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Consider the n-dimensional projective space over a finite field with q elements. A spread in is a set of lines which partition the point set. A parallelism is a partition of the set of lines by spreads. A deficiency one parallelism is a partial parallelism with one spread less than the parallelism. A transitive deficiency one parallelism corresponds to a parallelism possessing an automorphism group which fixes one spread and is transitive on the remaining spreads. Such parallelisms have been considered in many papers. As a result, an infinite family of transitive deficiency one parallelisms of has been constructed for odd q, and it has been proved that the deficiency spread of a transitive deficiency one parallelism must be regular, and its automorphism group should contain an elation subgroup of order . In the present paper we construct parallelisms of invariant under an elation group of order 49 with some additional properties, and thus we succeed to obtain all (46) transitive deficiency one parallelisms of . The three parallelisms from the known infinite family are among them. As a by-product, we also construct a much bigger number (55,022) of parallelisms which have the same spread structure, but are not transitive deficiency one.
We consider the n-dimensional projective space over the finite field . A spread in this finite projective space is a set of lines which partition the point set. Spreads exist when is even. Two spreads are isomorphic if there is an automorphism (collineation) of the projective space which maps one to the other.
A partition of the set of lines by spreads is a parallelism. Two parallelisms are isomorphic if there is an automorphism of which maps the spreads of one parallelism to spreads of the other. An automorphism of a parallelism is an automorphism of which preserves the parallelism, namely it maps each of its spreads to a spread of the same parallelism.
Spreads and parallelisms of are of interest for problems from projective geometry , design theory , network coding , error-correcting codes , and cryptography . The relation of spreads of to translation planes of order is well known . Every regular parallelism in is connected to a spread in , and, hence, to a translation plane of order [7,8] and vice versa . Recent intensive research is also motivated by the relation to subspace codes  and Grassmannian codes  involved in random network coding . Modern data centers use PIR codes to reduce the storage overhead and one of the ways they can be constructed is based on parallelisms of . These links of the research topic to problems from other research fields make it very important. More details and data on spreads and parallelisms can be found in [13,14,15].
Projective spaces of dimension are the most studied ones. The existence of parallelisms of follows from the constructions of Denniston  and Beutelspacher . The first constructed and best known parallelisms are in . Recently, Betten  classified all parallelisms of by computer. For bigger parameters, n and q, the classification of all parallelisms of the corresponding projective space becomes infeasible. During the years many authors have studied parallelisms and a lot of theoretical results have been obtained, for example [7,13,17,20,21]. There are many results found by computational methods too, for instance [19,22,23,24,25]. Authors often restrict the search space by assuming admissible automorphism groups or additional properties of the constructed objects. Examples of such properties follow.
A parallelism is transitive if it has an automorphism group which is transitive on the spreads. It can be defined by one spread and the corresponding automorphism group. A transitive parallelism is cyclic if there is an automorphism which permutes its spreads in one cycle. The easiest case for construction of parallelisms is to consider cyclic parallelisms. White  proved the non-existence of cyclic parallelisms of with which holds for too.
A regulus of is a set of mutually skew lines, such that any line intersecting three elements of the regulus intersects all its elements. Such a line is called a transversal. All the transversals of a regulus form its opposite regulus. A spread of is regular if it contains the unique regulus determined by any three of its elements. The regular spread is unique up to isomorphism . A spread is called Hall spread if it can be obtained from a regular spread by a replacement of one regulus by its opposite.
A parallelism is uniform if all its spreads are isomorphic. A parallelism is regular if all its spreads are regular. There exists an infinite class of cyclic regular parallelisms of for each due to Pentilla and Williams . It incorporates regular parallelisms in  and in . Parameters of do not match to this class.
The number of points equals the number of hyperplanes of . The dual space of can be obtained by reversing the inclusion relation, namely, in the dual space hyperplanes become points, points become hyperplanes, and lines stay lines. A spread is a spread and a parallelism is a parallelism in the dual space of . A parallelism which is isomorphic to its dual is self-dual.
A deficiency one parallelism is a partial parallelism with one spread less than the parallelism. Each deficiency one parallelism can be uniquely extended to a parallelism. A transitive (partial) parallelism possesses an automorphism group acting transitively on its spreads. A transitive deficiency one parallelism is a parallelism with an automorphism group that fixes the deficiency spread and is transitive on the remaining spreads. Therefore, a transitive deficiency one parallelism can be completely defined by the deficiency spread, one spread from the transitive part, and the corresponding automorphism group.
Biliotti, Jha, and Johnson  and Diaz, Johnson, and Montinaro  determine the properties of transitive deficiency one parallelisms of finite projective spaces. They prove that the deficiency spread must be regular and the automorphism group should contain an elation subgroup (a subgroup which fixes all points of one line) of order ( Theorem 267).
There is an infinite class of transitive deficiency one parallelisms in for , p an odd prime, due to Johnson. Their deficiency spread is regular, the remaining spreads are Hall, and the parallelisms are invariant under the full central collineation group (all automorphisms of the projective space which fix the points of one line) ( Theorem 174). Let us call them Johnson type parallelisms. They comprise part of all transitive deficiency one parallelisms in . The rest are invariant under subgroups of the full central collineation group .
Johnson-type parallelisms have, in general, the same structure as the parallelisms of Denniston  and Beutelspacher , but the construction used by Johnson allows to consider the isomorphism classes of the parallelisms too. In ( Corollary 26), Johnson and Pomareda show that for , an odd prime, the number of nonisomorphic Johnson-type parallelisms is exactly . Theorem 27 in the same paper shows that the order of their full automorphism groups is 2 or 4 times the order of the full central collineation group of the regular spread.
All transitive deficiency one parallelisms of for are known [19,24,25]. The transitive deficiency one parallelisms of are 12 and two of them are of the Johnson type (which complies with the formula given in ).
The next open problem is the classification of transitive deficiency one parallelisms of . This is the main aim of our work. We use the above mentioned group-theoretic characterization ( Theorem 267) to construct all transitive deficiency one parallelisms of . A more general purpose of our work is to construct new examples of parallelisms of . It is of importance for the study of the properties of these objects.
Our approach to the problem is different from the one which was used in  to obtain all the transitive deficiency one parallelisms of because the construction of all parallelisms of with automorphisms of order is presently infeasible by our method. That is why we impose additional restrictions which follow from the transitivity requirement, and we obtain all parallelisms for which they hold. We then check which of them are transitive deficiency one.
The paper consists of five sections. Section 2 considers the construction of the parallelisms, and Section 3 the backtrack search algorithms that we use. The obtained results are presented in Section 4, and additional comments on their impact are made in Section 5.
We use -the 4-dimensional vector space over to construct . The points of are all 4-dimensional vectors over , such that if i is the maximum index for which . We sort these 400 vectors in ascending lexicographic order and then assign them numbers, such that is number 1, and number 400. There are 2850 lines (1-dimensional subspaces) in . We sort them in lexicographic order defined on the numbers of the points they contain and assign to each line a number according to this order. The first line , while .
A spread in has 50 lines which partition the point set and a parallelism has 57 spreads. Each invertible matrix over defines an automorphism of this projective space by the map .
The deficiency spread of a transitive deficiency one parallelism of must be regular . In , there are 1347 spreads up to isomorphism. The regular spread is unique up to isomorphism and the construction of transitive deficiency one parallelisms can start with the deficiency spread . Without loss of generality we choose for the smallest in the considered lexicographic order regular spread which begins with the first line .
2.2. Automorphism Groups
The authomorphism (collineation) group of is isomorphic to the projective general semi-linear group . Let us denote it by . It is of order . We use permutation representation of the groups. To find the generators of G we use the "Isomorphism and automorphism group" module of the Q-Extension program . For the other computations on the needed groups and subgroups, we use the computer algebra system GAP  and the software of the first author where needed.
It follows from the investigations of Biliotti, Diaz, Jha, Johnson, and Montinaro ( Theorem 267) that a transitive deficiency one parallelism of is invariant under an elation group of order which fixes a regular spread. That is why we construct parallelisms invariant under the elation group of order 49 which fixes the regular spread ().
We start with constructing . We are interested in to be the deficiency spread. Since should act as an elation on , it must fix one of its lines pointwise. Therefore, we find the generators of the automorphism group of which fixes the line pointwise. This is the full central collineation group with axis . It is of order and has up to conjugacy only one subgroup of order 49. We let be this subgroup, and try to construct transitive deficiency one parallelisms invariant under it. can be generated by the two automorphisms of order 7 of which are defined by the following two invertible matrices:
We need the normalizer of in G for the rejection of isomorphic solutions . It is defined as and . There are permutations in which do not preserve . After their removal, a group N of order remains. We want to construct parallelisms invariant only under this particular , therefore we use the group N to find all isomorphism classes.
Theorem 267  states that the subgroup of order of the full automorphism group of a transitive deficiency one parallelism in is a normal subgroup. Hence, we can find the factor group
2.3. A Parallelism Invariant under
To obtain a parallelism of we need 57 spreads. and the group which fixes it are already known. Next the remaining 56 spreads of the parallelism must be added. The group acts on the points and lines of as it is shown in Table 1. The first point is in and in eight other line orbits under . Hence a parallelism fixing and invariant under can be presented by eight spreads (orbit leaders) each one with a spread orbit of length 7 under . Let us denote the orbit leader by , and its spread orbit by .
has 8 subgroups of order 7 which are in one conjugacy class under G. The action of on the points and lines of is given in Table 1. A line orbit can entirely be included in a spread if it comprises pairwise disjoint lines. We call such an orbit spread-like orbit. The line orbits without this property are non-spread-like. The line orbits of length 49 are all spread-like. The number of the line orbits of length 7 are given as “spread-like + non-spread-like” in Table 1.
fixes pointwise and other 56 lines not pointwise. It also fixes all the 7 spreads of one of the spread orbits of length 7. Without loss of generality, we can assume that fixes and the other spreads of . Figure 1 shows the structure of . We assume that consecutive numbers have been assigned to the line orbits under and to the elements of each line orbit.
Each row in Figure 1 presents one of the 7 spreads of . Each spread has 8 lines which are fixed by and are in different orbits of length 7 under . We denote such a line by where denotes its orbits of length 7 under , and is the number of this line in its orbit under . The lines are all the 56 lines fixed by .
Apart from the 8 fixed lines each spread of contains 6 spread-like line orbits of length 7 under , where is the line orbit of length 49 to which the line belongs, and are the numbers of these lines in the line orbit of length 49. The action of and its subgroups coincide with the theoretical results about the automorphism group of the transitive deficiency one parallelisms in ( Theorem 13).
A parallelism invariant under has the following structure
A transitive deficiency one parallelism can be considered as a parallelism invariant under possessing an automorphism group which is transitive on the non-trivial spread orbits .
2.4. The Isomorphism of Solutions
For the construction of transitive deficiency one parallelisms we use an exhaustive backtrack search which leads to lexicographically ordered objects. This allows the rejection of partial solutions which are not minimal with respect to the chosen lexicographic order, because they already have been constructed. Our way to achieve this is by applying a minimality test to some of the partial solutions and to all full solutions . The minimality test checks if there exists an element of the group N which maps the current solution to a lexicographically smaller one. If such an element is found, the current partial solution is rejected.
Computer-aided classification of combinatorial structures requires methods for their construction, as well as methods for the rejection of equivalent ones. Often these two stages interleave and the objects can be constructed in a way that makes it possible to reduce the appearance of equivalent solutions. Depending on the theoretical requirements imposed on the objects different algorithms can be incorporated. For example, in , Bouyukliev uses the concept of canonical augmentation; for the classification of parallelisms in , Betten applies the Schmalz algorithm that proceeds along a chain of subobjects which are extended to larger subobjects until the target objects are classified .
The software used by the authors in the present work is based on the orderly generation technique  which implies exhaustive backtrack search, lexicographically ordered objects and a minimality test on partial solutions. The authors use their own programs written in C++. Each of the authors implements a slightly different construction algorithm.
Each point (the first one too) has to be in each spread of the parallelism. In the described lexicographic order point 1 is in the first 57 lines. Line is in and without loss of generality we assume that contains line . Thus, we start by searching for which begins with line .
3.2. Method 1
All possibilities for are constructed in advance. For that purpose the subgroup which fixes is used. It is defined by the matrix . One of the 14 line orbits under which contains (see Figure 1) is known (). The rest are chosen in all possible ways from the 400 spread-like line orbits under (see Table 1). This is performed by the function S1Construct(2).
Here, NotPossibleLineOrb returns true if some of the lines of the considered i-th line orbit under intersect lines of the already chosen Orb-1 orbits, or if no more orbits of this length can be added (there must be 8 orbits of length 1 and 6 orbits of length 7—Figure 1). Put adds the orbit to the spread and Take removes it. If the spread is ready, it is saved by WriteSpread, and if more orbits have to be added, S1Construct(Orb+1) is called to choose the next orbit.
There are spreads with fixed by . In this list remain spreads if the spreads not disjoint to are removed. Among them, there are 8084 spreads with a spread orbit of length 7 under .
In a transitive deficiency one parallelism the spread is mapped to the orbit leaders of , (see (2)) under the action of a subgroup of the group H defined in (1). We apply all the elements of H to each of the 8084 possibilities for and remove the spreads which are mapped by these elements to less than 7 different spreads. At this step, isomorphism check on the partial solutions with two spread orbits is performed. As a result 15 non-isomorphic partial solutions for the first two spread orbits remain. For each of them we save a list L of all possible to which is mapped under H. To construct , a backtrack search on the elements of L is performed by PConstr(2), where allS=|L| and SpreadOK is called to check if the i-th spread possibility does not have common lines with the obtained until this moment partial parallelism.
At the end isomorphism check is applied to the obtained parallelisms. The extension of one of the 15 partial solutions takes about a week on a 3 GHz PC. That is why the resources of the National Centre for High Performance and Distributed Computing were used.
3.3. Method 2
By this method, the backtrack search to obtain parallelisms is not applied on lists of spreads constructed in advance, but on the lines of the projective space. The lines of the spread orbit leaders are constructed in consecutive order by exhaustive backtrack search on the lines that are not in the partial solution yet. Details on this ’line by line’ construction approach can be found in . The algorithm used here is generally the same as in , but more restrictions on the constructed spreads are added because our aim is to obtain transitive deficiency one parallelisms.
We know the first spread , fix one line in each spread orbit leader , and choose their other spread lines by MakePar(2, 1).
void MakePar(int Line, int Spr)
int Point = FirstMissingPoint(Line, Spr);
for(int i = FirstLine[Point]; i<=LastLine[Point]; i++)
if(Possible(Line, i, Spr))
PutLine(Line, i, Spr);
else MakePar(2, Spr+1);
else MakePar(Line+1, Spr);
FirstMissingPoint returns the number of the first missing point in spread Spr. Each point must be in one line of the spread. Therefore, we try to add only lines containing the first missing point. Their numbers are between FirstLine[Point] and LastLine[Point]. Possible returns true if the considered line i has no common points with the lines of the current partial spread. If this is the case, PutLine adds it to the current solution. We continue to add lines until all points are covered (i.e., the number of lines is 50) and when this happens, RestrictionsOK checks if the imposed restrictions hold. If so, MakePar(2, Spr+1) starts adding the next spread, or WriteParallelism saves a ready parallelism.
We obtain 15 non-isomorphic partial solutions for the first two spread orbits after rejecting by RestrictionsOK solutions for for which there exists a line such that no element of H maps any line of to (because in this case transitivity on is not possible). We also save a list L of the different spreads to which is mapped by the elements of H. From the solutions for we reject by RestrictionsOK those which are not in L and those which are in L, but for which there exists a line not contained in the partial solution and not contained in any of the spreads from L that can be added after .
Method 2 was implemented by the first author, and Method 1 by the second one. The number of the transitive deficiency one parallelisms constructed by them is the same. With this technique we also construct some parallelisms which are not transitive deficiency one.
4.1. The Transitive Deficiency One Parallelisms of
The main aim of this work is to construct transitive deficiency one parallelisms of . We obtain 46 transitive deficiency one parallelisms of . The distribution of the constructed parallelisms by the order of their automorphism groups is presented in Table 2, where All is the number of parallelisms with a full automorphism group of order Auts. The rows below the double line show how many of these parallelisms are invariant under elation groups of given orders, namely E A is the number of parallelisms invariant under an elation group of order A.
Three of the transitive deficiency one parallelisms are invariant under the full central collineation group of order 2352 and are, therefore, of Johnson type. The full automorphism group orders and the number of these parallelisms comply with the results obtained in . The remaining 43 transitive deficiency one parallelisms have the same spread structure as the parallelisms from Johnson’s infinite family, but do not belong to it.
We construct the duals of these 46 parallelisms. The dual parallelisms are transitive deficiency one parallelisms with the same order of the automorphism groups, but they are not isomorphic to the original parallelisms. The duals of the Johnson-type parallelisms are not of the Johnson type.
4.2. Uniform Deficiency One Parallelisms of
By our construction methods we obtain 55,022 non-isomorphic uniform deficiency one parallelisms. All transitive deficiency one parallelisms are among them, but not all the constructed parallelisms are transitive deficiency one. These parallelisms are of particular interest too, because not many parallelisms of are known. The order of their full automorphism groups and the number of their spread orbits are presented in Table 3, where All is the number of parallelisms with a full automorphism group of order Auts. The rows below the double line show how many of these parallelisms have a given number of spread orbits under their full automorphism groups, namely k orbits is the number of parallelisms whose spreads are in k orbits under the full automorphism group of the parallelism.
The parallelisms are constructed assuming invariance under and, thus, their spreads are in at most nine orbits (see Section 2.3). Under the full automorphism group the number of spread orbits can decrease. The last row in the table corresponds to the obtained transitive deficiency one parallelisms of . In that case the deficiency spread comprises the first orbit, and the other spreads the second one.
The number of reguli in the regular spread of is
The number of reguli in the other spreads of is smaller. To recognize the type of the spreads in the constructed parallelisms, we use invariants based on the intersection of the spread with the reguli of the projective space as in . These invariants can be represented by the triple , where s is the number of whole reguli in the spread and r (w) is the number of reguli which share exactly seven (six) lines (out of all lines) with the spread. For the regular spread of this invariant is and for the Hall spread it is .
All the constructed parallelisms have the same structure: one regular spread and the remaining spreads are Hall spreads. They are available online and can be used for further investigations as well as in suitable applications.
5. Discussion on the Obtained Results
Only several parallelisms of were known before the present work, while their number is at least now. They include all transitive deficiency one parallelisms of this projective space. All the parallelisms constructed in this work are available online and can be used for further investigations, as well as in suitable applications.
Our results comply with previous theoretical results, namely:
We obtain three parallelisms of Johnson type and it is shown in  that for , an odd prime, the number of non-isomorphic Johnson-type parallelisms is exactly .
The order of the full automorphism groups of the parallelisms of Johnson type that we construct is either 4704, or 9408 and it is shown in  that the order should be two or four times the order of the full central collineation group of the regular spread.
None of the transitive deficiency one parallelisms of is self-dual, as shown in .
It is proved in previous papers on transitive deficiency one parallelisms that their deficiency spread is regular, and they are invariant under an elation group of order . Our investigation on the transitive deficiency one parallelisms of is based on these results. We construct transitive deficiency one parallelisms with one regular spread which are invariant under an elation group of order 49.
Our computer-aided results show that:
There are 46 transitive deficiency one parallelisms of which are invariant under an elation group of order 49 and have a regular deficiency spread. The three parallelisms from Johnson’s infinite class are among them.
All the spreads of the constructed transitive deficiency one parallelism of are Hall spreads and the deficiency spread is regular by assumption.
The dual of a transitive deficiency one parallelism of is a transitive deficiency one parallelism with the same order of the full automorphism group as the original parallelism, but it is not isomorphic to the original parallelism.
The duals of the Johnson type parallelisms of are not of the Johnson-type.
The observations from the last three items can be made on the computer-aided classification of transitive deficiency one parallelisms of too . It is possible that they hold in each for an odd prime p, and this is an interesting open problem, which we think is a challenge for future theoretical considerations. If the duals of the Johnson-type parallelisms of are always transitive deficiency one, but always not of the Johnson-type, then Johnson’s infinite family will yield a twice bigger number of transitive deficiency one parallelisms (the family members and their duals).
We believe that our present computer-aided investigation will be helpful to further theoretical and computer-aided considerations of these interesting objects and their applications.
Conceptualization, S.T. and S.Z.; methodology, S.T. and S.Z.; software, S.T. and S.Z.; validation, S.T.; formal analysis, S.T. and S.Z.; investigation, S.Z.; resources, S.T.; data curation, S.Z.; writing—original draft preparation, S.T. and S.Z.; writing—review and editing, S.T. and S.Z. All authors have read and agreed to the published version of the manuscript.
S.T. was partially supported by the Bulgarian National Science Fund under Contract No. KP-06-H62/2/13.12.2022, and S.Z. was partially supported by the Bulgarian National Science Fund under Contract No. KP-06-N32/2-2019.
Institutional Review Board Statement
Data Availability Statement
The data obtained and presented in this study are openly available at http://www.moi.math.bas.bg/moiuser/~stela (accessed on 24 May 2023). Transitive deficiency one parallelisms are in a separate file with the point-line incidence matrix of . For each parallelism the order of its full automorphism group, the order of the elation subgroup, and the name of its dual parallelism is given.
We are grateful to the anonymous referees for the suggestions which have improved the presentation of the results. We acknowledge the access to the e-infrastructure provided by the Grant No. D01-168/28.07.2022 “National Centre for High Performance and Distributed Computing” of the Ministry of Education and Science of Bulgaria.
Conflicts of Interest
The authors declare no conflict of interest.
The following abbreviations and notations are used in this manuscript:
the finite field with q elements with addition and
multiplication defined on them
the n-dimensional projective space over the finite field
the projective general semi-linear group
a group of automorphisms of which fixes all points of one line
elation (automorphism of which fixes all points of one line)
the group of automorphisms of the considered projective space ,
an elation subgroup of G of order 49
the eight subgroups of order 7 of
the regular deficiency spread of the constructed parallelisms
the normalizer of in G,
the subgroup of which preserves
representatives (orbit leaders) of the spread orbits of length 7
the spread orbits of length 7 of a parallelism invariant under
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The structure of .
The structure of .
Action of and on the lines of .
Action of and on the lines of .
343 + 56
0 + 64
Transitive deficiency one parallelisms of -order of the full automorphism groups and their elation subgroups.
Transitive deficiency one parallelisms of -order of the full automorphism groups and their elation subgroups.
Spread orbits and order of the full automorphism groups of the constructed uniform deficiency one parallelisms.
Spread orbits and order of the full automorphism groups of the constructed uniform deficiency one parallelisms.
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Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely
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the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas,
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