# Preprints

## Intersections of the Hermitian surface with irreducible quadrics in \(PG(3,q^2)\), \(q\) odd

In \(PG(3,q^2)\), with q odd, we determine the possible intersection sizes of a Hermitian surface H and an irreducible quadric Q having the same tangent plane at a common point P.

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Finite Fields Appl. 30: 1-13 (2014), ISSN: 1071-5797, doi:10.1016/j.ffa.2014.05.005

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## Codes and caps from orthogonal Grassmannians

In this paper we investigate linear error correcting codes and projective caps related to the Grassmann embedding \(\varepsilon_k^{gr}\) of an orthogonal Grassmannian \(\Delta_k\). In particular, we determine some of the parameters of the codes arising from the projective system determined by \(\varepsilon_k^{gr}(\Delta_k)\). We also study special sets of points of \(\Delta_k\) which are met by any line of \(\Delta_k\) in at most \(2\) points and we show that their image under the Grassmann embedding \(\varepsilon_k^{gr}\) is a projective cap.

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Finite Fields Appl. 24: 148-169 (2013), ISSN: 1071-5797, doi:10.1016/j.ffa.2013.07.003

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## Looking for ovoids of the Hermitian surface: a computational approach

In this note we present a computational approach to the construction of ovoids of the Hermitian surface and show some related experimental results.

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## BIB-Designs from circular nearrings

Let \((N,\Phi)\) be a finite circular Ferrero pair. We define the disk with center \(b\) and radius \(a\), \(\mathcal{D}(a;b)\), as

$$ \mathcal{D}(a;b)=\{x\in \Phi(r)+c\mid r\neq 0,\ b\in \Phi(r)+c,\ |(\Phi(r)+c)\cap (\Phi(a)+b)|=1\} .$$

Using this definition we introduce the concept of interior part of a circle, \(\Phi(a)+b\), as the set \(\mathcal{I}(\Phi(a)+b)=\mathcal{D}(a;b)\setminus (\Phi(a)+b)\).

Moreover, if \(\mathcal{B}^{\mathcal{D}}\) is the set of all disks, then,
in some interesting cases, we show that the incidence structure \((N,\mathcal{B}^{\mathcal{D}},\in)\) is actually a balanced
incomplete block design and we are able to calculate its parameters
depending on \(|N|\) and \(|\Phi|\).

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## Codes and combinatorial structures from circular planar nearrings

Nearrings are generalized rings in which addition is not in general abelian and only one distributive law holds. Some interesting combinatorial structures, as tactical configurations and balanced incomplete block designs (BIBDs) naturally arise when considering the class of planar and circular nearrings. In a previus article the authors define the concept of disk and prove that in the case of field-generated planar circular nearrings it yields a BIBD, called "disk-design". In this paper we present a method for the construction of an association scheme which makes the disk-design, in some interesting cases, an union of partially incomplete block designs (PBIBDs). Such designs can be used in the construction of some classes of codes for which we are able to calculate the parameters and to prove that in some cases they are also cyclic.

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## Some observations on Fr"ohlich's composition rings

Sub-composition rings and full ideals of Fröhlich's composition rings (1968) are determined. Moreover, a generalization of these structures is introduced by consideration of non-commuting sets of indeterminates.

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## Sampling complete designs

In the present paper, complete designs of graphs are considered. The notion of (regular) sampling is introduced and analyzed in detail, showing that the trivial necessary condition for its existence is actually sufficient. Some examples are also provided.

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Discrete Math. 312 (3): 488-497 (2012), ISSN: 0012-365X, doi:10.1016/j.disc.2011.02.034

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## Unitals in \(PG(2,q^2)\) with a large \(2\)-point stabiliser

Let \(\mathcal U\) be a unital embedded in the Desarguesian projective plane \(PG(2,q^2)\). Write \(M\) for the subgroup of \(PGL(3,q^2)\) which preserves \(\mathcal U\). We show that \(\mathcal U\) is classical if, and only if, it has two distinct points \(P, Q\) for which the stabiliser \(G=M_{P,Q}\) has order \(q^2-1\).

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Discrete Math. 312 (3): 532-535 (2012), ISSN: 0012-365X, doi:10.1016/j.disc.2011.03.017

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## Down-linking \((K_v,\Gamma)\)-designs to \(P_3\)-designs

Let \(\Gamma'\) be a subgraph of a graph \(\Gamma\). We define a down-link from a \((K_v,\Gamma)\)-design \(B\) to a \((K_n,\Gamma')\)-design \(B'\) as a map \(f:B\to B'\) mapping any block of \(B\) into one of its subgraphs. This is a new concept, closely related with both the notion of metamorphosis and that of embedding. In the present paper we study down-links in general and prove that any \((K_v,\Gamma)\)-design might be down-linked to a \((K_n,\Gamma')\)-design, provided that \(n\) is admissible and large enough. We also show that if \(\Gamma'=P_3\), it is always possible to find a down-link to a design of order at most \(v+3\). This bound is then improved for several classes of graphs \(\Gamma\), by providing explicit constructions.

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Util. Math., 90: 3-21 (2013), ISSN: 0315-3681

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## On \(d\)-divisible graceful \(\alpha\)-labelings of \(C_{4k}\times P_m\)

In a previous paper the concept of a \(d\)-divisible graceful \(\alpha\)-labeling has been introduced as a generalization of classical \(\alpha\)-labelings and it has been shown how it is useful to obtain certain cyclic graph decompositions. In the present paper it is proved the existence of \(d\)-divisible graceful \(\alpha\)-labelings of \(C_{4k}\times P_m\) for any integers \(k\geq1\), \(m\geq2\) for several values of \(d\).

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## On \(d\)-graceful labelings

In this paper we introduce a generalization of the well known concept of a graceful labeling. Given a graph \(G\) with \(e=dm\) edges, we call \(d\)-graceful labeling of \(G\) an injective function from \(V(G)\) to the set \(\{0,1,2,\dots, d(m+1)-1\}\) such that \(\{|f(x)-f(y)| | [x,y]\in E(G)\} =\{1,2,3,\dots,d(m+1)-1\}-\{m+1,2(m+1),...,(d-1)(m+1)\}\). In the case of \(d=1\) and of \(d=e\) we find the classical notion of a graceful labeling and of an odd graceful labeling, respectively. Also, we call \(d\)-graceful \(\alpha\)-labeling of a bipartite graph \(G\) a \(d\)-graceful labeling of \(G\) with the property that its maximum value on one of the two bipartite sets does not reach its minimum value on the other one. We show that these new concepts allow to obtain certain cyclic graph decompositions. We investigate the existence of \(d\)-graceful \(\alpha\)-labelings for several classes of bipartite graphs, completely solving the problem for paths and stars and giving partial results about cycles of even length and ladders.

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## Decompositions of complete multipartite graphs via generalized graceful labelings

We prove the existence of infinite classes of cyclic \(G\)-decompositions of the complete multipartite graph, \(G\) being a caterpillar, a hairy cycle or a cycle. All the results are obtained by the construction of \(d\)-divisible \(\alpha\)-labelings of \(G\), introduced in [A. Pasotti, On \(d\)-graceful labelings, to appear on Ars Combin.] as a generalization of classical \(\alpha\)-labelings, whose existence implies that one of graph decompositions.

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## A new result on the problem of Buratti, Horak and Rosa

The conjecture of Peter Horak and Alex Rosa (generalizing that of Marco Buratti) states that a multiset \(L\) of \(v-1\) positive integers not exceeding \(\lfloor v/2\rfloor\) is the list of edge-lengths of a suitable Hamiltonian path of the complete graph with vertex-set \(\{0,1,...,v-1\}\) if and only if the following condition (here reformulated in a slightly easier form) is satisfied: for every divisor \(d\) of \(v\), the number of multiples of \(d\) appearing in \(L\) is at most \(v-d\). In this paper we do some preliminary discussions on the conjecture, including its relationship with graph decompositions. Then we prove, as main result, that the conjecture is true whenever all the elements of \(L\) are in \(\{1,2,3,5\}\).

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## On the Buratti-Horak-Rosa Conjecture about Hamiltonian paths in complete graphs

In this paper we investigate a problem proposed by Marco Buratti, Peter Horak and Alex Rosa (denoted by BHR-problem) concerning Hamiltonian paths in the complete graph with prescribed edge-lengths. In particular we solve \(BHR(\{1^a,2^b,t^c\})\) for any even integer \(t\geq4\), provided that \(a+b\geq t-1\). Furthermore, for \(t=4,6,8\) we present a complete solution of \(BHR(\{1^a,2^b,t^c\})\) for any positive integer \(a,b,c\).

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## Loops, regular permutation sets and colouring of directed graphs

We establish a correspondence among loops, regular permutation sets and directed graphs with a suitable edge colouring and characterize regular permutation sets and, respectively, colored graphs giving rise to the same loop, to isomorphic loops and to isotopic loops.

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## Slid product of loops: a generalization

In this paper, following the slid product construction for loops with inverses that we have introduced in a previuous paper, we present the general setting in order to build up a new loop (L,⊕) starting from loops (K,+) equipped with a well ordering "⪯", (P,+ˆ) and (P,+) with the same neutral element. The results established in the aforementioned note are generalized as well. Moreover we investigate the nuclei of L, the normality of subloops isomorphic to (K,+) and (P,+ˆ) and discuss some examples.

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## On \(k\)-symmetries of hyperbolic spaces

We determine all the possible pointwise \(k\)-symmetric spaces of negative
constant curvature. In general such spaces are not \(k\)-symmetric.
In fact we show that, for all \(k\) different from \(2\), the \(3\)-dimensional
hyperbolic space \(H^3\) is not \(k\)-symmetric, i.e. for any possible choice
of a suitable set of selected \(k\)-symmetries, one for each point of \(H^3\),
the regularity condition does not hold.

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## Kikkawa left loops and symmetric \(2\)-structures

We approach the existence of class 1 symmetric \(2\)-structures by studying a class of left loops related to them. Such left loops turn out to satisfy a relation very similar to the left Bol property, and they are of exponent \(2\). Then we give a characterization of these left loops by means of invariant involution sets, and classify them according to the cardinality of the point stabilizers, comparing then our results with the classification of symmetric \(2\)-structures given in [3].

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## \(\alpha\)-labelings of generalized Petersen graphs

An \(\alpha\)-labeling of a bipartite graph \(\Gamma\) of size \(e\) is an injective function \(f: V(\Gamma)\rightarrow \{0,1,2,\ldots,e\}\) such that \(\{|f(x)-f(y)| \ |\ [x,y]\in E(\Gamma)\}=\{1,2,\ldots,e\}\) and with the property that its maximum value on one of the two bipartite sets does not reach its minimum value on the other one. We prove that the generalized Petersen graph \(P_{8n,3}\) admits an \(\alpha\)-labeling for any integer \(n\geq1\) confirming that the conjecture posed by A. Vietri in [A. Vietri, Graceful labellings for an infinite class of generalised Petersen graphs, Ars Combin. 81 (2006), 247-255.]is true. In such a way we obtain an infinite class of decompositions of complete graphs into copies of \(P_{8n,3}\).

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## On optimal \((v,5,2,1)\)-optical orthogonal codes

The size of a \((v,5,2,1)\) optical orthogonal code (OOC) is shown to be at most equal to \(\lceil{v\over12}\rceil\) when \(v\equiv11\) (mod 132) or \(v\equiv154\) (mod 924), and at most equal to \(\lfloor{v\over12}\rfloor\) in all the other cases. Thus a \((v,5,2,1)\)-OOC is naturally said to be optimal when its size reaches the above bound. Many direct and recursive constructions for infinite classes of optimal \((v,5,2,1)\)-OOCs are presented giving, in particular, a very strong indication about the existence of an optimal \((p,5,2,1)\)-OOC for every prime \(p\equiv1\) (mod 12).

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## A new class of fibered loops related to hyperbolic planes

In this paper we introduce a new class of fibered loops arising from a suitable selected subset of the set of all limit rotations of a hyperbolic plane and we inspect their algebraic and geometric properties.

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## On some subvarieties of the Grassmann variety

Let \(\mathcal S\) be a Desarguesian \((t-1)\)--spread of \(PG(rt-1,q),\) \(\Pi\) a \(m\)-dimensional subspace of \(PG(rt-1,q)\) and \(\Lambda\) the linear set consisting of the elements of \(\mathcal S\) with non-empty intersection with \(\Pi.\) It is known that the Plücker embedding of the elements of \(\mathcal S\) is a variety of \(PG(r^t-1,q)\), say \({\mathcal V}_{rt}\). In this paper, we describe the image under the Plücker embedding of the elements of \(\Lambda\) and we show that it is an \(m\)-dimensional algebraic variety, projection of a Veronese variety of dimension \(m\) and degree \(t\), and it is a suitable linear

section of \({\mathcal V}_{rt}.\)

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Linear Multilinear Algebra 63 (11): 2121-2134 (2015), ISSN 0308-1087, doi:10.1080/03081087.2014.983449

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## Line Polar Grassmann Codes of Orthogonal Type

Polar Grassmann codes of orthogonal type have been introduced in I. Cardinali and L. Giuzzi, Codes and caps from orthogonal Grassmannians, *Finite Fields Appl.* **24** (2013), 148-169. They are subcodes of the Grassmann code arising from the projective system defined by the Plücker embedding of a polar Grassmannian of orthogonal type. In the present paper we fully determine the minimum distance of line polar Grassmann Codes of orthogonal type for \(q\) odd.

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J. Pure Appl. Algebra 220 (5): 1924-1934 (2016), doi:10.1016/j.jpaa.2015.10.007

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## Intersections of the Hermitian Surface with irreducible Quadrics in even Characteristic

We determine the possible intersection sizes of a Hermitian surface \(\mathcal H\) with an irreducible quadric of \({\mathrm PG}(3,q^2)\) sharing at least a tangent plane at a common non-singular point when \(q\) is even.

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Electron. J. of Combin. 23 (4): P4.13 (2016)

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## Cyclic kite-designs of order \(v\) that are cyclically embedded into a cyclic \((v,4,2)\)-design,

By means of a direct and a recursive construction we prove that there exists a cyclic kite-design of order \(v\) that is cyclically embedded into a cyclic \((v,4,2)\)-BIBD whenever the exponent of \(5\) in the prime factorization of \(v\) is even and for any other prime factor \(p\) of \(v\) we have \(p\equiv1 (mod 24)\) and \(-3\) is not an 8th power of \(Z_p\).

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## A generalization of the problem of Mariusz Meszka

Mariusz Meszka has conjectured that given a prime \(p=2n+1\) and a list \(L\) containing \(n\) positive integers not exceeding n there exists a near 1-factor in \(K_p\) whose list of edge-lengths is \(L\).

In this paper we propose a generalization of this problem to the case in which \(p\) is an odd integer not necessarily prime. In particular, we give a necessary condition for the existence of such a near 1-factor for any odd integer \(p\). We show that this condition is also sufficient for any list L whose underlying set \(S\) has size \(1, 2,\) or \(n\). Then we prove that the conjecture is true if \(S=\{1,2,t\}\) for any positive integer t not coprime with the order p of the complete graph. Also, we give partial results when \(t\) and \(p\) are coprime. Finally, we present a complete solution for \(t\leq 11\).

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## Enumerative Coding for Line Polar Grassmannians with applications to codes

A \(k\)-polar Grassmannian is the geometry having as pointset the set of all \(k\)-dimensional subspaces of a vector space \(V\) which are totally isotropic for a given non-degenerate bilinear form \(\mu\) defined on \(V.\) Hence it can be regarded as a subgeometry of the ordinary \(k\)-Grassmannian. In this paper we deal with orthogonal line Grassmannians and with symplectic line Grassmannians, i.e. we assume \(k=2\) and \(\mu\) a non-degenerate symmetric or alternating form. We will provide a method to efficiently enumerate the pointsets of both orthogonal and symplectic line Grassmannians. This has several nice applications; among them, we shall discuss an efficient encoding/decoding/error correction strategy for line polar Grassmann codes of both types.

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## Minimum distance of Symplectic Grassmann codes

We introduce the Symplectic Grassmann codes as projective codes defined by symplectic Grassmannians, in analogy with the orthogonal Grassmann codes introduced in [4]. Note that the Lagrangian-Grassmannian codes are a special class of Symplectic Grassmann codes. We describe the weight enumerator of the Lagrangian--Grassmannian codes of rank 2 and 3 and we determine the minimum distance of the line Symplectic Grassmann codes.

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Linear Algebra Appl. 488: 124-134 (2016), doi:10.1016/j.laa.2015.09.031

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## Intersection sets, three-character multisets and associated codes

In this article we construct new minimal intersection sets in \({\mathrm{AG}}(r,q^2)\) sporting three intersection numbers with hyperplanes; we then use these sets to obtain linear error correcting codes with few weights, whose weight enumerator we also determine. Furthermore, we provide a new family of three-character multisets in \({\mathrm{PG}}(r,q^2)\) with \(r\) even and we also compute their weight distribution.

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Des. Codes Cryptogr., doi:10.1007/s10623-016-0302-8

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## A geometric approach to alternating \(k\)-linear forms

Given an \(n\)-dimensional vector space \(V\) over a field \({\mathbb K}\), let \(2\leq k < n\). A natural one-to-one correspondence exists between the alternating \(k\)-linear forms of \(V\) and the linear functionals of \(\bigwedge^kV\), an alternating \(k\)-linear form \(\varphi\) and a linear functional \(f\) being matched in this correspondence precisely when \(\varphi(x_1,\ldots, x_k) = f(x_1\wedge\cdots\wedge x_k)\) for all \(x_1,\ldots, x_k \in V\). Let \(\varepsilon_k:{\mathcal G}_k(V)\rightarrow {\mathrm{PG}}(\bigwedge^kV)\) be the Pl\"{u}cker embedding of the \(k\)-Grassmannian \({\mathcal G}_k(V)\) of \(V\). Then \(\varepsilon_k^{-1}(\ker(f)\cap\varepsilon_k({\mathcal G}_k(V)))\) is a hyperplane of the point-line geometry \({\mathcal G}_k(V)\). It is well known that all hyperplanes of \({\mathcal G}_k(V)\) can be obtained in this way, namely every hyperplane of \({\mathcal G}_k(V)\) is the family of \(k\)-subspaces of \(V\) where a given alternating \(k\)-linear form identically vanishes. For a hyperplane \(H\) of \({\mathcal G}_k(V)\), let \(R^\uparrow(H)\) be the subset (in fact a subspace) of \({\mathcal G}_{k-1}(V)\) formed by the \((k-1)\)-subspaces \(A\subset V\) such that \(H\) contains all \(k\)-subspaces that contain \(A\). In other words, if \(\varphi\) is the (unique modulo a scalar) alternating \(k\)-linear form defining \(H\), then the elements of \(R^\uparrow(H)\) are the \((k-1)\)-subspaces \(A = \langle a_1,\ldots, a_{k-1}\rangle\) of \(V\) such that \(\varphi(a_1,\ldots, a_{k-1},x) = 0\) for all \(x\in V\). In principle, when \(n-k\) is even it might happen that \(R^\uparrow(H) = \emptyset\). When \(n-k\) is odd then \(R^\uparrow(H) \neq \emptyset\), since every \((k-2)\)-subspace of \(V\) is contained in at least one member of \(R^\uparrow(H)\), but it can happen that every \((k-2)\)-subspace of \(V\) is contained in precisely one member of \(R^\uparrow(H)\). If this is the case, we say that \(R^\uparrow(H)\) is \emph{spread-like}. In this paper we obtain some results on \(R^\uparrow(H)\) which answer some open questions from the literature and suggest the conjecture that, if \(n-k\) is even and at least \(4\), then \(R^\uparrow(H) \not= \emptyset\) but for one exception with \({\mathbb K}\leq{\mathbb R}\) and \((n,k) = (7,3)\), while if \(n-k\) is odd and at least \(5\) then \(R^\uparrow(H)\) is never spread-like.

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## On transparent embeddings of point-line geometries

We introduce the class of transparent embeddings for a point-line geometry \(\Gamma = ({\mathcal P},{\mathcal L})\) as the class of full projective embeddings \(\varepsilon\) of \(\Gamma\) such that the preimage of any projective line fully contained in \( \varepsilon({\mathcal P})\) is a line of \(\Gamma\). We will then investigate the transparency of Plücker embeddings of projective and polar grassmannians and spin embeddings of half-spin geometries and dual polar spaces of orthogonal type. As an application of our results on transparency, we will derive several Chow-like theorems for polar grassmannians and half-spin geometries.

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## Minimum distance of Line Orthogonal Grassmann Codes in even characteristic

In this paper we determine the minimum distance of orthogonal line-Grassmann codes for \(q\) even. The case \(q\) odd was solved in "I. Cardinali, L. Giuzzi, K. Kaipa, A. Pasini, Line Polar Grassmann Codes of Orthogonal Type, J. Pure Applied Algebra (doi:10.1016/j.jpaa.2015.10.007 )" We also show that for \(q\) even all minimum weight codewords are equivalent and that symplectic line-Grassmann codes are proper subcodes of codimension \(2n\) of the orthogonal ones.

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