Sets of subspaces with restricted hyperplane intersection numbers

On sets of subspaces with restricted hyperplane intersection numbers T im Alderson Simeon Ball Abstract Let X be a set of ( h − 1) -dimensional subspaces of PG( k h − 1 , q ) with the property that e very hyperplane contains at most t elements of X . W e pro ve the upper bound |X | ≤ ( t − k + 2) q h + t , and characterise the structure of X in the case of equality . W e call sets attaining this bound length- maximal . For k = 3 , such sets are known as maximal arcs and have been well-studied. They are kno wn to exist for t < q h if and only if q is ev en and t di vides q h . For k = 4 and q > 2 , we sho w that any length-maximal set must satisfy t = q h + 1 and that ev ery hyperplane is either a t -secant or a 1 -secant. For k ≥ 5 and q > 2 , no length-maximal set exists. In the language of additi ve codes, these results assert that additiv e two-weight codes ov er F q h attaining the natural Griesmer-type bound do not e xist when the code dimension is 5 or more and q > 2 . K eywords. maximal arc, additi ve code, two-weight code, projectiv e space, subspaces, Griesmer bound 1 Intr oduction A maximal ar c X in PG(2 , q ) is a set of points with the property that e very line is incident with 0 or t points of X . It is straightforward to prove that |X | = ( t − 1) q + t and that t divides q . Denniston [7] provided e xamples for q e ven and all possible v alues of t . Further e xamples were discov ered by Thas [10, 12], Mathon [9], and Hamilton and Mathon [8]. When q is odd, maximal arcs were sho wn not to exist by Ball, Blokhuis and Mazzocca [2]; see also [3]. The parameter set ( t, q ) = (3 , 9) was ruled out earlier by Cossu [5] and the parameter set ( t, q ) = (3 , 3 h ) by Thas [11]. Simeon Ball Departament de Matem ` atiques, Uni versitat Polit ` ecnica de Catalunya, Barcelona, Spain. The author acknowledges the support of the Spanish Ministry of Science, Innov ation and Univ ersities grant PID2023-147202NB-I00. e-mail: simeon.michael.ball@upc.edu T im Alderson Department of Mathematics and Statistics, Univ ersity of New Brunswick Saint John, Saint John, NB, Canada e-mail: tim.alderson@unb .ca 29 March 2026. Mathematics Subject Classification (2010): 51E20, 94B05 1 2 T im Alderson, Simeon Ball In this article we consider the natural generalisation of maximal arcs to higher dimensions. A maximal arc in PG(2 , q h ) yields, by field reduction, a set of ( h − 1) -dimensional subspaces in PG(3 h − 1 , q ) with the property that e very h yperplane contains either 0 or t subspaces. This leads us to study sets X of ( h − 1) -dimensional subspaces in PG( kh − 1 , q ) with the property that e very hyperplane contains at most t subspaces of X . There is an equiv alent formulation in terms of additiv e codes. An additi ve [ n, k , d ] h q code C is an F q - linear subspace of F n q h of size q kh in which e very non-zero vector has at least d non-zero coordinates. Let G be a k h × n matrix whose ro w span is C . By fixing a basis { e 1 , . . . , e h } for F q h ov er F q we can write G = h X j =1 e j G j , where G j is a k h × n matrix with entries from F q . For each coordinate i ∈ { 1 , . . . , n } we define a subspace π i as the span of the i -th column of G j , j ∈ { 1 , . . . , h } . Thus π i is a subspace of PG( k h − 1 , q ) of projecti ve dimension at most h − 1 . The property that e very non-zero v ector of C has at least d non-zero coordinates is equiv alent to the property that ev ery hyperplane of PG( k h − 1 , q ) contains at most t = n − d elements of the multi-set X = { π 1 , . . . , π n } . In this setting the length bound |X | ≤ ( t − k + 2) q h + t is a Griesmer-type bound for additi ve codes. Codes attaining this bound are additiv e tw o-weight codes whose non-zero weights are n and n − t . An additi ve permutation on each coordinate produces an equi valent code containing the all-one v ector . Although maximal arcs do not exist in planes of odd characteristic, Mathon [6] discov ered an ex- ample of 21 lines in PG(5 , 3) (equi valently , k = 3 , h = 1 , q = 3 , t = 3 ) with the property that e very hyperplane contains 0 or 3 lines. This example shows that a direct generalisation of the Ball– Blokhuis–Mazzocca nonexistence theorem to higher dimensions is not possible. T o our knowledge it is the only kno wn example in odd characteristic, and the uniqueness question remains open. The main results of this paper are as follo ws. Theorem 2 establishes the bound |X | ≤ ( t − k + 2) q h + t and characterises equality . The divisibility condition t − k + 3 | q h (Corollary 5) provides a further necessary condition for length-maximal sets. Lemma 7 sho ws that for k = 4 , any length-maximal set must satisfy t = q h + 1 and ha ve no zero-secant hyperplanes. Lemma 8 sho ws that no length-maximal set exists for k = 5 and q > 2 , and by induction this extends to all k ≥ 5 with q > 2 . The case q = 2 is discussed in Section 4. Throughout, k ≥ 2 , and dimension refers to projecti ve dimension. 2 An upper bound on the size of X W e first prov e a divisibility condition on t in the extremal case. Lemma 1. Let X be a multi-set of ( h − 1) -dimensional subspaces in PG( k h − 1 , q ) of size ( t − k + 2) q h + t with the pr operty that every hyperplane contains either 0 or t subspaces of X . Let π be an Sets of subspaces with restricted hyperplane intersection numbers 3 ( s − 1) -dimensional subspace skew to X . Then π is contained in exactly q h − q kh − s − h q − 1 + ( k − 2)( q kh − s − q h ) t ( q − 1) zer o-secant hyperplanes. In particular , if ther e exists a point sk ew to X , then t divides ( k − 2) q h . Pr oof. Let a 0 and a t denote the number of zero-secant and t -secant hyperplanes containing π , respec- ti vely . Counting hyperplanes through π , a 0 + a t = q kh − s − 1 q − 1 , and counting pairs ( x, H ) with x ∈ X and H a hyperplane through both π and x , ta t =  ( t − k + 2) q h + t  q kh − s − h − 1 q − 1 . Solving for a 0 gi ves the stated formula. Applying this for s = 0 and s = 1 yields t | ( k − 2) q h (1 + q + · · · + q kh − h − 1 ) and t | ( k − 2) q h (1 + q + · · · + q kh − h − 2 ) , from which t divides ( k − 2) q h . Theorem 2. Let X be a multi-set of ( h − 1) -dimensional subspaces in PG( kh − 1 , q ) with the pr operty that every hyperplane contains at most t subspaces of X . Then |X | ≤ ( t − k + 2) q h + t, (1) with equality if and only if 1. every set of ( k − 1) elements of X spans a (( k − 1) h − 1) -dimensional subspace, and 2. every hyperplane contains either 0 , 1 , . . . , k − 3 or t elements of X . Pr oof. Let x 1 , . . . , x k − 2 be elements of X and let π = ⟨ x 1 , . . . , x k − 2 ⟩ . The dimension of π is ( k − 2) h − 1 − e 1 for some e 1 ≥ 0 . Let ( k − 1) h − 1 − e 1 − e 2 denote the maximum dimension of ⟨ π , x ⟩ ov er x ∈ X \ { x 1 , . . . , x k − 2 } . W e count pairs ( x, H ) where x ∈ X \ { x 1 , . . . , x k − 2 } and H is a hyperplane containing both x and π . By hypothesis each such hyperplane contains at most t − k + 2 elements of X \ { x 1 , . . . , x k − 2 } , so this count is at most q 2 h + e 1 − 1 q − 1 ( t − k + 2) . Since ( k − 1) h − 1 − e 1 − e 2 is the maximum dimension of ⟨ π , x ⟩ , each element of X \ { x 1 , . . . , x k − 2 } lies in at least q h + e 1 + e 2 − 1 q − 1 4 T im Alderson, Simeon Ball hyperplanes through π , so the count is at least q h + e 1 + e 2 − 1 q − 1 ( n − k + 2) . Combining these inequalities, n ≤ q 2 h + e 1 − 1 q h + e 1 + e 2 − 1 ( t − k + 2) + k − 2 . This bound is maximised when e 1 = e 2 = 0 , giving n ≤ ( t − k + 2) q h + t . Equality requires e 1 = e 2 = 0 , which means e very ( k − 1) elements of X span a (( k − 1) h − 1) - flat (part 1 of Theorem 2), and ev ery hyperplane containing k − 2 elements of X contains exactly t elements of X (part 2 of Theorem 2). The bound (1) is called the maximal ar c bound , and a set X attaining it is called length-maximal . A length-maximal X corresponds to an additi ve two-weight code o ver F q h with weights n and n − t . Lemma 3. If X is a length-maximal set of ( h − 1) -dimensional subspaces in PG( k h − 1 , q ) with q > 2 and k ≥ 4 , then t ≥ k . Pr oof. If t = k − 1 then |X | = q h + k − 1 and the corresponding additi ve code is an [ q h + k − 1 , k , q h ] MDS code. Since k ≥ 4 and q > 2 , a classical result of Bruen and Silv erman [4] then requires 36 | q , whence t ≥ k . Lemma 4. Let X be a set of ( h − 1) -dimensional subspaces in PG(3 h − 1 , q ) with the pr operty that every hyperplane contains at most t elements of X . If |X | = ( t − 1) q h + t then t ≤ q h + 1 . If t < q h + 1 then ther e exists a hyperplane disjoint fr om X . If t = q h + 1 then e very hyperplane contains e xactly t elements of X . Pr oof. Counting pairs ( x, H ) where x ∈ X and H is a hyperplane containing x , |X | · q 2 h − 1 q − 1 ≤ t · q 3 h − 1 q − 1 , with equality if and only if e very hyperplane contains exactly t elements of X . Substituting |X | = ( t − 1) q h + t and simplifying, ( t − 1) q h + t ≤ t q 3 h − 1 q 2 h − 1 = t  q h + 1 q h + 1  . This rearranges to q h ( q h − t + 1) ≥ 0 , which holds if and only if t ≤ q h + 1 , with equality precisely when t = q h + 1 and e very hyperplane is a t -secant. Theorem 5. Let X be a length-maximal set of ( h − 1) -dimensional subspaces in PG( k h − 1 , q ) with k ≥ 3 and t < q h + k − 2 . Then t − k + 3 divides q h . Sets of subspaces with restricted hyperplane intersection numbers 5 Pr oof. For k = 3 , part 2 of Theorem 2 giv es that ev ery hyperplane contains 0 or t elements of X . By Lemma 4, since t < q h + 1 , there e xists a hyperplane disjoint from X . By Lemma 1, t divides q h . For k ≥ 4 , we argue inductiv ely . By part 1 of Theorem 2, taking the quotient at any element x ∈ X yields a set X ′ of ( h − 1) -dimensional subspaces in PG(( k − 1) h − 1 , q ) of size ( t − 1 − ( k − 1) + 2) q h + ( t − 1) = (( t − 1) − ( k − 1) + 2) q h + ( t − 1) , with e very hyperplane containing at most t − 1 elements of X ′ . Thus X ′ is length-maximal with parameter t − 1 in PG(( k − 1) h − 1 , q ) . Since t − 1 < q h + k − 3 , the induction hypothesis gi ves ( t − 1) − ( k − 1) + 3 = t − k + 3 di vides q h . 3 Nonexistence of length-maximal sets f or k ≥ 4 W e enumerate secant hyperplanes of a length-maximal set X . A hyperplane containing precisely s elements of X is called an s -secant . For X length-maximal we write N ( t, h, k , s ) for the number of s -secants of X in PG( k h − 1 , q ) . Lemma 6. Let X be a length-maximal set of ( h − 1) -dimensional subspaces in PG( k h − 1 , q ) . Set n = |X | = ( t − k + 2) q h + t . Then 1. N ( t, h, 3 , t ) = q 3 h − 1 q − 1 − q h − 1 q − 1 ·  q h ( q h + 1) t − q h  2. N ( t, h, 3 , 0) = q h − 1 q − 1 ·  q h ( q h + 1) t − q h  3. N ( t, h, 4 , t ) =  q h + 1 − 2 q h t   q h + 1 − q h t − 1  q 2 h − 1 q − 1 4. N ( t, h, 4 , 1) = n · N ( t − 1 , h, 3 , 0) =  ( t − 2) q h + t  · q h − 1 q − 1 ·  q h ( q h + 1) t − 1 − q h  5. N ( q h + 2 , h, 5 , 2) = n 2 · N ( q h + 1 , h, 4 , 1) = q 2 h + 2 2 · ( q 2 h + 1) · q h − 1 q − 1 6. N ( q h + 2 , h, 5 , q h + 2) = ( q 2 h + 2)( q 2 h + 1) q h ( q h − 1) ( q h + 2)( q − 1) Pr oof. Set n = ( t − k + 2) q h + t and we use the term ( h − 1) -flat for an ( h − 1) -dimensional subspace. P art 1. By part 2 of Theorem 2, ev ery hyperplane is a 0 -secant or a t -secant when k = 3 . Counting pairs ( x, H ) with x ∈ X and H a t -secant through x , and using the fact that each ( h − 1) -flat in PG(3 h − 1 , q ) lies in q 2 h − 1 q − 1 hyperplanes, N ( t, h, 3 , t ) · t = n · q 2 h − 1 q − 1 . 6 T im Alderson, Simeon Ball Substituting n = ( t − 1) q h + t and rearranging giv es the formula. P art 2 follo ws since N ( t, h, 3 , 0) + N ( t, h, 3 , t ) = | PG(3 h − 1 , q ) | . P art 3. By part 1 of Theorem 2, any two elements of X span a (2 h − 1) -flat; each such flat lies in q 2 h − 1 q − 1 hyperplanes of PG(4 h − 1 , q ) . By part 2 of Theorem 2, each hyperplane is a 0 -, 1 -, or t -secant when k = 4 . Counting triples ( x, y , H ) with x  = y in X and H a t -secant through both, N ( t, h, 4 , t ) ·  t 2  =  n 2  · q 2 h − 1 q − 1 . Substituting n = ( t − 2) q h + t and simplifying gi ves the formula. P art 4. For k = 4 , a 1 -secant hyperplane contains exactly one element x i ∈ X . The quotient of X at any x i ∈ X is a length-maximal set X ∗ i in PG(3 h − 1 , q ) with parameter t − 1 . Each 0 -secant of X ∗ i corresponds to a 1 -secant of X through x i , so summing ov er all x i , N ( t, h, 4 , 1) = n · N ( t − 1 , h, 3 , 0) . Substituting the formula from part 2 of Lemma 6 with t replaced by t − 1 giv es the result. P art 5. No w let k = 5 and t = q h + 2 , so n = q 2 h + 2 . By part 2 of Theorem 2, e very hyperplane is a 0 -, 1 -, 2 -, or t -secant. Since an y hyperplane containing 3 or more elements of X must contain t elements (as any 3 elements span a (3 h − 1) -flat, and any hyperplane through a (3 h − 1) -flat that contains a fourth element of X must contain all t by part 2 of Theorem 2), the secant types are restricted to 0 , 1 , 2 , and t . For k = 5 , each element x i ∈ X lies in a 2 -secant hyperplane H if and only if H contains exactly one further element x j  = x i . The number of 2 -secant hyperplanes through x i equals the number of 1 -secant hyperplanes of the quotient set X ∗ i in PG(4 h − 1 , q ) , which has parameter t − 1 = q h + 1 . Summing ov er all x i and di viding by 2 (since each 2 -secant is counted twice), N ( q h + 2 , h, 5 , 2) = n 2 · N ( q h + 1 , h, 4 , 1) . By Lemma 7 (prov ed belo w), the quotient set X ∗ i has t − 1 = q h + 1 and has no 0 -secants, so N ( q h + 1 , h, 4 , 1) = | PG(4 h − 1 , q ) | − N ( q h + 1 , h, 4 , q h + 1) . Alternati vely , substituting t − 1 = q h + 1 directly into part 4 of Lemma 6 and simplifying gi ves N ( q h + 2 , h, 5 , 2) = q 2 h + 2 2 · ( q 2 h + 1) · q h − 1 q − 1 . P art 6. W ith k = 5 , t = q h + 2 , n = q 2 h + 2 , an y three elements of X span a (3 h − 1) -flat (since any k − 1 = 4 elements span a (4 h − 1) -flat by part 1 of Theorem 2, which implies any three also span a (3 h − 1) -flat). Each (3 h − 1) -flat in PG(5 h − 1 , q ) lies in q 2 h − 1 q − 1 hyperplanes. Counting quadruples ( x 1 , x 2 , x 3 , H ) with distinct x i ∈ X and H a t -secant through all three, N ( q h + 2 , h, 5 , q h + 2) ·  t 3  =  n 3  · q 2 h − 1 q − 1 . Sets of subspaces with restricted hyperplane intersection numbers 7 Substituting n = q 2 h + 2 and t = q h + 2 and simplifying, N ( q h + 2 , h, 5 , q h + 2) = ( q 2 h + 2)( q 2 h + 1) q 2 h ( q h + 2)( q h + 1) q h · q 2 h − 1 q − 1 = ( q 2 h + 2)( q 2 h + 1) q h ( q h − 1) ( q h + 2)( q − 1) . Theorem 7. Let X be a length-maximal set of ( h − 1) -dimensional subspaces in PG(4 h − 1 , q ) with q = p m > 2 . Then t = q h + 1 and e very hyperplane is either a t -secant or a 1 -secant of X . Pr oof. By part 2 of Theorem 2, ev ery hyperplane is a 0 -, 1 -, or t -secant. The counts N ( t, h, 4 , 0) , N ( t, h, 4 , 1) , N ( t, h, 4 , t ) are non-negati ve inte gers summing to | PG(4 h − 1 , q ) | . Using parts 3 and 4 of Lemma 6, N ( t, h, 4 , 0) = | PG(4 h − 1 , q ) | − N ( t, h, 4 , t ) − N ( t, h, 4 , 1) = q h ( q h − 1)( t − q h − 1)  t ( q h + 1) − 4 q h − 2  t ( q − 1) . From Lemma 3 and Theorem 5, we hav e 4 ≤ t ≤ q h + 1 . W e show that N ( t, h, 4 , 0) ≥ 0 forces t = q h + 1 . For t ≤ q h , the factor ( t − q h − 1) ≤ − 1 < 0 . For t ≥ 4 , the factor t ( q h + 1) − (4 q h + 2) ≥ 4( q h + 1) − (4 q h + 2) = 2 > 0 . Hence N ( t, h, 4 , 0) < 0 for 4 ≤ t ≤ q h , a contradiction. Therefore t = q h + 1 , and substituting gi ves N ( t, h, 4 , 0) = 0 . Theorem 8. Ther e is no length-maximal set of ( h − 1) -dimensional subspaces in PG(5 h − 1 , q ) for any prime power q = p m > 2 . Pr oof. Suppose for a contradiction that X is such a set, with |X | = ( t − 3) q h + t . The quotient of X at any element is a length-maximal set in PG(4 h − 1 , q ) with parameter t − 1 . By Theorem 7, we obtain t − 1 = q h + 1 , that is t = q h + 2 . Hence n = |X | = q 2 h + 2 . The secant types are 0 , 1 , 2 , and t = q h + 2 . Consider the quantity δ = N ( q h + 2 , h, 5 , q h + 2) + N ( q h + 2 , h, 5 , 2) − | PG(5 h − 1 , q ) | , (2) which satisfies δ ≤ 0 since N ( q h + 2 , h, 5 , 0) and N ( q h + 2 , h, 5 , 1) are non-negati ve. Substituting parts 5 and 6 of Lemma 6 and expanding, we obtain δ ( q − 1) = ( q h ) 6 − 5( q h ) 5 + 7( q h ) 4 − 3( q h ) 3 = ( q h ) 3 ( q h − 1) 2 ( q h − 3) . (3) Since q > 2 , we have q h ≥ 3 . If q h > 3 then δ ( q − 1) > 0 , so δ > 0 , contradicting δ ≤ 0 . If q h = 3 (that is, q = 3 and h = 1 ) then δ = 0 , so all of N ( q h + 2 , h, 5 , 0) and N ( q h + 2 , h, 5 , 1) v anish, and X corresponds to a projectiv e system equiv alent to a linear [11 , 5 , 6] 3 code. No such code exists [1, Corollary 30]. In all cases we hav e a contradiction. Remark 9. Restricting to the case h = 1 , Theor em 8 resolves Conjectur es 1 and 2 of [1]. Remark 10. By induction on the quotient construction, Theor em 8 extends to all k ≥ 5 when q > 2 : for k ≥ 6 , any length-maximal set in PG( k h − 1 , q ) would yield, by taking the quotient at an element, a length-maximal set in PG(( k − 1) h − 1 , q ) , and none xistence pr opagates upwar d fr om k = 5 . 8 T im Alderson, Simeon Ball 4 Remarks on small cases and open questions For k = 2 , a length-maximal set is a set of ( h − 1) -dimensional subspaces of PG(2 h − 1 , q ) such that e very hyperplane contains at most t elements, with |X | = tq h + t = t ( q h + 1) . This corresponds to a partition of PG(2 h − 1 , q ) into ( h − 1) -flats (a spread), which exists for all q and h . For k = 3 , length-maximal sets correspond to the higher-dimensional analogues of maximal arcs. The field reduction of a maximal arc in PG(2 , q h ) yields an example for ev ery parameter t | q h and all q ev en. Mathon’ s example [6] provides an instance with k = 3 , h = 1 , q = 3 , t = 3 in odd characteristic, and to our knowledge is the only such example in odd characteristic. Whether this example is unique, and whether further e xamples exist in odd characteristic for lar ger parameters, are questions that remain open. In the language of additiv e codes, our results assert that additiv e two-weight [ n, k , d ] h q codes meeting the maximal arc bound do not exist for k ≥ 5 when q > 2 . For k = 4 and q > 2 , any such code must satisfy t = q h + 1 and ha ve exactly tw o nonzero weights n and n − t ; existence remains open. Refer ences [1] T .L. Alderson and Z. Zhang, Projectiv e systems and bounds on the length of codes of non-zero defect, J . Algebra Combin. Discr ete Struct. Appl. , to appear . [2] S. Ball, A. Blokhuis and F . Mazzocca, Maximal arcs in Desarguesian planes of odd order do not exist, Combinatorica , 17 (1997) 31–41. [3] S. Ball and A. 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