Basic references on Hermitian polarities, Hermitian curves and the classical Hermitian unital are [Hir98, Section II.5] and [BE08, Chapter 2]. On Buekenhout's construction of unitals embedded in translation planes, see [BE08, Chapter 4]. The cyclic unital of order six is due to Bagchi and Bagchi [BB89].
‣ HermitianAbstractUnital( q ) | ( function ) |
Returns: The classical unital object, which is the abstract unital of order q isomorphic to the Hermitian curve in the classical projective plane.
The Hermitian curve has the following canonical equation: \(X_0^{q + 1} + X_1^{q + 1} + X_2^{q + 1} = 0\). The function computes the blocks of the unital with the help of PGU(3,q) and calls AbstractUnitalByDesignBlocks. The Name of the unital is set as HermitianAbstractUnital(q).
‣ AllBuekenhoutMetzAbstractUnitalParameters( q ) | ( function ) |
Returns: All the pairs of parameters over \(GF(q^2)\) which result non-isomorphic (orthogonal) Buekenhout-Metz unitals of order q.
The argument q must be a prime power (if even, then at least 4).
If q is an odd prime power and \((\alpha, \beta)\) is 2-tuple of \(GF(q^2)\), then this pair is a suitable parameter of an orthogonal Buekenhout-Metz unital, if \((\beta^q - \beta)^2 + 4 \alpha^{q + 1}\) is a nonsquare in \(GF(q)\).
If q is an even prime power and \((\alpha, \beta)\) is 2-tuple of \(GF(q^2)\), then this pair is a suitable parameter of an orthogonal Buekenhout-Metz unital, if \(\beta \not\in GF(q)\) and \(\alpha^{q + 1}(\beta^q + \beta)^2\) has absolute trace 0.
In both cases \(\alpha = 0\) yields the Hermitian classical unital, hence we omit the tuples with \(\alpha = 0\).
‣ OrthogonalBuekenhoutMetzAbstractUnital( q, alpha, beta ) | ( function ) |
Returns: The unital object, which is the abstract unital of order q isomorphic to the orthogonal Buekenhout-Metz unital with parameters alpha and beta in the classical projective plane.
The argument q must be a prime power (if even, then at least 4), the other arguments alpha and beta - elements of \(GF(q^2)\) - must be a pair from AllBuekenhoutMetzAbstractUnitalParameters(q).
The point set \(U_{\alpha, \beta} = \left\{ ( x, \alpha x^2 + \beta x^{q + 1} + r, 1) \colon x \in GF(q^2), r \in GF(q) \right\} \cup \left\{ (0, 1, 0) \right\}\) in \(PG(2,q^2)\) is a unital (called the orthogonal Buekenhout-Metz unital) if the pair of parameters \((\alpha, \beta)\) satisfies the conditions explained in the description of AllBuekenhoutMetzAbstractUnitalParameters(q).
‣ BuekenhoutTitsAbstractUnital( q ) | ( function ) |
Returns: The unital object, which is the abstract unital of order q isomorphic to the Buekenhout-Tits unital in the classical projective plane.
The argument q must be a power of 2, such that the exponent is an odd integer at least 3. The point set \(U_T = \left\{ ( x_0 + x_1 \delta, y_0 + (x_0^{\tau + 2} + x_1^\tau + x_0x_1)\delta, 1) \colon x_0, x_1, y_0 \in GF(q)\right\} \cup \left\{ (0,1,0) \right\}\) in \(PG(2,q^2)\) is a unital (called the Buekenhout-Tits unital) if \(\delta \in GF(q^2) \setminus GF(4)\) and \(\delta^q = 1 + \delta\). This \(\delta\) is just a basis element along with 1 in \(GF(q^2)\) over \(GF(q)\), hence we can omit it as a parameter. The function \(\tau \colon GF(q) \rightarrow GF(q)\) assigns to the field element \(x\) the following: \(x \mapsto x^{2^\frac{k + 1}{2}}\), where \(q = 2^k\).
‣ BagchiBagchiCyclicUnital( n ) | ( function ) |
Returns: A unital object of order n, with a cyclic automorphism group acting on the points.
The construction method needs a positive integer \(n\) such that \(n+1\) and \(n^2-n+1\) are primes. For \(n\leq 20\), only the parameters \(n=4\) and \(n=6\) yield an abstract unital.
The package contains the following libraries of abstract unitals:
Class BBT: 909 unitals of order 3 by Betten, Betten and Tonchev [BBT03].
Class Krcadinac: 4466 unitals of order 3 with nontrivial automorphism groups by Krčadinac [Kr{č}06]. 722 of the BBT unitals appear in this class.
Class KNP: 1777 unitals of order 4 by Krčadinac, Nakić and Pavčević [KNP11].
Class P3M: 173 unitals of order 3, constructed by paramodification from the BBT and Krcadinac libraries [MN20].
Class P4M: 25641 unitals of order 4, constructed by paramodification from the KNP libraries [MN20].
Class SL28inv: 6 \(SL(2,8)\)-invariant unitals of order 8 with many translation centers, constructed by Möhler [Möh20] using the Grundhöfer-Stroppel-Van Maldeghem method [GSVM16].
‣ BBTAbstractUnital( n ) | ( function ) |
Returns: The nth (abstract) unital of order 3 of the unitals by Betten, Betten and Tonchev.
‣ KNPAbstractUnital( n ) | ( function ) |
Returns: The nth (abstract) unital of order 4 of the unitals by Krčadinac, Nakić and Pavčević.
‣ KrcadinacAbstractUnital( n ) | ( function ) |
Returns: The nth (abstract) unital of order 3 of the unitals by Krčadinac.
‣ P3MAbstractUnital( n ) | ( function ) |
Returns: The nth (abstract) unital of order 3, constructed by paramodification from the BBT and Krcadinac libraries.
‣ P4MAbstractUnital( n ) | ( function ) |
Returns: The nth (abstract) unital of order 4, constructed by paramodification from the KNP libraries.
‣ SL28InvariantAbstractUnital( n ) | ( function ) |
Returns: The nth (abstract) \(SL(2,8)\)-invariant unital of order 8, constructed by the Grundhöfer-Stroppel-Van Maldeghem method.
‣ DisplayUnitalLibraryInfo( ) | ( function ) |
The function prints the information about the available libraries of unitals.
‣ NumberOfAbstractUnitalsInLibrary( name ) | ( function ) |
Returns: The number of abstract unitals in the library name.
‣ ReadLibraryDataFromFiles@( data ) | ( function ) |
Returns: The list of boolean incidence matrices of size \((q^3 + 1) \times q^2(q^2 - q + 1)\) read from filename.
The argument data must be a record with fields filename, order, nr. The file data.filename must be gzipped and must contain data.nr matrices of dimension mentioned above. The matrices must be 0-1 matrices without any whitespace between the entries in one row and there must not be any empty lines between matrices.
‣ ReadAbstractUnitalFromLibraryNC@( name, n ) | ( function ) |
Returns: The nth abstract unital from the library name. Non-checking version.
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