CHARACTERIZATION RESULTS ON WEIGHTED MINIHYPERS AND ON LINEAR CODES MEETING THE GRIESMER BOUND

. We present characterization results on weighted minihypers. We prove the weighted version of the original results of Hamada, Helleseth, and Maekawa. Following from the equivalence between minihypers and linear codes meeting the Griesmer bound, these characterization results are equivalent to characterization results on linear codes meeting the Griesmer bound.

Considering this lower bound on the length n for given values k, d, and q, the question arises whether there exists a linear [n, k, d]-code whose length n is equal to the lower bound g q (k, d). For some values of the parameters k, d, and q, linear codes of length equal to g q (k, d) are known to exist, for other values it is proved that no such codes exist.
Let PG(N, q) be the N -dimensional projective space over the finite field of order q. For i ≥ 0, put v i = (q i − 1)/(q − 1), which is the number of points in PG(i − 1, q). A weight function w of PG(N, q) is a mapping from the point set of PG(N, q) to the set of non-negative integers. For a point P , the integer w(P ) is called the weight of the point P , and for a set M of points, its weight is the sum of the weights of its points. The sum of the weights of all points is the total weight of w. In principle, a minihyper is nothing else than such a weight function, but usually the definition is in the following way, which gives some information on w. Of course, the set F is determined by the weight function w. When the range of w is {0, 1}, the converse is true and then the minihyper is identified with F and called a non-weighted minihyper. Thus, a non-weighted {f, m; N, q}-minihyper of PG(N, q) is a set F of f points of PG(N, q) such that m is the minimum weight of the hyperplanes. This is the definition of a minihyper given by Hamada and Tamari in [15] and it was generalized to the definition of a weighted minihyper in [7].
Linear [n, k, d]-codes meeting the Griesmer bound can be linked with non-weighted minihypers in PG(k − 1, q) when d ≤ q k−1 and with (weighted) minihypers in PG(k − 1, q) when d > q k−1 . We explain first of all the link for 1 ≤ d ≤ q k−1 . Then d can be written uniquely as d = q k−1 − h i=1 q λi such that: (a) 0 ≤ λ 1 ≤ · · · ≤ λ h < k − 1, (b) at most q − 1 of the values λ i are equal to a given value.
Using this expression for d, the Griesmer bound for a linear [n, k, d]-code over F q can be expressed as: Hamada showed that in the case d = q k−1 − h i=1 q λi , there is a one-to-one correspondence between the set of all non-equivalent [n, k, d]-codes meeting the Griesmer bound and the set of all projectively distinct [12]. More precisely, the link is described in the following way.
Let G = (g 1 · · · g n ) be a generator matrix for a linear [n, k, d]-code C, d ≤ q k−1 , meeting the Griesmer bound. It can be shown that g i = ρg j , ρ ∈ F * q , for i = j. Then the set PG(k − 1, q) \ {g 1 , . . . , g n } is the minihyper linked to the code C meeting the Griesmer bound.
For d > q k−1 , the link between linear codes meeting the Griesmer bound and weighted minihypers is as follows.
Let G = (g 1 · · · g n ) be a generator matrix for a linear [n, k, d]-code over F q , d > q k−1 , meeting the Griesmer bound. We again look at a column of G as being the coordinates of a point in P G(k − 1, q). Let the point set of P G(k − 1, q) be {s 1 , . . . , s v k }. Let m i (G) denote the number of columns in G defining s i . Let m(G) = max{m i (G) i = 1, 2, . . . , v k }. Then θ = m(G) is uniquely determined by the code C and we call it the maximum multiplicity of the code. Define the weight function w : Now the question arises how to construct linear codes meeting the Griesmer bound. The standard construction method is of Belov, Logachev, and Sandimirov [2]. This construction method is easily described by using the corresponding minihypers. We first of all describe the construction for non-weighted minihypers.
Consider in PG(k − 1, q) a union of pairwise disjoint ǫ 0 points P 1 , P 2 , . . . , P ǫ0 , If one allows subspaces that are not mutually disjoint, then the union has to be replaced by the weight function, that is, the sum of the characteristic functions of the subspaces. These minihypers will correspond to linear codes with d > q k−1 .
Now that the standard examples of minihypers, or equivalently, of linear codes meeting the Griesmer bound, are known, the characterization problem on minihypers, or equivalently, on linear codes meeting the Griesmer bound, arises: Fundamental research on this characterization problem was performed by Hamada and Helleseth who studied minihypers thoroughly, and who developed many techniques that have proven to be very useful in the study of minihypers. Their main result can be formulated as follows. [13,14])

Result 1.2. (Hamada, Helleseth and Maekawa
. . , ǫ 1 lines, and ǫ 0 points, which all are pairwise disjoint, so is of Belov-Logachev-Sandimirov type. The main result of this paper generalizes this result of Hamada, Helleseth, and Maekawa to weighted minihypers. To state the result, we define a concept which generalizes the Belov, Logachev, and Sandimirov construction. Our definition is a generalization of a similar definition from [9]. Let S 1 , . . . , S u be subspaces of PG(k − 1, q). Define as follows a weight function from the point set of PG(k − 1, q) to the set of integers: for each point P , its weight w(P ) is the number of indices i with P ∈ S i .
In other words, w is the sum of the characteristic functions of the subspaces S i . If F is the union of the subspaces S i , then (F, w) is a minihyper, and we call it the sum of the subspaces S 1 , . . . , S u . We explicitly note that it is allowed that the list S 1 , . . . , S u contains a subspace S i several times.
With this definition, the original Belov, Logachev, and Sandimirov construction can be described as a sum of mutually skew subspaces. Our main theorem of this paper is the following.

Technical lemmas
Let q be a prime power. Every integer f > 0 can be uniquely written in the form f = i≥k ǫ i v i+1 for some integer k ≥ 0 where 1 ≤ ǫ k ≤ q and 0 ≤ ǫ i ≤ q − 1 for i > k. This enables us to define (just for this paper) the q-successor of f as the integer T q (f ) := i≥k ǫ i v i . We also define T q (0) := 0. Thus T q is a map from the set of the non-negative integers to itself. When q is clear from the context, we simply write T . Applying T more than once to a number, results in numbers T i (f ) = T (T i−1 (f )). The {f, h; N, q}-minihypers related to Griesmer codes often have the property that h = T (f ). It is also important to note that many of these minihypers have the property that T j (f ) is the minimum weight of the subspaces of codimension j. This motivates the following lemmas. We note that variants of the lemmas, either of smaller total weight or non-weighted versions or statements without proofs, can be found in the literature, e.g. in [9], [11], and [13].
Lemma 2.1. Let w be a weight function from the point set of PG(N, q) to the set of non-negative integers of total weight f at most qv N +1 . Then some hyperplane has weight at most T (f ).
Proof. This is trivial, if the total weight f is zero.
Let h j be the minimum weight of the subspaces of codimension j and put h j := T j (f ) + δ j for an integer δ j . Then h 0 = f , so δ 0 = 0. Also h N +1 = T N +1 (f ) = δ N +1 = 0. Considering for j ≥ 1 all subspaces of codimension j on a fixed subspace of codimension j + 1 and weight h j+1 gives If one substitutes for f , h j , and h j+1 , this reads as follows Then we find recursively that δ N +1 ≥ δ N ≥ · · · ≥ δ 1 > 0. As δ N +1 = 0, this is impossible.
(a) For each j with 0 ≤ j ≤ N , the minimum weight of the subspaces of codimen- Suppose that ∆ is a subspace of codimension two of weight T 2 (f ). Then for each of the q + 1 hyperplanes π 0 , . . . , π q on ∆, the restriction of w to π j is a where the δ j are non-negative integers such that q j=0 δ j = ǫ 0 . Proof. (a) Using the same notations as in the previous lemma, we again find the inequalities (1). However, as As every subspace of codimension 2 has weight at least T 2 (f ) and as w(∆) = T 2 (f ), the restriction of w to π i gives a minihyper with the stated parameters.
Proof. We prove this by induction on the codimension of U . If N = u, the statement is trivial. Now we study the case that U is a hyperplane.
As N −1 i=0 ǫ i < q, this implies that w(S) ≥ T (h). Then Lemma 2.1 applied to the restriction of w onto U shows that T (h) is the minimum weight of the (N − 2)subspaces of U . Considering such a subspace S in (2), we find For the induction step, we now consider the case that U has codimension at least two and a point of weight zero. Let H be a hyperplane on U . As we have proved the assertion for hyperplanes, we know that As U , and hence H, contains a point of weight zero, Lemma 2.2 applied to the minihyper in H shows that n N −1 = 0. Therefore we can apply the induction hypothesis to U considered as a subspace of H. This gives the desired minihyper in U where not only We need one more definition. If w is a weight function on PG(N, q), then for any subspace S of PG(N, q), we call the number m := min{w(P )|P ∈ S} the multiplicity of the subspace S; we also say that the subspace S occurs with multiplicity m in the minihyper. Of course, if S has dimension s and multiplicity m, then the weight w(S) of S is at least v s+1 m.
Lemma 2.4. Suppose that w is a weight function on the point set of PG(N, q), which is the sum of the characteristic functions of r ≤ q − 1 non-empty subspaces S 1 , . . . , S r . Then for any non-empty subspace T , the following results hold.
(a) The multiplicity of T is equal to the number of subspaces S i that contain T .
(b) If dim(T ) ≥ 1, then there exists an (N − 1)-subspace ∆ that does not contain any of the subspaces T, S 1 , . . . , S r and such that T ∩∆ has the same multiplicity as T .
Proof. (a) Let m be the number of subspaces S i containing T . Then the remaining r − m ≤ q − 1 subspaces S i cannot cover T . Hence, T possesses a point that has weight m. Therefore, T has multiplicity m.
(b) At most r ≤ q − 1 non-empty and proper subspaces of T have the form T ∩ S i for some i. Thus some hyperplane S of T will not contain any of these. Then every subspace S i that contains S also contains T , so S and T have the same multiplicity. Also S is a proper subspace of all subspaces S, S i and as r ≤ q−1, this implies that some hyperplane ∆ on S will not contain any of the subspaces T, S 1 , . . . , S r .
In this section, we prove the basic results on minihypers in PG(2, q). Since an {f, t; 2, q}-minihyper meets every line in at least t points, there is a strong connection to the theory of blocking sets in projective planes. Our first lemma generalizes a result on blocking sets of [1], in which the following lemma is proved for nonweighted minihypers without restriction on t.
Proof. We assume in the proof that t ≥ 2. For, if t = 1, then B defines a blocking set w.r.t. the lines of PG(2, q). It follows from [4,5] that every blocking set B, not containing a line of PG(2, q), contains at least q + √ q + 1 points. Defining m := P ∈B (w(P ) − 1) and s = f − tq − 1, we have |B| = f − m = qt + 1 + s − m. Suppose that B contains no line. Then there exist points not in B. As the q + 1 lines on such a point all have weight at least t, we find f ≥ (q + 1)t, that is, s ≥ t − 1. Also, if l is a line, then considering a point of l that is not in B and the other q lines on this point, we find w(l) ≤ f − tq = s + 1.
Consider a point X. The sum of the weights of the lines on X is f + q · w(X). Hence, the sum of the numbers w(l) − t for the lines l on X is f + qw(X) − (q + 1)t = s + 1 − t + qw(X). This we use to estimate the number where P Q denotes the line through the points P and Q. First of all, for P / ∈ B, we have since this is a sum of weights of lines l on P , where each line l on P occurs w(l) ≤ s+1 times. As |B| = f − m, we find ∆ ≤ (q 2 + q + 1 − f + m)(s + 1)(s + 1 − t). If Q ∈ B, then each line on Q has at least q − s points not in B and thus we find similarly where the sum is over all lines l on Q. Using we find a lower bound on ∆. Comparing both bounds yields We may assume that s ≤ √ qt, since otherwise we are done. Using 2 ≤ t < q − 1 and s ≤ √ tq < (q + t)/2, we see that the coefficient of m on the left hand side is smaller than the coefficient of m on the right hand side. Hence, the inequality remains true when deleting the m-terms. Using f = qt+ 1 + s, the remaining inequality simplifies to . If s ≥ t, then we find s ≥ √ tq as desired. Assume finally that s < t. Then s = t − 1, f = tq + t, and every line on a point not in B must have exactly weight t. Since B contains no line, then every line has weight exactly t. But counting the total weight using the lines on a point X, we find f = (q + 1)t − qw(X), which implies w(X) = 0 for all points, that is, f = 0. As t ≥ 2, this is a contradiction.
For weighted multiple blocking sets having at least one point of weight one, we can obtain better results using polynomial techniques. These improvements are described in [6], where they are used to prove characterization results on nonweighted minihypers. In [8], using polynomial techniques, the following corollary is proved. Together with Lemma 3.1, it will play an important role in characterizing planar minihypers.
For ǫ 1 = 0, this lemma is trivial. So suppose that ǫ 1 > 0. By Lemma 3.1, F contains a line l. We shall show in the next paragraph that this line has weight at least q + ǫ 1 . Thus, removing l from (F, w) (that is, reducing the weight of every point of l by one) yields an {(ǫ 1 − 1)(q + 1) + ǫ 0 , ǫ 1 − 1; 2, q}-minihyper. Then an induction argument on ǫ 1 completes the proof.
Assume that l has only weight q + ǫ 1 − δ for some δ > 0. As l is contained in F , then δ ≤ ǫ 1 − 1. Now we reduce only the weight of q + 1 − δ different points from l by one; this results in a {(q + 1)ǫ ′ 1 + ǫ ′ 0 , ǫ ′ 1 ; 2, q}-minihyper with ǫ ′ 1 = ǫ 1 − 1 and ǫ ′ 0 = ǫ 0 + δ, in which l will have weight ǫ ′ 1 . If we consider a point of l whose weight has not been reduced, then the other q lines on this point will have the same weight as before, that is, at least weight ǫ 1 = ǫ ′ 1 + 1. This contradicts Result 3.2. Note that Result 3.
We now generalize this to arbitrary dimensions.
Proof. We use induction on N ; the case N = 2 being handled in the previous lemma. Now assume that N ≥ 3 and that the statement holds for N − 1. Consider a point P ∈ PG(N, q) \ F . Projecting (F, w) from P onto a hyperplane π of PG(N, q), with P / ∈ π, yields in the following way a minihyper (F ′ , w ′ ) in PG(N − 1, q) = π. The set F ′ is defined as the projection of F from P , i.e. P ′ ∈ F ′ if and only if the line P P ′ intersects the set F . For any point P ′ ∈ F ′ , we define w ′ (P ′ ) = w(P P ′ ). It is clear that (F ′ , w ′ ) is an {ǫ 1 (q + 1) + ǫ 0 , ǫ 1 ; N − 1, q}minihyper. By the induction hypothesis on N , (F ′ , w ′ ) is a sum of ǫ 1 lines and ǫ 0 points.
By Lemma 3.3, (F ′′ , w ′′ ) is a sum of m lines and n points. Thus, the line l ′ is the projection of m lines contained in F . Since this holds for all lines l ′ contained in (F ′ , w ′ ), we thus find ǫ 1 (not necessarily distinct) lines l 1 , . . . , l ǫ1 contained in F that are projected to the lines of (F ′ , w ′ ). Our argument also shows the following. If a line l occurs x times in the list l 1 , . . . , l ǫ1 , then every point of l has weight at least x. We want to show that the sum of the lines l 1 , . . . , l ǫ1 is contained in the minihyper (F, w). For this we have to show for each point X that w(X) is equal or larger to the number of indices i with X ∈ l i .
To see this, we select a line h on X that meets F only in X, and we project again as before but using this time for P a point of h that is different from X. As already noticed, the projection (F ′ , w ′ ) of (F, w) contains a sum of ǫ 1 lines. As ǫ 0 + ǫ 1 < √ q + 1, it is readily seen that these ǫ 1 lines are the images of the lines l 1 , . . . , l ǫ1 . As h meets F only in X, then the point X ′ onto which X is projected satisfies w ′ (X ′ ) = w(X). Now, if X lies on x of the lines l 1 , . . . , l ǫ1 , then X ′ lies on x of the projected lines, and as (F ′ , w ′ ) contains the sum of these lines, we have Now we have shown that (F, w) contains in fact the sum of the lines l 1 , . . . , l ǫ1 , and is therefore the sum of these lines and of ǫ 0 points.

Proof of the main theorem
We start with a lemma that will be used to find large subspaces in minihypers.
Suppose that P is a point of F lying on two subspaces S 1 and S 2 of multiplicity m 1 and m 2 such that m 1 + m 2 > w(P ). Then the subspace S 1 , S 2 is completely contained in F .
Proof. As m 1 , m 2 ≤ w(P ), then m 1 and m 2 are positive and thus S 1 and S 2 are contained in F . Therefore the statement is trivial, if one of the subspaces S 1 and S 2 contains the other. Otherwise, S 1 , S 2 is the union of planes l 1 , l 2 with lines l i on P and in S i . It suffices thus to show that these planes are contained in F .
Assume that such a plane π := l 1 , l 2 is not contained in F . Then Proposition 2.3 gives w(π) = a 1 (q + 1) + a 0 , with integers a 0 , a 1 ≥ 0 and a 1 + a 0 < √ q + 1, and such that every line of π has weight at least a 1 . Since each point of l i has weight at least m i and since the other q − 1 lines of π on P each have weight at least a 1 , we find a 1 (q + 1) + a 0 = w(π) ≥ w(P ) + qm 1 + qm 2 + (q − 1)(a 1 − w(P )).
, with 0 ≤ t < N and t i=0 ǫ i < √ q + 1. We want to show that it is a sum of subspaces, where the sum consists of ǫ i subspaces of dimension i for i = 0, . . . , t. The proof is by induction on t. The case when t = 0 is trivial. The case t = 1 was proved in the last section. In the rest of this section, we prove the induction step. We thus suppose that t ≥ 2 and that the assertion is true for smaller values of t. We also assume that ǫ t > 0, because otherwise we can immediately apply the induction hypothesis. Proof. We prove this by induction on m(U ). If m(U ) = 0, this is trivial. Suppose now that m(U ) > 0. By the induction hypothesis of this section, we see that the restriction of w to π 0 is a sum of γ 0 + ǫ 1 + · · · + ǫ t < √ q + 1 subspaces. Lemma 2.4 shows that π 0 has a hyperplane ∆ not containing any of the subspaces of this sum, that is, the weight of ∆ is and such that S := U ∩ ∆ has dimension t − 2 and multiplicity m(U ). This shows already that U is the only subspace of the sum π 0 ∩ (F, w) passing through S.
Consider the remaining hyperplanes π 1 , . . . , π q on ∆. By Lemma 2.2, the restriction of w to the hyperplanes π i produces {γ i + t j=1 ǫ j v j , t j=2 ǫ j v j−1 ; N − 1, q}minihypers in π i , where γ i ≥ 0 and q i=0 γ i = ǫ 0 . The global induction hypothesis of this section shows that these minihypers can be uniquely written as a sum of subspaces. The number of subspaces in this sum is γ i + ǫ 1 + · · · + ǫ t < √ q + 1. As w(∆) = δ, we see that ∆ does not contain any of the subspaces that occurs in the sum for π i . In particular, S is not one of the subspaces occurring in the sum that makes up the minihyper in π i .
Let U j i , j = 1, . . . , r i , be the different (t − 1)-subspaces of π i on S that have positive multiplicity m i,j (thus r 0 = 1 and U 1 0 = U and m 0,1 = m(U ) = m(S)). Lemma 2.4 gives j m i,j = m(S) for each i.
First consider the case that r i = 1 for all i, that is, U 1 i is the only (t − 1)subspace of positive multiplicity of π i on S, and m(U 1 i ) = m(S). By Lemma 4.1, the t-subspace T := U 1 1 , U 1 2 is contained in F . Then T ∩ π i is a (t − 1)-subspace of π i of positive multiplicity containing S. Hence, T ∩π i = U 1 i for all i and thus T is the union of the subspaces U 1 0 , . . . , U 1 q . As all these subspaces have multiplicity m(S), we see that all points of T have weight at least m(S), that is, T has multiplicity at least m(S) = m(U ). As the multiplicity of T cannot exceed the one of U , then m(T ) = m(U ). By construction, no other t-subspace of positive multiplicity contains U (or S).
Now consider the case that some r i > 1, say r 1 > 1. As j m(U j 1 ) = m(S), it follows that m(U j 1 ) < m(S) = m(U ) for all j. Thus, the induction hypothesis applied to U j 1 gives m(T ) = m(U j 1 ) where the sum runs over all t-subspaces T containing U j 1 . As π 1 does not contain a t-subspace of (F, w), we see that different subspaces U j 1 produce different subspaces T , in fact that every t-subspace of positive multiplicity contains exactly one of the subspaces U j 1 . Therefore m(T ) = j m(U j 1 ) = m(S) where the first sum runs over all t-subspaces containing S. Every t-subspace of positive multiplicity occurring in this sum meets π 0 in a (t − 1)-subspace of positive multiplicity and containing S. As U is the only such (t − 1)-subspace of π 0 , all these t-subspaces contain U and so we are done as m(U ) = m(S). Proof. Using Lemma 2.2, we find a subspace ∆ of codimension two with the following property. If π 0 , . . . , π q are the hyperplanes on ∆, then the restriction of w to π j is a The global induction hypothesis of this section shows that the minihyper in π j is a sum of subspaces, containing ǫ i subspaces of dimension i − 1 for i = 2, . . . , t, and ǫ 1 + γ j points.
Consider π 0 . Let U 1 , . . . , U s be the different (t − 1)-subspaces occurring in the sum that make up the minihyper in π 0 and let U i occur m i times in this sum. Then s i=1 m i = ǫ t . By Lemma 2.4, the subspace U i has multiplicity m i ; also the U i are all (t − 1)-subspaces of π 0 of positive multiplicity. As w(π 0 ) < v t+1 , then π 0 contains no t-subspace of positive multiplicity. Lemma 4.2 applied to the U i now shows the following. If T is the set of all t-subspaces of positive multiplicity, then T ∈T m(T ) = ǫ t ; moreover, we have m(U ) = U⊂T ∈T m(T ) for every (t − 1)subspace U of π 0 . Now, if P is a point of π 0 , then the sum of the m(U ) over all (t − 1)-subspaces U of π 0 on P is less than or equal to w(P ) (because we know the structure of the minihyper in π 0 ). Therefore, the sum of the m(T ) over all t-subspaces T ∈ T that contain P is less than or equal to w(P ). Since the same property can be proved for the points of π 1 , . . . , π q , it holds for all points. Thus using each T ∈ T exactly m(T ) times, then the sum of these ǫ t t-subspaces is contained in (F, w).
Proof. By the preceding lemma, w contains a sum of ǫ t non-necessarily distinct subspaces T 1 , . . . , T ǫt of dimension t. This means for each point P that w(P ) is at least as large as the number of subspaces T i that contain P . Let S i be a (fixed) hyperplane of T i . We define as follows a new weight function w ′ . For every point P , we define w ′ (P ) to be equal to w(P ) minus the number of affine spaces T i \ S i that contain P . Note that w ′ (P ) ≥ 0 for all points P . Clearly the sum of the weights w ′ (P ) over all points is ǫ t q t less than the corresponding sum for the original function w, and thus it is We analyze the w ′ -weight of the hyperplanes π. If π does not contain any of the S i , then it also does not contain any of the t-subspaces T i ; in this case we have If π contains a subspace S i , then we have w ′ (π) ≥ v t , which is even better. Therefore Lemma 2.1 shows that w ′ defines an {f, h; N, q}-minihyper. The induction hypothesis in this section shows that this minihyper is a sum of subspaces, namely ǫ t + ǫ t−1 subspaces of dimension t − 1 and, for i < t − 1, another ǫ i subspaces of dimension i. As t i=0 ǫ i < √ q +1, the subspaces S 1 , . . . , S ǫt must occur in this sum. It follows that (F, w) is the sum obtained from the previous sum when replacing S i by T i .
The following corollary now follows immediately, which is in fact the known result of Hamada, Helleseth, and Maekawa (Result 1.2).
Proof. In a non-weighted minihyper, we can define the weight function w by giving the points of F weight one, and the points not belonging to F weight zero. Then F can be described as a sum of the subspaces mentioned in the statement of this corollary, but since the points of F have weight one, these subspaces must be pairwise disjoint.