The Known Maximal Partial Ovoids of Size  of

Coolsaet, Kris; Beule, Jan De; Siciliano, Alessandro

doi:10.1002/jcd.21307

Cited by 7 publications

(12 citation statements)

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“…It is our feeling that such a characterisation could be helpful in proving the non-existence for larger q. We note that in [3], where a completely different approach is used, a comparable conclusion on the pairs point-antipode is made. Finally, we also mention the work in [6], where the non-existence for larger q is shown under the extra assumption that (q 2 − 1) 2 divides the order of the automorphism group of the maximal partial ovoid.…”

Section: Proofmentioning

confidence: 86%

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On large maximal partial ovoids of the parabolic quadric Q(4, q)

Beule

2012

Des. Codes Cryptogr.

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We use the representation T 2 (Ø) for Q(4, q) to show that maximal partial ovoids of Q(4, q) of size q 2 − 1, q = p h , p odd prime, h > 1, do not exist. Although this was known before, we give a slightly alternative proof, also resulting in more combinatorial information of the known examples for q odd prime. keywords: maximal partial ovoid, generalized quadrangle, parabolic quadric.MSC (2010): 05B25, 51D20, 51E12, 51E20, 51E21.

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Section: Proofmentioning

confidence: 86%

“…We used an explicit description of the known examples ( [3]) to compute the intersection numbers with all elliptic quadrics. We list the results.…”

Section: Proofmentioning

confidence: 99%

On large maximal partial ovoids of the parabolic quadric Q(4, q)

Beule

2012

Des. Codes Cryptogr.

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“…Next we give a construction for maximal partial spreads of W 3 (q) of size q 2 − 1. Note that, by duality, such a partial spread is equivalent to a maximal partial ovoid of Q 4 (q) of size q 2 − 1 and the construction and examples given in Example 5.4 are hence equivalent to those given in [11] (see, in particular, Theorem 4 and the Table on page 5 of [11]). Note also that for q = 5, 7, 11 these examples were first found by Penttila with the aid of a computer.…”

Section: Codes In W 3 (Q)mentioning

confidence: 98%

“…Example 5.4. The following is the dual of the construction considered in [11]. Let V ∼ = F 4 q be equipped with the symplectic form f (x, y) = x 1 y 2 − x 2 y 1 − x 3 y 4 + x 4 y 3 .…”

Section: Codes In W 3 (Q)mentioning

confidence: 99%

Neighbour-Transitive Codes and Partial Spreads in Generalised Quadrangles

Crnković¹,

Hawtin²,

Švob³

2021

Preprint

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A code C in a generalised quadrangle Q is defined to be a subset of the vertex set of the point-line incidence graph Γ of Q. The minimum distance δ of C is the smallest distance between a pair of distinct elements of C. The graph metric gives rise to the distance partition {C, C 1 , . . . , C ρ }, where ρ is the maximum distance between any vertex of Γ and its nearest element of C. Since the diameter of Γ is 4, both ρ and δ are at most 4. If δ = 4 then C is a partial ovoid or partial spread of Q, and if, additionally, ρ = 2 then C is an ovoid or a spread. A code C in Q is neighbour-transitive if its automorphism group acts transitively on each of the sets C and C 1 . Our main results i) classify all neighbour-transitive codes admitting an insoluble group of automorphisms in thick classical generalised quadrangles that correspond to ovoids or spreads, and ii) give two infinite families and six sporadic examples of neighbourtransitive codes with minimum distance δ = 4 in the classical generalised quadrangle W 3 (q) that are not ovoids or spreads. * This work has been supported by the Croatian Science Foundation under the projects 6732 and 5713.1. C is equivalent to a regular spread if and only if C has covering radius 2.2. If |C| = q 2 then C has covering radius ρ = 4 and is equivalent to a spread minus a line.3. If |C| = q 2 − 1 then q = 2, 3, 5, 7 or 11 and C is equivalent to one of the codes in Example 5.4, each of which has covering radius ρ = 3.4. If |C| = q + 1 and C has covering radius ρ = 3 then C is equivalent to the set of points on a hyperbolic line.5. If q = 3 and |C| = 5 then C has covering radius ρ = 3 and is equivalent to the code given in Example 5.3.Note that an example of a code as in part 2 of Theorem 1.2 is given, for each q, by a regular spread of W 3 (q) with one line removed; see Lemma 4.3. We do not prove here that these are all such examples. However, since the automorphism group of such a code must have order divisible by q 2 (q + 1), [32, Table 1] suggests that perhaps this is the case. Hence, we pose the following question.

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“…We consider the case when U is extendible as the typical one: otherwise N has a very restricted (strong) structure; although note that there exist examples of maximal point sets U , of size q 2 − 2, q ∈ {3, 5, 7, 11}, not determining the points of a conic at infinity. These examples occur in the theory of maximal partial ovoids of generalized quadrangles, and where studied in [12], [4], and [6]. Non-existence of such examples for q = p h , p an odd prime, h > 1, was shown in [7].…”

Section: Now We Generalize Theoremmentioning

confidence: 99%