THE NON-EXISTENCE OF (104 , 22; 3 , 5) -ARCS

. Using some recent results about multiple extendability of arcs and codes, we prove the nonexistence of (104 , 22)-arcs in PG(3 , 5). This implies the non-existence of Griesmer [104 , 4 , 82] 5 -codes and settles one of the four remaining open cases for the main problem of coding theory for q = 5 ,k = 4 ,d = 82.


Introduction
A central problem in coding theory is to optimize one of the three main parameters of a linear code over a fixed field F q given the other two [7]. The most popular version of this problem is to find the minimal length, denoted by n q (k, d), of a linear code for fixed values of the dimension k and the minimum distance d. There is a well-known lower bound on n q (k, d), due to Griesmer [5,15]: Codes meeting the Griesmer bound are called Griesmer codes. For fixed k and q, Griesmer codes exist for all sufficiently large d [6]. Thus the problem of determining the exact value of n q (k, d) for fixed q and k for all d is a finite one. It has been solved in the following cases: • for q = 2, k ≤ 8 for all d; • for q = 3, k ≤ 5 for all d; • for q = 4, k ≤ 4 for all d; • for q = 5, 7, 8, 9, k ≤ 3 for all d.
In the case of q = 5, k = 4, there exist four values of d for which n q (k, d) is not decided. They are given in the table below. In this paper, we present a new method for proving the non-existence of arcs and codes with prescribed parameters which provides a solution of certain instances of the main problem of coding theory. The method is based on the extendability of a special family of codes called quasidivisible codes [10]. In particular, we prove the non-existence of [104, 4, 82] 5 -codes, which implies that n 5 (4, 82) = 105 and settles one of the four open cases for q = 5, k = 4.
The approach to this problem is geometric. It is known that the existence of a linear [n, k, d] q code is equivalent to that of a (multi)arc with parameters (n, n − d) in PG(k − 1, q) [3]. We consider a hypothetical (104, 22)-arc K in PG (3,5). It is known (cf. [13]) that an arc with these parameters is not extendable. This implies certain restrictions on a special arc in the dual space related to K. The existence of the latter is ruled out using recent results from [9,10].
The paper is structured as follows. In section 2, we recall some basic definitions on arcs in PG(k − 1, q). We introduce the notions of induced and dual arc and prove a necessary condition for the existence of an arc with given parameters based on analogues of the MacWilliams identities for arcs. At the end of the section, we present the spectra of some arcs in PG (2,5) that are needed in the proof. In section 3, we introduce the notion quasidivisible arc and relate the structure of a quasidivisible arc K to the structure of a certain dual arc K of K. We prove a theorem which relates the size of K to the structure of the restriction of K to a hyperplane. We present a characterization of the (3 mod 5)-arcs of small sizes in PG(2, 5) and PG (3,5). The non-existence of (104, 22)-arcs is proved in section 4. We show that such an arc is necessarily quasidivisible. Furthermore, we rule out the existence of planes of certain multiplicities and show that K is not "too big". This implies that the only valid possibility for K is a sum of three planes, which in turn is a contradiction to earlier classification results.

Preliminary results
Let P be the set of points of the projective geometry Σ = PG(k − 1, q). Every mapping K : P → N 0 is called a multiset in PG(k − 1, q). This mapping is extended additively to the subsets Q of P by K(Q) = P ∈Q K(P ). The integer n := K(P) is called the cardinality of K. The support of K is the set of all points of positive multiplicity. The sequence (a i ), where a i is the number of hyperplanes H with K(H) = i, is called the spectrum of K.
A multiset K in the geometry PG(k − 1, q) is called an (n, w)-multiarc (or simply an (n, w)-arc) if (a) K(P) = n, (b) K(H) ≤ w for every hyperplane H, and (c) there exists a hyperplane H 0 with K(H 0 ) = w. A multiset K PG(k − 1, q) is called an (n, w)-blocking set (or (n, w)-minihyper ) if (a) K(P) = n, (b) K(H) ≥ w for every hyperplane H, and (c) there exists a hyperplane H 0 with K(H 0 ) = w.
Denote by H the set of all hyperplanes in Σ and let K be a multiset in Σ. Consider a function σ such that σ(K(H)) is a non-negative integer for all hyperplanes H. The arc is called the σ-dual of K. If σ is a linear function, the parameters of K σ are easily computed from the parameters of K [11].
An (n, w)-arc K in PG(k − 1, q) is called t-extendable, if there exists an (n + t, w)arc K in PG(k − 1, q) with K (P ) ≥ K(P ) for every point P ∈ P. An arc is called simply extendable if it is 1-extendable. An (n, w)-blocking set K in PG(k − 1, q) is called reducible (also minimal ) if there exists an (n − 1, w)-blocking set K with K (P ) ≤ K(P ).
An (n, w)-arc K with spectrum (a i ) is said to be divisible with divisor ∆ > 1 if ). In this paper, the subspace U will be always a point.
Let K be an (n, w)-arc in PG(k − 1, q) with spectrum (a i ). Denote by λ j the number of points P with K(P ) = j. Simple counting arguments yield the following identities, which are equivalent to the first three MacWilliams identities for linear codes: . The above identities imply: Note that the sum on the left-hand side can be written as H , where the sum is taken over all hyperplanes H of PG(k − 1, q). Fix a hyperplane H 0 and consider all hyperlines S (i.e. over all subspaces of co-dimension 2) contained in H 0 . For such a subspace S, denote by H 1 , . . . , H q the remaining hyperplanes through S and set Here the maximum is taken over all hyperlines S and multiplicity i contained in H 0 . Assume that the spectrum (b i ) of the restriction K| H0 is known. Then we obtain from (2) For projective arcs, i.e. arcs with point multiplicities 0 and 1, this becomes Note that the right-hand side depends only on the parameters of K. Inequality (6) is a necessary condition for the existence of an (n, w)-arc in PG(k − 1, q). We finish by presenting the spectra of the arcs in PG(2, 5) with parameters (9,3), (10,3), (11,3), and (22,5) which will be needed in the sequel. The classification of the arcs with the first three parameter sets is given in [8]. The (22,5)-arcs are obtained as complements of the (9,1)-blocking sets. There exist three non-equivalent (9,1)blocking sets: two reducible consisting of a line and three further points and one irreducible -the projective triangle.

Quasidivisible arcs and the extendability of arcs
Let K be a t-quasidivisible (n, w)-arc with divisor q in Σ = PG(k − 1, q), t < q. Denote by K the σ-dual to K in the dual geometry Σ, where σ(x) = n + t − x (mod q). In other words, we have (7) K : where H is the set of all hyperplanes in Σ. This means that hyperplanes of multiplicity congruent to n + a (mod q) become (t − a)-points in the dual geometry.
In particular, maximal hyperplanes are 0-points with respect to K. Let us note that in the general case the cardinality of K cannot be obtained from the parameters of K. Define the sum of two multisets K and K in the same geometry by (K + K )(P ) = K (P ) + K (P ). For a set of points Q ⊆ P, χ Q denotes the characteristic function of Q. The following theorem is straightforward.
Theorem 3.1 ([9, 10]). Let K be an (n, w)-arc in Σ = PG(k − 1, q) which is t-quasidivisible modulo q with t < q. Let K be defined by (7). If for some multiset K in Σ and c not necessarily different hyperplanes P 1 , . . . , P c in Σ, then K is c-extendable. In particular, if K contains a hyperplane in its support, then K is extendable.
Theorem 3.1 links the extendability of t-quasidivisible arcs with the structure of the multiset K defined in the dual geometry. Note that this theorem gives a sufficient but not a necessary condition since 0-points with respect to K can correspond to hyperplanes that are not of maximal multiplicity.

K( S) ≡ t (mod q).
An arc F in PG(k − 1, q) is called a (t mod q)-arc if (a) all points have multiplicity at most t; (b) for all subspaces S of positive dimension F(S) ≡ t (mod q). The importance of (t mod q)-arcs is due to the fact that every t-quasidivisible arc K gives rise to a unique (t mod q)-arc K.
(3) the σ-dual of an ((m − t)q + m, m − t)-blocking set in PG(2, q) with line multiplicities m − t, . . . , m, and with σ(x) = (x − t)/q (cf. [10]). Lifted arcs are arcs obtained by the following construction. Let F 0 be a (t mod q)-arc in PG(k − 2, q). Let H be a hyperplane in PG(k − 1, q). For a fixed point P in PG(k − 1, q), not incident with H, define an arc F in PG(k − 1, q) as follows: F(P ) = t; for each point Q = P , F(Q) = F 0 (R), where R = P, Q ∩ H. It turns out that F is a (t mod q)-arc in PG(k − 1, q) of size q · |F 0 | + t. We call this arc a lifted arc from F 0 .
A plane (1 mod q)-arc is either a hyperplane or the complete space. The (2 mod q)-arcs with q ≥ 5, q odd, in the plane were characterized by Maruta [12]. Such an arc is either a lifted arc from a (2 + iq)-line, i = 0, 1, 2, or the sum of an oval, a tangent to the oval, and twice the internal points to the oval. It can be easily proved by induction that for higher dimensions K is a lifted arc and hence contains a hyperplane without 0-points. This implies that the arc K is again extendable (in fact, in many cases even 2-extendable).
For (3 mod q)-arcs in PG(2, q), the situation is far more complicated. We have many (3 mod q)-arcs obtained as the sum of a (2 mod q)-and a (1 mod q)-arc, but also some non-trivial indecomposable arcs. For instance, in the case of q = 5, a (3 mod 5)-arc of size 18 is the sum of three lines. There exists a unique (3 mod 5)-arc of size 23 which has as 2-points the vertices of quadrangle, as 3-points -ite s diagonal points and as 1-points the intersections of the diagonal lines with the sides of the quadrangle [14]. There exists also a unique (3 mod 5)-arc of size 28. It has as 3-points the points of an oval and as 1-points -the internal points of this oval [14]. Note that in the last two cases there is always a 3-line incident with one 3-point only. This fact will be used later in Lemma 4.3. Consider a Griesmer t-quasidivisible arc K, t < q, with parameters (n, w) in PG(k − 1, q). Set d = n − w and write d as Denote by w j the maximal multiplicity of a subspace S of co-dimension j of PG(k − 1, q): w j = max codim S=j K(S), j = 1, . . . , k − 1. We have By convention, w 0 = n. As already noted, the size of K cannot be obtained from the parameters of K. Nevertheless knowledge of the structure of the restriction of K to some hyperplane gives an upper bound on the size of K.

Lemma 3.4 ([10]
). Let K be a t-quasidivisible (n, n − d)-Griesmer arc with d given by (8). Let S be a hyperline contained in the hyperplane H 0 with K(H 0 ) = w 1 − aq where a ≥ 0 is an integer.
4. The nonexistence of (104, 22)-arcs Assume that K is a (104, 22)-arc in PG (3,5). Let us denote by γ i , i = 0, 1, 2, the maximal multiplicity of an i-dimensional subspace in PG (3,5). In the following lemma, we summarize the straightforward properties of (104, 22)-arcs. (d) This follows by the fact that there is no (21, 3)-blocking set in AG (3,5) with respect to the planes (cf. Corollary 2.3 in [1]) (e) This follows by counting the number of points through a 1-line in a 1-plane and using the fact that a (22, 5)-arc does not have a 1-line (cf. the spectra of (22, 5)-arcs).
(f) Assume H 0 is a 4-plane in PG (3,5). By Lemma 4.1(b), each line from H 0 meets K in at most two points, i.e. K| H0 is a plane (4, 2)-arc. It is easily seen that such an arc can be extended to an oval -(6, 2)-arc. In other words, there exist at least two points which is incident only with 1-lines from K| H0 . Let X be such a point.
Consider a projection ϕ from X onto some plane δ that does not contain X. Set Now it is easily checked that K ϕ (L) ≤ 21, for every line L in δ. This means that the projection point X is not contained in a 22-plane. Therefore K can be extended to a (105, 22)-arc in PG (3,5). But such an arc does not exist (cf. [13]). This is a contradiction which rules out the existence of 4-planes.
By Lemma 4.1, a (104, 22)-arc K is 3-quasidivisible. Moreover, 0-points with respect to the dual arc K must come necessarily from maximal planes. This forces certain restrictions on the structure of K described in the lemma below.
Lemma 4.2. Let K be a (104, 22)-arc in PG (3,5). Then there exists no plane P in the dual space such that K| P is 3χ L for some line L in the dual space.
Proof. Let X be a point in PG (3,5). Summing up the multiplicities of all planes through X, we have   We can use Lemma 4.3 together with the necessary condition (6) to restrict further the possible multiplicities of planes. Our key observation is that if a 5-tuple of planes through a line L in H 0 gives a high contribution to the left-hand side of (6) then K( L) is small. Proof. Let H 0 be a 6-plane. the K| H0 is a (6, 2)-arc and has spectrum a 2 = 15, a 1 = 6, a 0 = 10. Consider an arbitrary line L in H 0 . By Lemma 3.4, if L is a 2-line with respect to K, then it is an 8-line with respect to K; similarly, if L is a 1-line it is a 3-line with respect to K (since 22-planes do not have 1-lines) and, finally, if it is a 0-line with respect to K, it is a 3-, 8-or 13-line with respect to K.