds on the 2-rank of the incidence matrices are derived. A lower bound on the minimum distance of the dual codes is proved, and it is shown that the bound is achieved if and only if the related maximal arc contains a hyperoval of the plane.
In Section 3, we analyze the binary linear codes of length 52 spanned by the incidence matrices of 2-(52, 4, 1) designs associated with maximal arcs of degree 4 in projective planes of order 16. The codes associated with maximal arcs in P G(2, 16) are distance optimal, while one code associated with an arc in the semi-field plane is shown to be optimal with respect to each of its parameters: minimum distance, dimension, and length. A conjecture concerning the codes of maximal arcs in P G(2, 2 m ) is formulated.
The codes are classified according to their dimension, and all codes having the same dimension are further classified up to equivalence. The classification shows that some designs associated with maximal arcs in nonisomorphic planes generate equivalent codes. This phenomenon establishes new links between several of the known planes that are discussed in Section 4.
Section 5 lists explicitly eleven new maximal arcs of degree 4 found recently in the planes DEMP, SEMI2, LMRH, HALL, and BBH1.

Binary codes of designs arising from maximal arcs
An oval of a Steiner 2-(v, k, 1) design D with k 2 is a set S that meets every block in at most two points. Proof. Let n i (i = 0, 1, 2) denote the number of blocks meeting S in i points. Counting in two ways the incident pairs of points from S and blocks, and incidence pairs of pairs of points from S and blocks, we have and the statement follows.
An oval of size r + 1 is called a hyperoval.
the electronic journal of combinatorics 27(1) (2020), #P1.62 Note 3. If D is a symmetric Steiner design, that is, a projective plane, a hyperoval is simply a maximal arc of degree two.
Theorem 4. Let C be a binary linear code of length n = 2 m+s − 2 m + 2 s , spanned by the rows of the incidence matrix A of a Steiner 2-(2 m+s − 2 m + 2 s , 2 s , 1) design D with m s 1.
The dual code C ⊥ admits majority-logic decoding that corrects up to t = 2 m−1 errors. (iii) The minimum distance d ⊥ of C ⊥ is an even number equal to 2 m + 2 if D contains hyperovals, and d ⊥ 2 m + 4 if D has no hyperovals.
(iv) The minimum distance d of C is an even number smaller than or equal to 2 s .
(v) The dimension k of C, or equivalently, the 2-rank of A, rank 2 A, satisfies the inequalities Proof. (i) Since all rows of A are of even weight 2 s , every row is orthogonal to1 over GF (2), hence1 ∈ C ⊥ . Every point of D is contained in r = 2 m + 1 blocks. Thus, every column of A contains 2 m + 1 nonzero entries, and the binary sum of all rows of A is equal to1, hence1 ∈ C.
(ii) The rows of A provide a set of checks that can be used to correct up to errors in C ⊥ by using the Rudolph majority-logic decoding algorithm (cf. [33], [37, Theorem 8.1, page 1252]).
(iii) It follows from (ii) that d ⊥ 2t + 1 = 2 m + 1. Assume that d ⊥ = 2 m + 2, and let S be the support of a minimum weight codeword in C ⊥ . Clearly, S meets every block of D in an even number of points. Let n 2i denote the number of blocks that meet S in 2i points, 0 i 2 s−1 . Counting in two ways the occurrences of single points, and ordered pairs of points of S in blocks of D, we have Subtracting the first of the above equations from the second gives the electronic journal of combinatorics 27(1) (2020), #P1.62 hence n 2i = 0 for i > 1, and S is a hyperoval of D. Consequently, by Lemma 2, the number of codewords of C ⊥ having weight 2 m + 2 is equal to the number of hyperovals of D. If D has no hyperovals, then d ⊥ > 2 m + 2, and since all weights in C ⊥ are even by part (i), it follows that d ⊥ 2 m + 4.
Part (iv) is obvious. The upper bound in Part (v) follows from applying the sphere packing bound (cf., e.g. [24, 1.12]), to a punctured [n − 1, k, d − 1] code C of C. The lower bound in Part (v) follows from applying the sphere packing bound to a punctured Note 5. According to Theorem 4 (ii) and (iii), if D contains hyperovals then the dual code C ⊥ can correct the maximum number of errors guaranteed by its minimum distance, by using majority-logic decoding, and in addition, the number of codewords in C ⊥ having minimum weight is equal to the number of hyperovals of D.
As a corollary of Theorem 4, we have the following.
Theorem 6. Let C be a binary code of length n = 2 m+s − 2 m + 2 s , spanned by the incidence matrix A of a Steiner 2-(2 m+s − 2 m + 2 s , 2 s , 1) design D with m s 1, associated with a maximal arc S of degree 2 s in a projective plane Π of even order q = 2 m . Let d and d ⊥ denote the minimum distance of C and C ⊥ respectively.
The minimum distance d of C is an even number smaller than or equal to 2 s . If d = 2 s then 3 Binary codes associated with maximal arcs of degree 4 A Steiner design D associated with a maximal arc of degree 4 in a projective plane of order 2 m has parameters 2-(3·2 m +4, 4, 1), and its binary code C is of length n = 3·2 m +4 and has minimum distance d = 2 or d = 4. In this section, we consider 2-(3 · 2 m + 4, 4, 1) designs and their binary codes when m = 2, 3 and 4.
If m = 2, D is the unique (up to isomorphism) 2-(16, 4, 1) design, being isomorphic to the affine plane of order 4, and can be viewed as the design associated with a maximal arc of degree 4 in the projective plane of order 4. The 2-rank of its incidence matrix A is 9, hence the code C spanned by A has dimension 9. The minimum distance of C is 4, and there are exactly 20 codewords of minimum weight 4, being the rows of the incidence matrix A. The design D contains ovals, hence d ⊥ = 6 by Theorem 4, (iii). The lower bound, and the upper bound on rank 2 A from Theorem 4, are 8 and 11 respectively. The code C is distance optimal in the sense that 4 is the largest possible minimum distance for a binary linear code of length 16 and dimension 9. The dual [16,7,6] code C ⊥ is also distance optimal (see [16]).
In the next case, m = 3, the design parameters are 2- (28,4,1). There are at least 4653 known nonisomorphic designs with these parameters [4,26], having 2-ranks ranging from 19 to 26 [4], with the exception of 2-rank 20. It was shown in [25] that there are no 2-(28, 4, 1) designs with 2-rank 20. It was proved in [29] that the minimum 2-rank of any 2-(28, 4, 1) design is 19, and up to isomorphism, there is only one design of minimum 2-rank 19, being isomorphic to the design D associated with a maximal (28, 4)-arc in the projective plane of order 8, P G(2, 8) (also referred to as the Ree unital [7]). The minimum distance of the code C of the Ree unital D is 4, and every codeword of minimum weight corresponds to a block of D. The Ree unital contains ovals that are hyperovals in the projective plane of order 8, hence the minimum distance of the dual code C ⊥ is d ⊥ = 10, and C ⊥ can correct up to (10 − 1)/2 = 4 errors by majority-logic decoding. The [28,19,4] code C and its dual [28,9,10] code C ⊥ are both distance optimal (cf. [16]).
If m = 4, the parameters of the design D from Theorems 4 and 6 are 2-(52, 4, 1), and correspond to a design arising from a maximal (52, 4)-arc in a projective plane of order 16. Up to isomorphism, there are 22 known projective planes of order 16. The only projective plane of order 16 for which all inequivalent maximal arcs of degree 4 have been completely classified, is the Desarguesian plane P G(2, 16). There are exactly two inequivalent maximal (52, 4)-arcs in P G(2, 16) [1], with collineation groups of order 68 and 408. The maximal arc with group of order 408 admits a cyclic collineation of order 51 [12]. The maximal arcs of degree 4 have not been classified completely in any of the 21 known non-Desarguesian planes of order 16. Maximal arcs of degree 4 have been found in all but four of the known non-Desarguesian planes of order 16 [14,15,22]. All previously known maximal (52, 4)-arcs can be found in [15] and [22]. Eleven new maximal arcs of degree 4 are given in Section 5 of this paper. The line sets of the projective planes of order 16 and all known maximal arcs of degree 4, including the new arcs described in Section 5, are available online at https://www.combinatorics.org/ojs/index.php/eljc/article/view/ v27i1p62/8052 and https://www.combinatorics.org/ojs/index.php/eljc/article/ the electronic journal of combinatorics 27(1) (2020), #P1.62 view/v27i1p62/8051 respectively. Table 2 contains data about the 2-(52, 4, 1) designs associated with previously known and the eleven newly found maximal arcs of degree 4 described in Section 5. Column 3 lists the orders of the stabilizers of the maximal arcs, which happen to be also the full automorphism groups of the related designs. Column 4 gives the number of parallel classes of the design D associated with the given arc, followed by the number of parallel classes of the design D ⊥ associated with the dual arc. Column 5 lists the number of resolutions of D and D ⊥ . The last column lists the orders of the automorphism groups of the codes C(D) and C(D ⊥ ) respectively. Cliquer [31] and Magma [8] were used for these computations.
Since the number of parallel classes, resolutions, code dimension and minimum distance are invariant under isomorphism of designs, the data from Table 2 implies that the number of pairwise nonisomorphic resolvable 2-(52, 4, 1) designs associated with maximal arcs is greater than or equal to 55. Further computation of possible isomorphisms using Magma [8] shows that the number of nonisomorphic designs is exactly 55.
We note that the previously known lower bound on the number of nonisomorphic resolvable 2-(52, 4, 1) designs given in [10] is 30.
The 2-ranks of these 55 designs vary from 41 to 49, and the minimum 41 is achieved only by designs associated with maximal arcs in the Desarguesian plane P G (2,16).
The codes of the 55 nonisomorphic 2-(52, 4, 1) designs arising from maximal (52, 4)arcs were sorted according to their weight distributions, and codes having the same weight distribution were tested for equivalences using Magma. This classification shows that the 55 codes of the 55 nonisomorphic 2-(52, 4, 1) designs are partitioned into 27 equivalence classes. Table 3 lists the 2-ranks of the designs, and gives a partition of the codes into equivalence classes, the numbers A 2 and A 4 of codewords of weight 2 and 4 respectively, and the orders of the automorphism groups of the codes.
The main result implied by this classification of the codes up to equivalence is the surprising fact that in several instances the codes of designs arising from maximal arcs in different planes are equivalent, hence these codes provide new connections between the corresponding planes (see lines 4, 5, 6, 11, 13, 17, 18, 20, 24, 26 and 27 in Table  3). For example, the Mathon plane MATH is linked to the Johnson plane JOHN, the Lorimer-Rahilly plane LMRH, the semi-field plane SEMI2, and the Hall plane HALL. Table 4 lists the parameters of the codes and their dual codes, as well as the number A d ⊥ of codewords of minimum weight in C ⊥ . We note that according to Theorem 6, in the cases when d ⊥ = 18, A d ⊥ is equal to the number of hyperovals of the plane contained in the point set of D.
Four of the 55 codes associated with these designs have minimum distance d = 4: the codes C , C of the two maximal arcs in P G (2,16), and the codes C and C iv of dimension 45 associated with the maximal arcs SEMI4.1 and SEMI2.7 (see Table 3 and Table 4). Every codeword of minimum weight in C and C is the incidence vector of a block, while the number A 4 = 4469 of minimum weight codewords in C and C iv is larger than the number of blocks. These observations suggest that the following statement may be true in general.
the electronic journal of combinatorics 27(1) (2020), #P1.62 Conjecture 8. The binary code spanned by the incidence matrix A of a design D associated with a maximal arc of degree 4 in P G(2, 2 m ), where m 2, has minimum distance 4, and every codeword of minimum weight is a row of A.
We note that the codes C and C iv are equivalent (see Table 3 that lists the equivalence classes of codes), while the codes C , C of the two arcs in P G (2,16) are inequivalent, despite the fact that both codes have identical weight distributions. It follows from [12] that one of these codes, namely the code with full automorphism group of order 408, is an extended cyclic code. Both (52, 4)-arcs in P G(2, 16) contain hyperovals, hence d(C ) ⊥ = d(C ) ⊥ = 18 by Theorem 6. In addition, the number of codewords of minimum weight A d ⊥ in each of the codes (C ) ⊥ and (C ) ⊥ equals the number of hyperovals contained in the corresponding maximal arcs, namely A d ⊥ = 54 (see Table 4).
A comparison with the parameters of best known error-correcting codes [16] shows that the highest known minimum distance of a binary code of length 52 and dimension 41 is d = 4, while the best known theoretical upper bound is d 5.
In comparison, the [52, 45, 4] code C of the arc SEMI4.1 is distance optimal. The dimension 45 of C is just by one shorter from the upper bound 46 obtained from Theorem 6. However, the minimum distance of any binary [52, 46] code is at most 3 (which is seen by applying the sphere packing bound on a punctured [51, 46, 3] code), therefore the [52, 45, 4] code C is both distance and dimension optimal. This implies that if the binary code of a 2-(52, 4, 1) design D has minimum distance 4, the 2-rank of D cannot exceed 45. In addition, since a binary [51, 45, 4] code does not exist, because the parameters of a punctured [50, 45, 3] code violate the sphere packing bound, the [52, 45, 4] code C is also length optimal, that is, 52 is the smallest possible length for a binary code of dimension 45 and minimum distance 4. Table 1 lists all previously known connections, as well as some new connections described in this section, between nonisomorphic projective planes of order 16. An entry in a row and a column labeled by the same (non self-dual) plane indicates a connection between the plane and its dual plane that is based on designs associated with maximal arcs or their codes.

New connections between projective planes
An entry 1 indicates that the corresponding planes are connected by derivation [22,30], 2 indicates that the corresponding planes are connected by super-derivation [27], and 3 indicates that the planes share a semi-biplane [30].
An entry 5 indicates a new connection between a pair of nonisomorphic planes that share equivalent codes of non-isomorphic 2-(52, 4, 1) designs arising from maximal (52, 4)arcs.

New maximal (52, 4)-arcs
The specific line sets of the known projective planes of order 16 that we are using in this paper were graciously provided to the third author by Gordon F. Royle, and are available online at https://www.combinatorics.org/ojs/index.php/eljc/article/ view/v27i1p62/8052. All previously known maximal arcs of degree 4 are given in [15] and [22], and are available online at https://www.combinatorics.org/ojs/index.php/ eljc/article/view/v27i1p62/8051.
Our notation in this section follows [15]. Recently, the first author found eight new maximal arcs in some of the projective planes of order 16. The new arcs are unions of orbits of appropriate subgroups of the automorphism group of the corresponding plane. Two new arcs, stabilized by a nonabelian group of order 12, were found in the plane DEMP. Five new arcs in the plane SEMI2, and one new arc in the plane LMRH, were found as unions of orbits under subgroups of order 16 A probabilistic search algorithm developed by the second author was used to find three new maximal arcs in the planes DEMP, HALL and BBH1. Each of these new arcs has a stabilizer of order 4, so it is computationally unfeasible to find these arcs with the previous method that works well for stabilizers of order at least 12.
A randomized local search was performed to find sets in projective planes with prescribed line intersections. At the start of a new experiment, a set of points is selected at random. At each move, a neighborhood of the current subset, defined by a single interchange between a point in the set and a point outside, is examined in order to generate a list of swaps that minimize an objective function formed from the actual and desired line intersections, respectively. A locally optimal move is then randomly selected from that list and the set updated to reflect it. A tabu list of the most recent moves is maintained to prevent cycling, and occasional random moves are performed to improve the search efficiency. This search algorithm worked well for projective planes of order 16 and reproduced all the known (52,4)-arcs with frequencies inversely proportional to their group orders. We have run typically 10 4 experiments with 5 × 10 4 moves per experiment on a MacBook Pro. The same approach was used to verify the existence of 4-arcs P G (2,32) in [28].
The point sets of these three new maximal arcs are:  181,190,191,192,197,205,207,208