# On Translation Hyperovals in Semifield Planes

Kevin Allen and John Sheekey

###### Abstract

In this paper we demonstrate the first example of a finite translation plane
which does not contain a translation hyperoval, disproving a conjecture of
Cherowitzo. The counterexample is a semifield plane, specifically a
Generalised Twisted Field plane, of order $64$. We also relate this non-
existence to the covering radius of two associated rank-metric codes, and the
non-existence of scattered subspaces of maximum dimension with respect to the
associated spread.

##  1 Introduction

Hyperovals are extremal combinatorial objects in projecive planes; namely a
hyperoval is a set of $q+2$ points in which no three lie on a common line. We
refer to Section 2 for formal definitions.

Hyperovals have attracted much attention over the years, particularly in the
case of Desarguesian planes. Papers regarding existence, construction, and
classification abound. While full classifications appear out of reach, the
addition of extra assumptions on the symmetry of the plane, the hyperoval, or
both, leads to interesting questions and more potential for classification.

In particular, the most well-studied non-Desarguesian planes are the
translation planes, while the best understood hyperovals in Desarguesian
planes are the translation hyperovals. Therefore it is natural to consider the
question of existence and classification for translation hyperovals in
translation planes.

It is known that translation hyperovals exist in certain translation planes;
for example Desarguesian planes [18], André planes [8], Hall planes [14], and
Knuth’s binary semifield planes [10]. Cherowitzo [6] computationally
classified all hyperovals (translation and otherwise) in each of the nine
translation planes of order $16$. In particular he showed that every
translation plane of order $16$ contains a translation hyperoval, which lead
him to make the following statement.

######  Conjecture 1.

These results… lead one to the natural conjecture that translation hyperovals
exist in all translation planes of even order.

In this paper we will disprove this conjecture, by exhibiting a projective
plane of order $64$ containing no translation hyperoval. Specifically, we show
that the twisted field plane of order $64$, which is a semifield plane,
contains no translation hyperovals. We will also relate this problem to the
(non-)existence of so-called scattered subspaces with respect to a spread
associated to the translation plane, as well as the covering radius of the
rank-metric code (spread set) associated to the translation plane.

##  2 Definitions and Background

In this section we collect the necessary definitions and background for this
article. We refer to [7, 12, 6, 13, 15] for further details on these largely
well-known topics.

###  2.1 Projective Planes

A projective plane $\pi$ is an incidence structure
$(\mathcal{P},\mathcal{L},\mathcal{I})$ consisting of a set of points
$\mathcal{P}$, a set of lines $\mathcal{L}$, and an incidence relation
$\mathcal{I}\subset\mathcal{P}\times\mathcal{L}$ such that every two distinct
points are both incident with precisely one common line, and every pair of
distinct lines are incident with precisely one common point. The dual of
$\pi$, denoted $\pi^{d}$, is the incidence structure
$(\mathcal{L},\mathcal{P},\mathcal{I}^{d})$, where $\mathcal{I}^{d}$ is the
reverse relation of $\mathcal{I}$. Since the terms points and lines are
interchangeable in the definition of a projective plane, $\pi^{d}$ is again a
projective plane.

It is well known that for any finite projective plane there exists a positive
integer $q$, called the order of $\pi$, such that the plane contains
$q^{2}+q+1$ points and $q^{2}+q+1$ lines, every line is incident with $q+1$
points, and every point is incident with $q+1$ lines. If $q$ is a prime power,
there exists a projective plane of order $q$; the converse is a famous open
problem.

The classical examples of projective planes are the Desarguesian planes
$\mathrm{PG}(2,F)$ where $F$ is some (skew) field, in which points are one-
dimensional vector subspaces of $V(3,F)$, lines are two-dimensional vector
subspaces of $V(3,F)$, and incidence is given naturally by inclusion. This
plane can also be realised as the completion of the affine plane
$\mathrm{AG}(2,F)$, in which points are vectors in $V(2,F)$, and lines are
translations of one-dimensional subspaces; that is, cosets $u+\langle
v\rangle$ for some $u,v\in V(2,F),v\neq 0$. Then the addition of a line at
infinity consisting of the directions $\langle v\rangle$ returns a projective
plane isomorphic to the previous description.

###  2.2 Translation Planes

Translation planes are projective planes with extra symmetry. They arise from
affine planes sharing some natural properties of $\mathrm{AG}(2,F)$. In order
to introduce them formally, we need to recall some technical terminology.

An isomorphism from a plane
$\pi_{1}=(\mathcal{P}_{1},\mathcal{L}_{1},\mathcal{I}_{1})$ to another
$\pi_{2}=(\mathcal{P}_{2},\mathcal{L}_{2},\mathcal{I}_{2})$ is a bijection
$\phi$ from $\mathcal{P}_{1}\cup\mathcal{L}_{1}$ to
$\mathcal{P}_{2}\cup\mathcal{L}_{2}$ which preserves type and incidence; that
is, $\phi(\mathcal{P}_{1})=\mathcal{P}_{2}$,
$\phi(\mathcal{L}_{1})=\mathcal{L}_{2}$, and
$p\in\ell\Leftrightarrow\phi(p)\in\phi(\ell)$.

A collineation of a plane $\pi$ is an isomorphism from $\pi$ to itself; a
correlation is an isomorphism from $\pi$ to its dual $\pi^{d}$. An elation of
$\pi$ with centre $p$ and axis $\ell\ni p$ is a collineation of $\pi$ fixing
every point on $\ell$ and every line containing $p$.

A projective plane $\pi$ is said to be a translation plane if there exists a
line $\ell$ such that the group of elations with axis $\ell$ acts transitively
on the points of $\pi\backslash\ell$. If the plane is not Desarguesian, then
the line $\ell$ is unique, and is called the translation line of $\pi$. We
will usually denote the translation line by $\ell_{\infty}$. Any collineation
of $\pi$ must then fix $\ell_{\infty}$.

The name translation plane can be more easily understood in the affine
setting; in the affine plane $\mathrm{AG}(2,F)$, a translation
$\tau_{u}:v\mapsto v+u$ clearly maps lines to lines, preserves direction, and
fixes all lines with direction $\langle u\rangle$. The natural extension of
this map to $\mathrm{PG}(2,F)$ then satisfies the definition of an elation
given above, with $\ell$ the line at infinity, and $p$ the point at infinity
corresponding to $\langle u\rangle$. Furthermore, the group of translations
clearly acts transitively on the points of $\mathrm{AG}(2,F)$. Thus the
concept of an elation and a translation plane is a natural generalisation of
this example.

The dual of a translation plane is not necessarily a translation plane. If
both a plane $\pi$ and its dual $\pi^{d}$ are translation planes, then we call
it a semifield plane. If a semifield plane $\pi$ is not Desarguesian, then
there is a unique point $p_{\infty}\in\pi$ such that the dual of $p$ is the
translation line of $\pi^{d}$. We call this point the shears point. Any
collineation of $\pi$ must then fix $p_{\infty}$. It is known that we must
have $p_{\infty}\in\ell_{\infty}$, that is, the shears point must lie on the
translation line. Furthermore, the collineation group of a semifield plane has
precisely three orbits on points: the shears point, the points of the
translation line other than the shears point, and the remaining points of the
plane.

###  2.3 Hyperovals

A hyperoval in a finite projective plane $\pi$ (of even order $q$) is a set
$\mathcal{H}$ of $q+2$ points such that no three points of $\mathcal{H}$ are
incident with a common line. Hyperovals can only exist in planes of even
order; in planes of odd order, the maximum size of a set with this property is
$q+1$, and a famous result of Segre tells us that all sets attaining this
bound are equivelent to the set of points of a conic.

The study of hyperovals in planes $\mathrm{PG}(2,{\mathbb{F}}_{q})$, $q$ even,
has a long history, with connections to important objects in coding theory,
namely MDS codes with certain parameters and properties. We refer to [22] for
an up-to-date list of the known constructions for Desarguesian planes, as well
as the largest plane with a complete computer classification. Hyperovals in
general planes have also received much attention. It was conjectured that
every projective plane of even order contained a hyperoval; this was disproved
by the computer classification of [19], where a projective plane of order $16$
containing no hyperovals was exhibited.

We are now ready to formally introduce translation hyperovals.

######  Definition 1.

A translation hyperoval is a hyperoval $\mathcal{H}$ such that there exists a
line $\ell$ for which the group of elations with axis $\ell$ acts transitively
on $\mathcal{H}\backslash\ell$.

Payne [18] showed that every translation hyperoval in $\mathrm{PG}(2,2^{n})$
is equivalent to one defined by the set of vectors
$\\{(0,1,0),(0,0,1),(1,x,x^{2^{i}}):x\in\mathbb{F}_{2^{n}}\\}$ for some
positive integer $i$ relatively prime to $n$. These translation hyperovals
were first constructed by Segre [21].

It is known (see e.g. [6]) that for a translation hyperoval in a (non-
Desarguesian) translation plane, the line $\ell$ must be the translation line
$\ell_{\infty}$, and $|\mathcal{H}\cap\ell_{\infty}|=2$.

Since a semifield posesses a distinguished point at infinity, namely the
shears point, it makes sense to consider whether or not a translation
hyperoval contains the shears point.

######  Definition 2.

A translation hyperoval in a semifield plane is said to be of shears type if
it contains the shears point, and of non-shears type if it does not.

Translation hyperovals in translation planes were studied by Cherowitzo in [6]
where he computationally classified all hyperovals (translation and otherwise)
in each of the nine translation planes of order $16$. In particular he showed
that every translation plane of order $16$ contains a translation hyperoval,
which lead him to Conjecture 1.

##  3 Quasifields, Spreads, Spread Sets and Translation Planes

In this section we outline various well-known correspondences between
quasifields, spreads, spread sets and Translation Planes. We refer to [13] for
details, proofs, and further references.

###  3.1 Quasifields and Semifields

A quasifield is an algebraic structure similar to a finite field, without the
requirement that multiplication be associative, and assuming only one
distributive law. In the finite case, a quasifield must have order $q^{n}$ for
some prime power $q$ and some positive integer $n$. In this case we may take
the additive structure to be $(\mathbb{F}_{q^{n}},+)$, with multiplication
$\circ:\mathbb{F}_{q^{n}}\times\mathbb{F}_{q^{n}}\rightarrow\mathbb{F}_{q^{n}}$
satisfiying

  * •

$(x+x^{\prime})\circ y=x\circ y+x^{\prime}\circ y$ for all
$x,x^{\prime},y\in\mathbb{F}_{q^{n}}$;

  * •

For every $a,b\in\mathbb{F}_{q^{n}}$, $a\neq 0$, there exist unique
$x,y\in\mathbb{F}_{q^{n}}$ such that $x\circ a=a\circ y=b$.

A semifield is a quasifield in which the second distributive law also holds:

  * •

$x\circ(y+y^{\prime})=x\circ y+x\circ y^{\prime}$ for all
$x,y,y^{\prime}\in\mathbb{F}_{q^{n}}$.

Note that in the literature quasifields and semifields are assumed to contain
a multiplicative identity, with the terms prequasifield and presemifield used
to describe the case where an idenitity is not assumed. Since this distinction
does not have any relevance for this paper, we will abuse terminology and drop
the prefix.

There are many known constructions for quasifields and semifields. We refer to
[13, 16] for examples of constructions, and [20] for the classification of
semifields of order $64$.

One of the most well-studied families, and the example most relevant to this
paper, are the generalised twisted fields of Albert [1]. These are semifields
with multiplication

$x\circ y:=xy-jx^{q^{i}}y^{q^{k}},$

where $j$ is a fixed element of $\mathbb{F}_{q^{n}}$ satisfying
$N_{\mathbb{F}_{q^{n}}:\mathbb{F}_{q^{(n,i,k)}}}(j)\neq 1$, with $(n,i,k)$
denoting the greatest common divisor of these three integers. For example, for
$q=2,n=6,i=2,k=4$, the multiplication

$x\circ y:=xy-jx^{2^{2}}y^{2^{4}},$

defines a semifield if and only if
$N_{\mathbb{F}_{2^{6}},\mathbb{F}_{2^{2}}}(j)\neq 1$. Such elements certainly
exist, for example any $j$ satisfying $j^{6}+j+1=0$.

Two semifields with respective multiplications $\circ$ and $\star$ are
isotopic if there exist invertible additive maps $A,B,C$ from
$\mathbb{F}_{q^{n}}$ to itself such that $A(x\circ y)=B(x)\star C(y)$ for all
$x,y\in\mathbb{F}_{q^{n}}$. Every presemifield is isotopic to a semifield via
Kaplansky’s trick [16].

###  3.2 Spreads and Spread Sets

A spread (or $n$-spread) in $V=V(2n,F)$ is a set $\mathcal{D}$ of
$n$-dimensional vector subspaces of $V$ such that every nonzero element of $V$
is contained in precisely one element of $\mathcal{D}$.

Let us identify the elements of $V(2n,q)$ with elements of $V(2,q^{n})$, and
let $S_{\infty}=\\{(0,x):x\in\mathbb{F}_{q^{n}}\\}$. The $n$-dimensional
spaces meeting $S_{\infty}$ trivially are precisely those of the form

$S_{f}:=\\{(x,f(x)):x\in\mathbb{F}_{q^{n}}\\},$

where $f(x)\in\mathbb{F}_{q^{n}}[x]$ is a linearised polynomial, i.e. a
polynomial of the form $f(x)=\sum_{i=0}^{n-1}f_{i}x^{q^{i}}$. These are the
polynomials which define ${\mathbb{F}}_{q}$-linear maps from
$\mathbb{F}_{q^{n}}$ into itself. We denote the set of linearised polynomials
of degree at most $q^{n-1}$ as $\mathcal{L}$.

To a spread $\mathcal{D}$ containing $S_{\infty}$ we can associate a unique
set of linear maps (or linearised polynomials) $C(\mathcal{D})$ by

$C(\mathcal{D})=\\{f:S_{f}\in\mathcal{D}\\}.$

These satisfy the property that $C(\mathcal{D})=q^{n}$ and for any $f,g\in
C(\mathcal{D}),f\neq g$ , we have $\mathrm{rank}(f-g)=n$, where
$\mathrm{rank}$ denotes the usual linear algebra rank of an
${\mathbb{F}}_{q}$-linear map. This is called the spread set associated to
$\mathcal{D}$. The definition of a spread set coincides with that of a (not
necessarily linear) maximum rank distance (MRD) code. Conversely, any set $C$
satisfying this property defines a spread by

$\mathcal{D}(C):=\\{S_{\infty}\\}\cup\\{S_{f}:f\in C\\}.$

For a linear map $f$, we can define

$\phi_{f}:(x,y)\mapsto(x,f(x)+y).$

If $C$ is additively closed, then each of the maps in the set
$\phi_{C}:=\\{\phi_{f}:f\in C\\}$ fix the spread $\mathcal{D}(C)$. Moreover,
each $\phi_{f}$ fixes $S_{\infty}$ pointwise, and $\phi_{C}$ is an abelian
group acting transitively on $\mathcal{D}(C)\backslash\\{S_{\infty}\\}$. The
spread is then called a semifield spread, and $C$ a semifield spread set, for
reasons that will shortly become apparent. We refer to the distinguished
element $S_{\infty}$ as the shears element of $\mathcal{D}$.

Given a quasifield $Q$, we can define a spread set and a spread as follows.
Define

$R_{y}(x):=x\circ y.$

Then each $R_{y}$ is an additive map. Moreover $R_{y}-R_{y^{\prime}}$ is
invertible for all $y\neq y^{\prime}$, since otherwise there would exist some
nonzero $a$ such that $a\circ y=a\circ y^{\prime}$, contradicting one of the
axioms of a quasifield. Thus

$C(Q):=\\{R_{y}:y\in\mathbb{F}_{q^{n}}\\}$

defines a spread set, and $\mathcal{D}(Q):=\mathcal{D}(C(Q))$ is a spread. If
$Q$ is a semifield, then $R_{y+y^{\prime}}=R_{y}+R_{y^{\prime}}$, and so
$C(Q)$ is additively closed.

Conversely, from a spread or spread set we can define a quasifield; note
however that the quasifield is not uniquely determined by the spread or spread
set. However it is uniquely determined up to isotopy.

###  3.3 Translation Planes from Spreads

From a spread $\mathcal{D}$ we can define an affine plane as follows:

  * •

Points: elements of $V(2n,q)$;

  * •

Lines: translations of elements of $\mathcal{D}$, i.e. $u+S$ for $u\in
V(2n,q),S\in\mathcal{D}$.

We can complete this to a projective plane by adding a line at infinity
$\ell_{\infty}$, whose points are the elements of $\mathcal{D}$; to a line
$u+S$ we add the point at infinity $S$. We denote this plane as
$\pi(\mathcal{D})$.

It is straightforward to check that this is indeed a translation plane.
Moreover, André showed that every translation plane arises from a spread [2].

By the discussion in the previous subsections, we can define a translation
plane from a quasifield, and from a spread set. We may denote these naturally
as $\pi(Q)$, $\pi(C)$ respectively.

The dual of a translation plane defined by a semifield is again a translation
plane, and so the plane is a semifield plane. The shears point corresponds to
the shears element of the spread. As mentioned in Section 2.2, the
collineation group of a semifield plane has precisely three orbits on points:
the shears point, the points of the translation line other than the shears
point, and the remaining points of the plane. The transitivity on the points
of the translation line other than the shears point is demonstrated by the
maps $\phi_{f}$ defined in the previous section.

It is well known that two semifields define isomorphic planes if and only if
they are isotopic.

###  3.4 Scattered Subspaces

It is known (see e.g. [15, 9]) that translation hyperovals correspond to
scattered subspaces with respect to spreads. We recall the relevant notions
and demonstrate this fact here.

A subspace $U$ is said to be scattered with respect to a spread $\mathcal{D}$
if

$\mathrm{dim}(U\cap S)\leq 1\quad\mathrm{for~{}all}~{}S\in\mathcal{D}.$

The study of scattered subspaces with respect to a spread originates from [4];
we refer to [15] for a recent survey of the various applications of scattered
subspaces, including that of translation hyperovals. We note that the interest
in this notion is not restricted to spreads in $V(2n,q)$; the more general
case of spreads of $n$-dimensional subspaces of $V(kn,q)$ is also of interest.
However, here we deal only with the case $k=2$. In this case it is
straightforward to obtain an upper bound on the dimension of a scattered
subspace.

######  Proposition 1.

[4, Lemma 3.1 and Theorem 4.1] Let $\mathcal{D}$ be a spread in $V(2n,q)$.
Then the dimension of a scattered subspace is at most $n$. Moreover, if
$\mathcal{D}$ is Desarguesian, then there exists a scattered subspace of
dimension $n$.

Suppose $U$ is a scattered subspace of dimension $n$ with respect to a spread
in $V(2n,2)$. Then $U$ must intersect $2^{n}-1$ distinct elements of
$\mathcal{D}$ nontrivially, and hence since $|\mathcal{D}|=q^{n}+1$, there
exist precisely two elements of $\mathcal{D}$ which intersect $U$ trivially,
say $S_{1}$ and $S_{2}$. We claim that the set of points of the translation
plane $\pi(\mathcal{D})$ defined by the affine points $U$ and the two points
at infinity $S_{1},S_{2}$ form a translation hyperoval, which we will denote
by $\mathcal{H}_{U}$. Consider a line $v+S$ of $\pi(\mathcal{D})$ with
$S\notin\\{S_{1},S_{2}\\}$. Then the points of intersection of $v+S$ and
$\mathcal{H}_{U}$ are affine, and so the number of them is equal to
$|(v+S)\cap U|$, which is either $0$ or $2$, since $|U\cap S|=2$. Next
consider a line $v+S_{i}$, $i\in\\{1,2\\}$. Then this line meets
$\mathcal{H}_{U}$ in the point at infinity corresponding to $S_{i}$, as well
as unique affine point $v+s_{i}$, where $s_{i}+u_{i}=v$ for a unique $u_{i}\in
U$. The uniqueness here follows from the fact that $V(2n,2)=S_{i}\oplus U$.
Finally the line at infinity clearly meets $\mathcal{H}_{U}$ in the two points
$\\{S_{1},S_{2}\\}$, proving that $\mathcal{H}$ is a hyperoval. Then the group
of translations $\tau_{u}:u\in U$ clearly acts transitively on
$U=\mathcal{H}_{U}\backslash\ell_{\infty}$, showing that $\mathcal{H}_{U}$ is
indeed a translation hyperoval.

Conversely, it has been shown that (up to equivalence) every translation
hyperoval arises in this way from a scattered subspace.

######  Proposition 2.

[15, Theorem 1.7] Suppose $\mathcal{H}$ is a translation hyperoval in a
translation plane $\pi$ of order $2^{n}$, and suppose $\mathcal{D}$ is a
spread in $V(2n,2)$ such that $\pi$ is isomorphic to $\pi(\mathcal{D})$. Then
there exists a translation hyperoval in $\pi$ if and only if there exists an
$n$-dimensional scattered subspace with respect to $\mathcal{D}$.

Hence the existence of translation hyperovals corresponds to the existence of
scattered subspaces. Recently in [11] the following lower bound on the largest
dimension of a scattered subspace was shown.

######  Proposition 3.

[11, Proposition 4.15] Let $\mathcal{D}$ be a spread in $V(2n,q)$, $q\geq 8$.
Then there exists a scattered subspace of dimension at least $n/2-1$.

Clearly this does not guarantee the existence of translation hyperovals,
though it does indicate the dimension at which scattered subspaces become hard
to find, and impossible to guarantee by combinatorial methods.

Various generalisations of the notion of scattered subspaces have been put
forward in recent years, such as evasive subspaces [3]. One which is of
particular relevance to this paper is that of $(\mathcal{D},h)$-scattered
subspaces:

A subspace $U$ is said to be $(\mathcal{D},h)$-scattered if

$\mathrm{dim}(U\cap S)\leq h\quad\mathrm{for~{}all}~{}S\in\mathcal{D}.$

Clearly the property of a subspace being $(\mathcal{D},1)$-scattered coincides
with it being scattered with respect to $\mathcal{D}$. The following
specialisation of [11, Theorem 6.7] guarantees the existence of
$(\mathcal{D},h)$-scattered subspaces in certain circumstances.

######  Proposition 4.

Let $\mathcal{D}$ be a spread in $V(2n,q)$. Then there exists a
$(\mathcal{D},h)$-scattered subspace of dimension $n$ for any
$h\geq\lceil\sqrt{n+1}-1\rceil$ if $q\geq 4$, and for any
$h\geq\lceil\sqrt{n+2}-1\rceil$ if $q<4$.

For the case $n=6$, which is the case for spreads arising from semifields of
order $q^{6}$, we obtain that there exists a $2$-scattered subspace of
dimension $6$ with respect to any $6$-spread $\mathcal{D}$ in $V(12,q)$. In
particular, we note that the existence of a $1$-scattered subspace of
dimension $n$, and hence a translation hyperoval, is not guaranteed. Indeed,
we will demonstrate a counterexample.

###  3.5 Covering Radius

Let us assume that $\mathcal{D}=\mathcal{D}(\mathbb{S})$, where $\mathbb{S}$
is a semifield. Suppose $U$ is an $n$-dimensional scattered subspace with
respect to $\mathcal{D}$. Then without loss of generality we may assume that
one of the following two occur.

  * (Shears Type) $U\cap S_{\infty}=U\cap S_{0}=0$;

  * (Non-Shears Type) $U\cap S_{\infty}\neq 0,U\cap S_{0}=U\cap S_{y}=0$ for some $y\neq 0$.

These two cases correspond to whether or not the translation hyperorval
$\mathcal{H}_{U}$ contains the shears point $S_{\infty}$.

The covering radius of a set or subspace $C$ of the space of linear maps
$M=\mathrm{End}_{{\mathbb{F}}_{q}}(\mathbb{F}_{q^{n}})$ is a notion arising
naturally from coding theory in the rank metric; we refer to [5] for
background. We denote it by $\rho(C)$ and define it as

$\rho(C)=\min\\{i:\forall g\in M,\exists f\in
C\mathrm{~{}s.t.~{}}\mathrm{rank}(f-g)\leq i\\}.$

For a spread set $C\subset M$, which has cardinality $q^{n}$ and is such that
every nonzero element of $C$ is invertible, we define $C^{-1}=\\{f^{-1}:f\in
C,f\neq 0\\}\cup\\{0\\}$. If $C$ is a spread set, then $\rho(C)\leq n-1$,
since otherwise there would exist $g\notin C$ such that $\mathrm{rank}(g-f)=n$
for all $f\in C$. But for any nonzero $a\in\mathbb{F}_{q^{n}}$ there exists a
unique $y\in\mathbb{F}_{q^{n}}$ such that $a\circ y=g(a)$, and so there exists
$R_{y}\in C$ such that $\mathrm{rank}(g-R_{y})<n$.

######  Theorem 1.

Let $\mathbb{S}$ be a semifield of order $2^{n}$, $\pi(\mathbb{S})$ the
translation plane it defines, and $C=C(\mathbb{S})$ the spread set it defines
in $V(2n,2)$. Then there exists a translation hyperoval of shears type in
$\pi(\mathbb{S})$ if and only if $\rho(C)=n-1$, and there exists a translation
hyperoval of non-shears type in $\pi(\mathbb{S})$ if and only if
$\rho(C^{-1})=n-1$.

###### Proof.

The plane $\pi(\mathbb{S})$ contains a translation hyperoval of shears type if
and only if there exists an $n$-dimensional subspace $U$ which is scattered
with respect to $\mathcal{D}(\mathbb{S})$ such that $U\cap S_{\infty}=0$. For
any $n$-dimensional subspace such that $U\cap S_{\infty}=0$ there exists an
${\mathbb{F}}_{q}$-linear map $f$ from $\mathbb{F}_{q^{n}}$ to itself such
that

$U=\\{(x,f(x)):x\in\mathbb{F}_{q^{n}}\\}.$

Let $y\in\mathbb{F}_{q^{n}}$. Then $U\cap S_{y}=\\{(x,f(x))|f(x)=R_{y}(x)\\}$,
and so $\mathrm{dim}(U\cap S_{y})=n-\mathrm{rank}(f-R_{y})$. Hence there
exists an $n$-dimensional subspace which is scattered with respect to
$\mathcal{D}(\mathbb{S})$ if and only if there exists $f$ such that
$\mathrm{rank}(f-R_{y})\geq n-1$ for all $y\in\mathbb{F}_{q^{n}}$, if and only
if $\rho(C)\geq n-1$, if and only if $\rho(C)=n-1$, proving the first claim.

Similarly, $\pi(\mathbb{S})$ contains a translation hyperoval of shears type
if and only if there exists an $n$-dimensional subspace $U$ which is scattered
with respect to $\mathcal{D}(\mathbb{S})$ such that $U\cap S_{0}=0$. For any
such subspace there exists an ${\mathbb{F}}_{q}$-linear map $f$ from
$\mathbb{F}_{q^{n}}$ to itself such that
$U=\\{(f(x),x):x\in\mathbb{F}_{q^{n}}\\}$. Then for any nonzero
$y\in\mathbb{F}_{q^{n}}$ we have $\mathrm{dim}(U\cap
S_{y})=n-\mathrm{rank}(f-R_{y}^{-1})$, while $\mathrm{dim}(U\cap
S_{\infty})=n-\mathrm{rank}(f)$, and so arguing as before we obtain the second
claim. ∎

Note that we do not have such a result for general quasifields, since the lack
of a distinguished element at infinity nor transitivity on the remaining
points at infinity means that there is not a canonical choice for the spread
set.

Note also that the connection between the existence of scattered subspaces
with respect to semifield spreads and the covering radius of $C$ and $C^{-1}$
are also valid for $q>2$; however in this case we do not obtain translation
hyperovals.

###  3.6 Linearised Polynomials and Dickson Matrices

In order to explicitly determine whether or not translation hyperovals exist,
we need a practical method for determining the existence of a linear map $f$
such that $\mathrm{rank}(f-R_{y})\geq n-1$ for all $y\in\mathbb{F}_{q^{n}}$.
We do this by utilising linearised polynomials and Dickson matrices; this is
the approach used by Payne to classify translation hyperovals in Desarguesian
planes, and also used productively in recent years in the construction of MRD
codes.

To a linearised polynomial $f(x)=\sum_{i=0}^{n-1}f_{i}x^{q^{i}}$ we associate
the Dickson (or autocirculant) matrix $D_{f}$ defined as follows:

$D_{f}:=\begin{pmatrix}f_{0}&f_{1}&\cdots&f_{n-1}\\\
f_{n-1}^{q}&f_{0}^{q}&\cdots&f_{n-2}^{q}\\\ \vdots&\ddots&\ddots&\vdots\\\
f_{1}^{q^{n-1}}&f_{2}^{q^{n-1}}&\cdots&f_{0}^{q^{n-1}}\end{pmatrix}$

It is well known that the assignment $f\mapsto D_{f}$ is linear, and
$\mathrm{rank}(f)=\mathrm{rank}(D_{f})$ [17]. Hence we can translate Theorem 1
to this setting.

######  Lemma 1.

The plane $\pi(\mathbb{S})$ contains a translation hyperoval of shears type if
and only if there exists $f\in\mathcal{L}\backslash C(\mathbb{S})$ such that

$\mathrm{rank}(D_{R_{y}}-D_{f})\geq n-1$

for all $y\in\mathbb{F}_{q^{n}}$.

The plane $\pi(\mathbb{S})$ contains a translation hyperoval of non-shears
type if and only if there exists $g\in\mathcal{L}\backslash C(\mathbb{S})$
such that

$\mathrm{rank}(D_{R_{y}}^{-1}-D_{g})\geq n-1$

for all $y\in\mathbb{F}_{q^{n}}^{\times}$ and $\mathrm{rank}(D_{g})=n-1$.

Now the entries of $D_{R_{y}}$ are polynomials in $y$, and since
$\det(D_{R_{y}})=1$ for all non-zero $y$, we can regard $D_{R_{y}}^{-1}$ as a
matrix whose entries are polynomials in $y$. Hence for $f,g\in\mathcal{L}$,
the functions $d_{f}(y):=\det(D_{R_{y}}-D_{f})$ and
$d^{\mathrm{inv}}_{g}(y):=\det(D_{R_{y}}^{-1}-D_{g})$ are both polynomials in
$y$ (whose coefficients are expressions in the unknown coefficients of $f$ and
$g$ respectively). Note that $d_{g}(0)=\det(D_{g})$. If necessary we can
replace $d_{f}(y)$ and $d^{\mathrm{inv}}_{g}(y)$ with their reduction modulo
$y^{2^{n}}-y$.

######  Lemma 2.

The plane $\pi(\mathbb{S})$ of order $2^{n}$ contains a translation hyperoval
of shears type if and only if there exists $f\in\mathcal{L}\backslash
C(\mathbb{S})$ such that $d_{f}(y)=y^{2^{n}-1}+1$, and contains a translation
hyperoval of non-shears type if and only if there exists
$g\in\mathcal{L}\backslash C(\mathbb{S})$ such that $\mathrm{rank}(g)=n-1$ and
$d^{\mathrm{inv}}_{g}(y)=\frac{y^{2^{n}}+y}{y+a}$ for some
$a\in\mathbb{F}_{q^{n}}^{\times}$.

###### Proof.

Let $U=\\{(x,f(x)):x\in\mathbb{F}_{q^{n}}\\}$, and suppose $U$ defines a
translation hyperoval of shears type. Then we may assume without loss of
generality that $U\cap S_{0}=0$, and so $U\cap S_{y}\neq 0$ for all $y\neq 0$.
Then $\mathrm{rank}(D_{R_{y}}-f)=n-1$ for all nonzero
$y\in\mathbb{F}_{q^{n}}$, and so $d_{f}$ is zero at all nonzero elements of
$\mathbb{F}_{q^{n}}$ and nonzero at $y=0$. Clearly this implies that
$d_{f}(y)=y^{2^{n}-1}+1$.

Now let $W=\\{(g(x),x):x\in\mathbb{F}_{q^{n}}\\}$, and suppose $W$ defines a
translation hyperoval of non-shears type. Then we may assume without loss of
generality that $W\cap S_{0}=0$, and there exists a unique
$a\in\mathbb{F}_{q^{n}}^{\times}$ such that $W\cap S_{a}=0$ and $W\cap
S_{y}\neq 0$ for all nonzero $y\neq a$. Furthermore $W\cap S_{\infty}\neq 0$,
and so $d^{\mathrm{inv}}_{g}(0)=0$. Hence $d^{\mathrm{inv}}_{g}(y)\neq 0$ if
and only if $y=a$, and so $d^{\mathrm{inv}}_{g}(y)=\frac{y^{2^{n}}+y}{y+a}$ as
claimed. ∎

Note that this is the approach used by Payne in his classification of
translation hyperovals in $\mathrm{PG}(2,q^{n})$. He showed that for the case
$R_{y}(x)=xy$, the requirement that $d_{f}(y)=y^{2^{n}-1}+1$ implies that
$f(x)$ is monomial, that is, $f(x)=f_{i}x^{2^{i}}$. The classification of
translation hyperovals in $\mathrm{PG}(2,q^{n})$ then follows easily.

##  4 Translation Hyperovals in the Generalised Twisted Field plane of order
$64$

In this section we analyse the conditions from Lemma 1 and Lemma 2 for the
case of the Generalised Twisted Field plane of order $64$. We choose this
plane due to the fact that the Dickson matrices $D_{R_{y}}$ and
$D_{R_{y}^{-1}}$ are sparse, making the equations manageable. Furthermore the
known symmetries of these semifields allow us to further reduce the necessary
computation.

The multiplication in this presemifield, which we will denote by $\mathbb{T}$,
is given by

$x\circ y=xy-jx^{2^{2}}y^{2^{4}}=:R_{y}(x),$

where $j$ is a solution to $j^{6}+j+1=0$. We choose this representation to
match that in [20]. Any semifield isotopic to this presemifield has centre
isomorphic to $\mathbb{F}_{4}$. In particular, each of the maps $R_{y}$ are
$\mathbb{F}_{4}$-linear.

Note furthermore that we have the following identity, which will prove useful
in the subsequent calculations:

$\alpha R_{y}(\beta x)=R_{\alpha\beta y}(x)$

for all $\alpha,\beta\in\mathbb{F}_{2^{6}}$ such that
$\alpha\beta^{2^{2}}=\alpha^{2^{4}}\beta\neq 0$. Hence for any $\alpha,\beta$
satisfying this condition, the map
$\phi_{\alpha,\beta}:(x,y)\mapsto(\beta^{-1}x,\alpha y)$ fixes
$\mathcal{D}(\mathbb{T})$; in particular, it fixes $S_{\infty}$ and $S_{0}$,
and maps $S_{y}$ to $S_{\alpha\beta y}$. Note that any such
$\phi_{\alpha,\beta}$ fixes one further element of $\mathcal{D}(\mathbb{T})$
if and only if it fixes every element of $\mathcal{D}(\mathbb{T})$, and this
occurs precisely if $\alpha^{9}=1$ and $\alpha\beta=1$.

Furthermore, letting $U_{f}=\\{(x,f(x)):x\in\mathbb{F}_{q^{n}}\\}$ and
$W_{g}=\\{(g(x),x):x\in\mathbb{F}_{q^{n}}\\}$, we get that
$\phi_{\alpha,\beta}(U_{f})=U_{h}$ where $h(x)=\alpha f(\beta x)$, and
$\phi_{\alpha,\beta}(W_{g})=W_{k}$ where $k(x)=\beta^{-1}g(\alpha^{-1}x)$.

###  4.1 Shears Type

Suppose $\pi(\mathbb{T})$ contains a translation hyperoval of shears type. By
Lemma 2, we require the existence of some $f\in\mathcal{L}\backslash
C(\mathbb{S})$ such that $d_{f}(y):=\det(D_{R_{y}}-D_{f})=y^{2^{6}-1}+1$. This
leads to a system of equations in six unknowns $f_{0},\ldots,f_{5}$ over
$\mathbb{F}_{2^{6}}$. Note furthermore that for any
$\alpha,\beta\in\mathbb{F}_{2^{6}}$ such that
$\alpha\beta^{2^{2}}=\alpha^{2^{4}}\beta\neq 0$, if $h(x)=\alpha f(\beta x)$,
then

$d_{f}(y)=d_{h}(\alpha\beta y)=d_{h}(y),$

and so $f$ defines a translation hyperoval if and only if $h$ defines a
translation hyperoval. Note that $h_{0}=\alpha\beta f_{0}$, and the set
$\\{\alpha\beta:\alpha,\beta\in\mathbb{F}_{2^{6}},\alpha\beta^{2^{2}}=\alpha^{2^{4}}\beta\neq
0\\}$ is precisely the set of solutions to $x^{21}=1$. Thus we may assume
without loss of generality that $f_{0}\in\\{0,1,j,j^{2}\\}$. Furthermore if
$\alpha\beta=1$ then $\beta^{9}=1$ and $h_{1}=\beta f_{1}$, and so we can
assume without loss of generality that $f_{1}^{8}=f_{1}$, i.e.
$f_{1}\in\mathbb{F}_{8}$.

From the coefficients of $y^{62}$ and $y^{58}$ respectively, we get that

$\displaystyle 0$ $\displaystyle=j^{21}f_{0}+j^{38}f_{2}^{4},$ $\displaystyle
0$
$\displaystyle=j^{22}f_{0}^{16}f_{4}^{4}+j^{21}f_{0}^{5}+j^{22}f_{2}^{20}+j^{21}f_{2}f_{4}^{4}.$

Thus we have that either $f_{0}=f_{2}=0$, or $f_{0}=j^{17}f_{2}^{4}$ and
$f_{4}=j^{16}f_{2}^{52}$.

We plug these expressions into the coefficients of $y^{57}$ and $y^{54}$ and
set them equal to zero. It turns out that we get the same pair of equations
regardless of whether or not $f_{2}=0$, and we also observe that $f_{2}$ does
not appear in either of the resulting equations:

$\displaystyle 0$
$\displaystyle=j^{39}f_{1}^{16}f_{3}^{8}+f_{1}^{2}f_{5}^{4}+j^{5}f_{3}^{16}f_{5}^{2}+j^{34}f_{3}^{4}f_{5}^{8},$
$\displaystyle 0$
$\displaystyle=j^{10}f_{1}^{33}+j^{17}f_{1}^{12}+f_{3}^{9}+j^{27}f_{5}^{36}.$

Taking into account that $f_{1}\in\mathbb{F}_{8}$, and raising to an
appropriate power of $2$, we get that

$\displaystyle 0$
$\displaystyle=f_{1}(j^{51}f_{3}^{4}+f_{5}^{2})+j^{34}f_{3}^{8}f_{5}+j^{17}f_{3}^{2}f_{5}^{4},$
(1) $\displaystyle 0$
$\displaystyle=j^{36}f_{1}^{5}+f_{3}^{9}+j^{27}f_{5}^{36}.$

This leads to the following.

######  Theorem 2.

The Generalised Twisted Field plane of order $64$ does not contain a
translation hyperoval of shears type.

###### Proof.

The following MAGMA code verifies that the system (1) has no nontrivial
solutions.

>
>     q := 2;
>     F := GF(q);
>     P<x> := PolynomialRing(F);
>     L<j> := ext<F|x^6+x+1>;
>     F8 := {x:x in L|x^8 eq x};
>
>     S<f1,f3,f5> := PolynomialRing(L,3);
>
>     g := f1*(j^51*f3^4 + f5^2) + j^34*f3^8*f5 + j^17*f3^2*f5^4;
>     h := j^36*f1^5  + f3^9 + j^27*f5^36;
>
>     s1 := {[a,b,c]:a in F8,b,c in L|Evaluate(g,[a,b,c]) eq 0};
>     s2 := {[a,b,c]:a in F8,b,c in L|Evaluate(h,[a,b,c]) eq 0};
>
>     s1 meet s2 eq {[L|0,0,0]};
>

Hence we have that $f_{1}=f_{3}=f_{5}=0$, implying $f(x)$ is in fact an
$\mathbb{F}_{4}$-linear map. But since each $R_{y}$ is also
$\mathbb{F}_{4}$-linear, then the rank of $f-R_{y}$ as an
$\mathbb{F}_{2}$-linear map must be even; in particular it cannot be $n-1=5$,
contradicting Theorem 1.

Hence this plane does not contain a translation hyperoval of shears type. ∎

The MAGMA code used in this proof runs in less than one second.

###  4.2 Non-shears Type

Suppose $\pi(\mathbb{T})$ contains a translation hyperoval of non-shears type.
By Lemma 1, we require the existence of some $g\in\mathcal{L}\backslash
C(\mathbb{S})$ such that $\mathrm{rank}(D_{R_{y}}^{-1}-D_{g})\geq n-1$ for all
$y\in\mathbb{F}_{q^{n}}^{\times}$ and $\mathrm{rank}(D_{g})=n-1$.

Similar to the shears case, we may assume without loss of generality that
$g_{0}\in\\{0,1,j,j^{2}\\}$ and $g_{1}\in\mathbb{F}_{8}$.

Note if $R_{y}(x)=yx+jy^{2^{4}}x^{2^{2}}$ for $y\neq 0$, then $R_{y}$ is
$\mathbb{F}_{4}$-linear and so $R_{y}^{-1}$ must also be
$\mathbb{F}_{4}$-linear. It is straighforward then to calculate $R_{y}^{-1}$,
which we find to be

$R_{y}^{-1}(x)=y^{62}j^{21}x+y^{11}j^{22}x^{4}+y^{59}j^{26}x^{16}.$

Due to the complexity of the coefficients of $d^{\mathrm{inv}}_{g}(y)$ and the
unknown element $a$ such that $d^{\mathrm{inv}}_{g}(a)\neq 0$, there is little
that can be done from a theoretical point of view utilising Lemma 2, beyond
the above restrictions on the coefficients of $g$. Hence we must rely on a
long computation using Lemma 1.

######  Theorem 3.

The Generalised Twisted Field plane of order $64$ does not contain a
translation hyperoval of non-shears type.

###### Proof.

The following MAGMA code verifies that there are no tuples
$(g_{0},g_{1},g_{2},g_{3},g_{4},g_{5})$ with
$g_{0}\in\\{0,1,j,j^{2}\\},g_{1}\in\mathbb{F}_{8}$, and
$g_{i}\in\mathbb{F}_{64}$ for $i=3,4,5,6$ such that
$\mathrm{rank}(D_{R_{y}}^{-1}-D_{g})\geq n-1$ for all $y\in\mathbb{F}_{q^{n}}$
and $\mathrm{rank}(D_{g})=n-1$.

>
>     q := 2;
>     n := 6;
>
>     F := GF(q);
>     P<x> := PolynomialRing(F);
>     L<j> := ext<F|x^6+x+1>;
>     F8 := {x:x in L|x^8 eq x};
>
>     DicksonMatrix := function(v,n,q);
>     return Matrix([Rotate([a^(q^i):a in v],i):i in [0..n-1]]);
>     end function;
>
>     Cinv := {DicksonMatrix([j^21*y^62,0,j^22*y^11,0,j^26*y^59,0],n,q):y in
> L};
>
>     time nonshears := {<g0,g1,g2,g3,g4,g5>:g0 in {0,1,j,j^2},g1 in
> F8,g2,g3,g4,g5 in L|
>     forall{z:z in Cinv|Rank(z-f) ge n-1} where f is
> DicksonMatrix([g0,g1,g2,g3,g4,g5],n,q)};
>     #nonshears eq 0;
>
>

Hence by Lemma 1, there does not exist a translation hyperoval of non-shears
type in this plane. ∎

The calculation used in this proof takes approximately 8.5 hours on a single
CPU. We note that this computation could clearly be parallelised and optimised
further, but we do not attempt any improvements beyond the above restrictions
on $g_{0}$ and $g_{1}$. Without these restrictions, the computation would take
approximately three weeks.

##  5 Conclusion and Remarks

This culminates in the following theorem, disproving Cherowitzo’s conjecture.

######  Theorem 4.

There does not exist a translation hyperoval in the Twisted Field Plane of
order $64$.

######  Remark 1.

Due to the previously described equivalences, we have also demonstrated the
existence of a $6$-spread in $V(12,2)$ not admitting a scattered subspace of
dimension $6$, and an MRD code (semifield spread set) in
$M_{6}(\mathbb{F}_{2})$ with minimum distance $6$ and covering radius less
than $5$.

######  Remark 2.

The situation for the remaining semifield planes of order $64$ is more
difficult to analyse theoretically. Instead we would need to rely on
exhaustive computer searches. For translation hyperovals of shears type, this
can be done relatively efficiently by exploiting the additivity of
$C(\mathbb{S})$; a naive implementation can perform an exhaustive search in
about 8 hours (as opposed to less than a second for the generalised twisted
field). In fact, it turns out that many semifield planes of order $64$ do not
contain a translation hyperoval of shears type.

However, for the non-shears case we do not have additivity, and for the
majority of semifields we do not have enough symmetries to constrain the
coefficients $g_{i}$ as in Section 4.2, and so exhaustive computation takes
much longer. Hence further theoretical reductions, or a more significant
parallelised computation, would be necessary in order to determine the
existence or non-existence of translation hyperovals for these planes.

######  Remark 3.

Although hyperovals cannot exist in planes of odd order, scattered subspaces
of maximum dimension with respect to spreads can still exist. The
corresponding point set in the associated projective plane is a set of $q$
points not contained in the translation line meeting each line in $0,1$ or $q$
points upon which a group of translations acts transitively.

We can repeat the arguments from this paper in part; however, since
$\frac{q^{n}-1}{q-1}<q^{n}-1$ for $q>2$, we cannot conclude much about
$d_{f}(y)$. It remains an open question whether or not spreads defined by
generalised twisted fields possess a scattered subspace of dimension $n$ for
general $q$.

## References

  * [1] A. A. Albert.  Generalized twisted fields.  Pacific J. Math., 11:1–8, 1961.
  * [2] J. André.  Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe.  Math. Z., 60:156–186, 1954.
  * [3] D. Bartoli, B. Csajbók, G. Marino, and R. Trombetti.  Evasive subspaces.  Journal of Combinatorial Designs, 29(8):533–551, 2021.
  * [4] A. Blokhuis and M. Lavrauw.  Scattered spaces with respect to a spread in ${\rm PG}(n,q)$.  Geom. Dedicata, 81(1-3):231–243, 2000.
  * [5] E. Byrne and A. Ravagnani.  Covering radius of matrix codes endowed with the rank metric.  SIAM J. Discrete Math., 31(2):927–944, 2017.
  * [6] W. E. Cherowitzo.  Hyperovals in the translation planes of order $16$.  J. Combin. Math. Combin. Comput., 9:39–55, 1991.
  * [7] P. Dembowski.  Finite geometries. Springer-Verlag, Berlin-New York,, 1968.
  * [8] R. H. F. Denniston.  Some non-Desarguesian translation ovals.  Ars Combin., 7:221–222, 1979.
  * [9] J. D’haeseleer and G. Van de Voorde.  Translation hyperovals and $\mathbb{F}_{2}$-linear sets of pseudoregulus type.  Electron. J. Combin., 27(3):Paper No. 3.21, 19, 2020.
  * [10] N. Durante, R. Trombetti, and Y. Zhou.  Hyperovals in Knuth’s binary semifield planes.  European J. Combin., 62:77–91, 2017.
  * [11] A. Gruica, A. Ravagnani, J. Sheekey, and F. Zullo.  Generalised scattered subspaces.  CoRR, abs/2207.01027, 2022.
  * [12] D. R. Hughes and F. C. Piper.  Projective planes. Springer-Verlag, New York-Berlin,,, 1973.
  * [13] N. L. Johnson, V. Jha, and M. Biliotti.  Handbook of finite translation planes, volume 289 of Pure and Applied Mathematics (Boca Raton).  Chapman & Hall/CRC, Boca Raton, FL, 2007.
  * [14] G. Korchmáros.  Inherited arcs in finite affine planes.  J. Combin. Theory Ser. A, 42(1):140–143, 1986.
  * [15] M. Lavrauw.  Scattered spaces in Galois geometry.  In Contemporary developments in finite fields and applications, pages 195–216. World Sci. Publ., Hackensack, NJ, 2016.
  * [16] M. Lavrauw and O. Polverino.  Finite semifields.  In L. Storme and J. De Beule, editor, Current research topics in Galois geometry, Mathematics Research Developments, pages 127–155. Nova Science, 2011.
  * [17] G. Menichetti.  Roots of affine polynomials.  In Combinatorics ’84 (Bari, 1984), volume 123 of North-Holland Math. Stud., pages 303–310. North-Holland, Amsterdam, 1986.
  * [18] S. E. Payne.  A complete determination of translation ovoids in finite Desarguian planes.  Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 51:328–331 (1972), 1971.
  * [19] T. Penttila, G. F. Royle, and M. K. Simpson.  Hyperovals in the known projective planes of order $16$.  J. Combin. Des., 4(1):59–65, 1996.
  * [20] I. F. Rúa, E. F. Combarro, and J. Ranilla.  Classification of semifields of order 64.  J. Algebra, 322(11):4011–4029, 2009.
  * [21] B. Segre.  Sui $k$-archi nei piani finiti di caratteristica due.  Rev. Math. Pures Appl., 2:289–300, 1957.
  * [22] P. Vandendriessche.  Classification of the hyperovals in ${\rm PG}(2,64)$.  Electron. J. Combin., 26(2):Paper No. 2.35, 12, 2019.


#  The Parton Branching evolution package uPDFevolv2***This article is
dedicated the memory to M. Botje, author of the QCDnum package.

H. Jung<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron DESY, Germany
II. Institut für Theoretische Physik, Universität Hamburg, Germany  Elementary
Particle Physics, University of Antwerp, Belgium  0000-0002-2964-9845 A. Lelek
<EMAIL_ADDRESS>Elementary Particle Physics, University of
Antwerp, Belgium  0000-0001-5862-2775 K. Moral Figueroa
<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron DESY, Germany
0000-0003-1987-1554 S. Taheri Monfared<EMAIL_ADDRESS>Deutsches
Elektronen-Synchrotron DESY, Germany  0000-0003-2988-7859

DESY-24-083

uPDFevolv2 is a software package designed for evolving collinear and
Transverse Momentum Dependent (TMD) parton densities using the DGLAP evolution
equation. A comprehensive description of both the theoretical framework and
technical implementation is given, accompanied by a detailed guide on program
usage, focusing on customizable parameters.

This report is a technical release note for uPDFevolv2 version 2.5.03.

PROGRAM SUMMARY

Title of Program: uPDFevolv2 2.5.03
Computer for which the program is designed and others on which it is operable:
any with standard Fortran 77 (gfortran) and C++, tested on Linux, MAC
Programming Language used: FORTRAN 77, C++
High-speed storage required: No
Separate documentation available:  No
Keywords: QCD, DGLAP evolution equation, NLO and NNLO splitting functions,
transverse momentum dependent PDF (TMD)
Nature of physical problem: The evolution equations for parton densities
cannot be solved analytically and numerical methods need to be applied.
Transverse Momentum Dependent parton (TMD) densities can be obtained from the
inclusive DGLAP evolution equations once the evolution scale $q^{\prime}$ is
associated with the transverse momentum $q_{\rm t}$ of the emitted parton
Method of solution: The evolution equations for parton densities are solved
numerically using a formulation involving Sudakov form factors. The integral
equations are of Fredholm type and can be solved iteratively using a Monte
Carlo technique. The iterative solution allows for a treatment of the
kinematic relations at each individual branching process, and thus allows
directly to calculate the transverse momentum of the emitted partons, leading
to a direct calculation of the TMD parton densities.
Restrictions on the complexity of the problem: None
Other Program used: QCDnum for splitting functions and $\alpha_{s}$. Root for
plotting the result.
Download of the program: `https://updfevolv.hepforge.org`
Unusual features of the program: None

##  1 Introduction

uPDFevolv2 is a versatile software tool rooted in the Smallx [1, 2]
program†††The Smallx source code is available from Ref. [3] and built upon the
Parton Branching Method [4, 5]. It serves as a robust platform for evolving
parton densities, both collinear and Transverse Momentum Dependent (TMD),
utilizing the widely-used DGLAP evolution equation [6, 7, 8, 9] . With its
foundation in Smallx and the innovative Parton Branching Method (PB),
uPDFevolv2 offers researchers a powerful framework for studying the intricate
dynamics of parton evolution in high-energy physics.

The uPDFevolv2 package described here is a significant extension of uPDFevolv1
[10], which is based on the evolution of gluon densities using the CCFM
evolution equation [11, 12, 13, 14].

##  2 Theoretical Input

###  2.1 The Parton Branching solution of the DGLAP equation

The DGLAP evolution equation can be solved with the Parton Branching PB-
method, as detailed in Refs. [4, 5].

We begin by expressing the DGLAP evolution equation for the momentum-weighted
parton density $xf_{a}(x,\mu^{2})$ of parton $a$ with momentum fraction $x$ at
scale $\mu$:

$\mu^{2}\frac{\partial(xf_{a}(x,\mu^{2}))}{\partial\mu^{2}}=\sum_{b}\int_{x}^{1}dz\;P_{ab}\left(\alpha_{s}(\mu^{2}),z\right)\;\frac{x}{z}f_{b}\left(\frac{x}{z},\mu^{2}\right),$
(1)

where $P_{ab}$ represents the regularized DGLAP splitting functions governing
the transition of parton $b$ into parton $a$. The function $P_{ab}$ can be
decomposed as follows (following Ref. [4]):

$P_{ab}(z,\alpha_{s})=D_{ab}(\alpha_{s})\delta(1-z)+K_{ab}(\alpha_{s})\frac{1}{(1-z)_{+}}+R_{ab}(z,\alpha_{s}).$
(2)

Here, $D_{ab}$ and $K_{ab}$ are coefficients expressed as
$D_{ab}(\alpha_{s})=\delta_{ab}d_{a}(\alpha_{s})$ and
$K_{ab}(\alpha_{s})=\delta_{ab}k_{a}(\alpha_{s})$, respectively, while
$R_{ab}$ encompasses terms non-singular as $z\rightarrow 1$. Each coefficient
can be expanded in powers of $\alpha_{s}$:

$d_{a}(\alpha_{s})=\sum_{n=1}^{\infty}\left(\frac{\alpha_{s}}{2\pi}\right)^{n}d_{a}^{(n-1)},\quad
k_{a}(\alpha_{s})=\sum_{n=1}^{\infty}\left(\frac{\alpha_{s}}{2\pi}\right)^{n}k_{a}^{(n-1)},\quad
R_{ab}(z,\alpha_{s})=\sum_{n=1}^{\infty}\left(\frac{\alpha_{s}}{2\pi}\right)^{n}R_{ab}^{(n-1)}(z).$
(3)

Introducing the real-emission branching probabilities
$P_{ab}^{(R)}(\alpha_{s}(\mu^{2}),z)$:

$P_{ab}^{(R)}(\alpha_{s}(\mu^{2}),z)=K_{ab}(\alpha_{s}(\mu^{2}))\frac{1}{1-z}+R_{ab}(\alpha_{s}(\mu^{2}),z)\;,$
(4)

the solution to the evolution equation for the momentum-weighted parton
density $xf_{a}(x,\mu^{2})$ at scale $\mu$ is given by:

$xf_{a}(x,\mu^{2})=\Delta_{a}(\mu^{2})xf_{a}(x,\mu^{2}_{0})+\sum_{b}\int_{\mu^{2}_{0}}^{\mu^{2}}\frac{dq^{\prime
2}}{q^{\prime 2}}\frac{\Delta_{a}(\mu^{2})}{\Delta_{a}(q^{\prime
2})}\int_{x}^{z_{M}}dz\;\frac{\alpha_{s}}{2\pi}P^{(R)}_{ab}(z)\frac{x}{z}f_{b}\left(\frac{x}{z},q^{\prime
2}\right),$ (5)

where ${P}^{(R)}_{ab}(z)$ denotes the unregularized splitting functions and
$\mu_{0}$ is the starting scale and $\Delta_{a}$ is the Sudakov form factor
and ${\bf q}^{\prime}$ being a 2-dimensional vector with ${\bf
q}^{\prime\,2}=q^{\prime 2}$.

####  2.1.1 Determination of inclusive Parton Densities using the Parton-
Branching Method

The approach outlined above has been employed to determine collinear
($k_{t}$-integrated) parton densities.

Figure 1 shows predictions of parton densities evolved to a large scale using
the PB method [4] and compares them with calculations obtained using the
conventional tool, QCDnum [15]. The PB predictions are presented for various
values of $z_{M}$. Notably, when $z_{M}$ is sufficiently large, the
predictions align precisely with semi-analytical calculations, as seen in
prior studies [16, 17, 18, 19, 20].

Figure 1: Integrated gluon and down-quark distributions at $\mu^{2}=10$ GeV2
(left column) and $\mu^{2}=10^{5}$ GeV2 (right column) obtained from the
parton-branching solution for different values of $z_{M}$, compared with the
result from QCDnum. The ratio plots show the ratio of the results obtained
with the parton-branching method to the result from QCDnum. Figure taken from
[4].

These findings hold significance in two respects:

  * •

The DGLAP evolution equation, solved with the concept of resolvable
branchings, faithfully reproduces other DGLAP solutions when the "soft
resolution" scale $z_{M}$ is sufficiently large.

  * •

An iterative solution of the DGLAP equation using Monte Carlo techniques based
on resolvable branchings, such as the Parton Branching method, is equivalent
to other DGLAP solutions (e.g., QCDnum).

The PB-method has been applied to determine collinear parton densities by
fitting them to inclusive DIS measurements [21]. This resulted in two sets of
densities, PB-NLO-HERAI+II-set1 and PB-NLO-HERAI+II-Set2, depending on which
scale in $\alpha_{s}$ is used. In Set1, the evolution scale $q^{\prime}$ is
utilized, while in Set2, the transverse momentum $q^{\prime}(1-z)$ is
employed.

It’s crucial to note the significance of $z_{M}$: it must be close to one to
avoid omitting a significant part of the parton evolution. Here, we illustrate
the effect of $z_{M}=z_{\rm dyn}=1-q_{0}/q^{\prime}$, with $q_{0}$ being a
free resolution parameter of the order of a GeV, on inclusive collinear
distributions. The results, obtained using the same parameters and starting
distributions as in [21] but with a different cut on $z_{\rm dyn}$, are shown
in Fig. 2 (for details, refer to [22]).

Figure 2: Integrated down-quark distributions at $\mu=10$, $100$, and $500$
GeV obtained from the PB approach for different values of $z_{M}$: PB-NLO-
HERAI+II-set1 applies $z_{M}\to 1$, and PB-NLO-set1 applies $z_{M}=z_{\rm
dyn}$ with $q_{0}=1$ GeV. The ratio plots show the ratios to the one for
$z_{M}\to 1$ (figures from [22]).

Figure 2 depicts distributions for down-quarks at different scales,
demonstrating the impact when $z_{M}$ is smaller than one: soft gluons, with
transverse momenta even below a resolution scale of $q_{0}=1$ GeV, play a
significant role in collinear distributions and cannot be neglected.

###  2.2 Transverse Momentum Dependent parton densities: Parton Branching TMD

Below, we present an extension of the DGLAP evolution equation incorporating
the dependence on transverse momenta $k_{\rm T}$, as outlined by the Parton
Branching method [4]. Solving the evolution equation iteratively offers the
advantage of treating each splitting explicitly, allowing the application of
kinematic relations in every branching, akin to a parton shower process.
Consequently, parton distributions can be obtained not only depending on $x$
and $\mu$ (as in $f(x,\mu^{2})$), but also on the transverse momentum $k_{\rm
T}$ of the propagating parton (as in Transverse Momentum Dependent (TMD)
parton distributions ${\cal A}(x,k_{\rm T},\mu^{2})$).

####  2.2.1 The PB-TMD evolution equation

We can now extend the DGLAP evolution equation to incorporate the dependence
on transverse momenta as described by the Parton Branching method [4]. The
extended evolution equation for the transverse momentum dependent parton
density ${\cal A}(x,{\bf k},\mu^{2})$ is given by:

$\displaystyle{x{\cal A}}_{a}(x,{\bf k},\mu^{2})$ $\displaystyle=$
$\displaystyle\Delta_{a}(\mu^{2})\ {x{\cal A}}_{a}(x,{\bf
k},\mu^{2}_{0})+\sum_{b}\int{{d^{2}{\bf q}^{\prime}}\over{\pi{\bf q}^{\prime
2}}}\ {{\Delta_{a}(\mu^{2})}\over{\Delta_{a}({\bf q}^{\prime 2})}}\
\Theta(\mu^{2}-{\bf q}^{\prime 2})\ \Theta({\bf q}^{\prime 2}-\mu^{2}_{0})$
(6) $\displaystyle\times$
$\displaystyle\int_{x}^{z_{M}}{dz}\;\frac{\alpha_{s}}{2\pi}P_{ab}^{(R)}(z)\;\frac{x}{z}{\cal
A}_{b}\left(\frac{x}{z},{\bf k}+(1-z){\bf q}^{\prime},{\bf q}^{\prime
2}\right)\;\;,$

where the transverse momentum vectors (2-dimensional vectors) ${\bf k}$ and
${\bf q}$ are used to fully account for the transverse momentum dependence.
Here we implicitly assume angular ordering (the default option in uPDFevolv2)
which relates the transverse momentum of the emitted parton $q_{\rm t}$ to the
evolution scale $q^{\prime}$ via $q_{\rm t}^{2}=(1-z)^{2}q^{\prime\,2}$.

The final transverse momentum of the propagating parton is calculated as the
vectorial sum over all transverse momenta of the emitted partons $i$:

${\bf k}=-\sum_{i}{\bf q}_{i}\,$ (7)

where $|{\bf k}|=k_{\rm T}$. This enables the determination of the
corresponding Transverse Momentum Dependent (TMD) parton distribution ${\cal
A}(x,k_{\rm T},\mu^{2})$, in addition to the inclusive distribution
$f(x,\mu^{2})$, integrated over ${\bf k}$:

$\int{\cal A}(x,{\bf k},\mu^{2})d^{2}{\bf k}=f(x,\mu^{2})$ (8)

The evolution equation (6) can be solved iteratively using a Monte Carlo
method, with details provided in Ref. [4].

In the literature (e.g. [23]p.69), also $q^{2}$ ordering is being discussed,
which gives a slightly different relation between the evolution scale and the
transverse momentum $q_{\rm t}^{2}=(1-z)q^{\prime\,2}$ (available in
uPDFevolv2 as option `Iqord=1`). In Fig. 3 the transverse momentum
distribution for d-quarks is shown for angular ordering (default) as well as
for $q^{2}$-ordering for different values of the evolution scale $\mu$. It is
interesting to observe differences, while the integrated distributions are
identical.

Figure 3: Transverse momentum distributions of down quarks at $\mu=10,100$ GeV
(left, middle column) and $\mu=500$ GeV (right column). The red curve shows
PB-NLO-2018 Set1 (including intrinsic-$k_{\rm T}$), the blue curve shows a
prediction without including any intrinsic-$k_{\rm T}$ distribution
($q_{s}=0$), and the magenta curve shows a prediction applying $z_{M}=z_{\rm
dyn}$ with $q_{0}=1.0$ GeV without including intrinsic-$k_{\rm T}$
distributions.

####  2.2.2 Monte Carlo solution of the evolution equations

As described above, the evolution equations Eqs.(5,6) are integral equations
of the Fredholm type

$f(x)=f_{0}(x)+\lambda\int_{a}^{b}K(x,y)f(y)dy$

and can be solved by iteration as a Neumann series

$\displaystyle f_{1}(x)$ $\displaystyle=$ $\displaystyle
f_{0}(x)+\lambda\int_{a}^{b}K(x,y)f_{0}(y)dy$ $\displaystyle f_{2}(x)$
$\displaystyle=$ $\displaystyle
f_{0}(x)+\lambda\int_{a}^{b}K(x,y_{1})f_{0}(y_{1})dy_{1}+\lambda^{2}\int_{a}^{b}\int_{a}^{b}K(x,y_{1})K(y_{1},y_{2})f_{0}(y_{2})dy_{2}dy_{1}$
$\displaystyle\cdots$ (9)

using the kernel $K(x,y)$, with the solution

$f(x)=\lim_{n\to\infty}\sum_{i=0}^{n}f_{i}(x).$ (10)

Figure 4: Evolution by iteration

In a procedure termed forward evolution, a Monte Carlo (MC) technique [2, 1,
10, 4, 5] is employed to evolve from $\mu_{0}$ to a value $q^{\prime}$
determined by the Sudakov factor $\Delta(q^{\prime})$. This factor represents
the probability of evolving from $\mu_{0}$ to $q^{\prime}$ without
encountering any resolvable branching. The value $q^{\prime}$ is obtained by
solving the equation:

$\displaystyle R$ $\displaystyle=\Delta(q^{\prime}),$ (11)

where $R$ is a random number sampled from the interval $[0,1]$.

If $q^{\prime}>\mu$, the scale $\mu$ is reached, and the evolution halts,
leaving only the initial term without any resolvable branching. Conversely, if
$q^{\prime}<\mu$, a branching is generated at $q^{\prime}$ according to the
splitting function $\tilde{P}(z^{\prime})$. The evolution then continues using
the Sudakov factor $\Delta(q^{\prime\prime},q^{\prime})$. If
$q^{\prime\prime}>\mu$, the evolution stops, leaving a single resolvable
branching at $q^{\prime}$. Otherwise, the evolution continues as described
above. This iterative process repeats until $q$’s larger than $\mu^{2}$ are
generated. Through this method, all kinematically allowed contributions in the
series $\sum f_{i}(x,\mu^{2})$ are summed, yielding an MC estimate of the
parton distribution function.

Utilizing the Sudakov factor $\Delta$ and the relation

$\displaystyle\frac{\partial}{\partial q^{\prime}}\Delta(q^{\prime})$
$\displaystyle=$
$\displaystyle-\Delta(q^{\prime})\left[\frac{1}{q^{\prime}}\right]\int^{z_{M}}dz{P}(z),$

the first iteration of the evolution equation is expressed as

$\displaystyle f_{1}(x,\mu^{2})$ $\displaystyle=$ $\displaystyle
f_{0}(x,\mu^{2})$ (12) $\displaystyle+$
$\displaystyle\int_{x}^{1}\frac{dz^{\prime}}{z^{\prime}}\int_{\mu^{2}_{0}}^{\mu^{2}}(-d\Delta(q^{\prime\,2})){P}(z^{\prime}){f_{0}(x/z^{\prime},q^{\prime\,2}))}\left[{\int^{z_{M}}dz{P}(z)}\right]^{-1}.$

These integrals can be solved using a Monte Carlo method [24]: $z$ is sampled
from

$\displaystyle\int_{z_{min}}^{z}dz^{\prime}{P}(z^{\prime})$ $\displaystyle=$
$\displaystyle R_{1}\int_{z_{min}}^{z_{M}}dz^{\prime}{P}(z^{\prime}),$ (13)

where $R_{1}$ is a random number in $[0,1]$, and $q^{\prime}$ is determined by

$\displaystyle R_{2}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{x}f(x^{\prime})dx^{\prime}=F(x)$ (14)
$\displaystyle=$
$\displaystyle\int^{q^{\prime\,2})}\frac{\partial\Delta(q^{\prime\prime\,2}))}{\partial
q^{\prime\prime\,2}}dq^{\prime\prime\,2}$ $\displaystyle=$
$\displaystyle\Delta(q^{\prime\,2})$

solving for $q^{\prime}$ using the sampled $z$ and another random number
$R_{2}$ in the interval $[0,1]$.

This process continues until $q^{\prime}>\mu^{2}$, at which point the
evolution is terminated.

The first iteration of the evolution equation yields, with $z^{\prime}$ and
$q^{\prime\,2}$ selected as described above:

$\displaystyle xf_{1}(x,\mu^{2})$ $\displaystyle=$ $\displaystyle
xf_{0}(x)\Delta(\mu^{2})$ (15) $\displaystyle+$
$\displaystyle\sum_{i}\tilde{P}(z^{\prime}_{i}){x^{\prime}_{i}f_{0}(x^{\prime}_{i},q^{\prime\,2}_{i})}\left[{\int^{z_{M}}dz\tilde{P}(z)}\right]^{-1},$

where $x^{\prime}_{i}=x/z_{i}$.

###  2.3 Splitting functions, $\alpha_{s}$, and starting distributions

The evolution can be carried out at leading order (LO), next-to-leading order
(NLO), and next-to-next-to-leading order (NNLO). Splitting functions and
$\alpha_{s}$ are sourced from QCDnum [15], while the initial parton density
distributions are obtained in the same format. The NLO and NNLO splitting
functions are calculated in Refs [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35].

Alternatively, LHApdf [36] offers an alternative source for the starting
distributions, as well as using the parametrization of $\alpha_{s}$.

###  2.4 Computational Techniques: uPDFevolv2 Grid

In fitting programs where the DGLAP evolution is employed to determine the
starting distribution ${\cal A}_{0}(x)$, a full MC solution [10] becomes
impractical due to its time-consuming nature and susceptibility to numerical
fluctuations. Instead, a convolution method introduced in [37, 38] is
utilized.

The kernel ${\cal K}\left(x^{\prime\prime},k_{\rm T},\mu^{2}\right)$ is
determined from the Monte Carlo solution of the DGLAP evolution equation and
then convolved with the non-perturbative starting distribution ${\cal
A}_{0}(x)$:

$\displaystyle x{\cal A}(x,k_{\rm T},\mu^{2})$ $\displaystyle=$ $\displaystyle
x\int dx^{\prime}\int dx^{\prime\prime}{\cal A}_{0}(x^{\prime}){\cal
K}\left(x^{\prime\prime},k_{\rm
T},\mu^{2}\right)\delta(x^{\prime}x^{\prime\prime}-x)$ (16) $\displaystyle=$
$\displaystyle\int dx^{\prime}{{\cal
A}_{0}(x^{\prime})}\cdot\frac{x}{x^{\prime}}\ {{\cal
K}\left(\frac{x}{x^{\prime}},k_{\rm T},\mu^{2}\right)}.$

The kernel ${\cal K}$ encapsulates all dynamics of the evolution, including
Sudakov form factors and splitting functions. It is determined on a grid of
$50\otimes 50\otimes 50$ bins in $x,k_{\rm T},\mu^{2}$. The grid’s binning is
logarithmic, with 40 bins in logarithmic spacing below 0.1 for the
longitudinal variable $x$, and 10 bins in linear spacing above 0.1.

The starting distributions can be obtained from fits to measurements via the
xFitter platform [39, 40]. The user has to provide the grid files for gluons,
as well as for u- and d-type quarks obtained from uPDFevolv2, both collinear
and TMD grids are needed. These grid files are being used inside xFitter
convoluted with the starting distributions to provide evolved pdfs.

After a fit is performed, the resulting collinear pdfs are written in LHApdf
format, the TMD parton densities are written in TMDlib format [41]. Both
collinear and TMD parton distributions can be plotted using the graphical web
interface TMDplotter [41].

###  2.5 Determination of electroweak particle densities

The DGLAP evolution equation can be extended to include also photons and other
electroweak particles [42, 43, 44, 45, 46]. Since the quarks carry different
electric and weak charged, it is necessary to split the evolution into u-type
and d-type quarks.

The evolution of the photon density inside a hadron has been described in Ref.
[47, 48]. In Fig. 5 the collinear photon density as a function of $x$ is shown
for different values of the evolution scales $\mu$ (from Ref. [47]). For
comparison also the prediction from CT14qed-proton [49] is shown.

Figure 5: The collinear photon density at $\mu=10$ GeV and $\mu=100$ GeV as a
function of $x$. For comparison also shown is the prediction from CT14qed-
proton [49].

In Fig. 6 the TMD density of photons is shown (from Ref. [47]).

Figure 6: The TMD photon density at $\mu=10$ GeV and $\mu=100$ GeV as a
function of $k_{\rm T}$. For comparison also shown is the gluon density. Plot
from [47].

The determination of effective W densities has been discussed already in Refs.
[50, 51, 52, 53, 54, 55, 56, 57, 58]. In recent years, this ideas has been
picked up again in Refs. [45, 43, 59, 42, 60].

The approach to determine the photon densities within the PB-method can be
easily extended to calculate the collinear and TMD densities of Z and W. The
straight forward application of the method gives the collinear and TMD
densities as shown in Figs. 7,8, compared also to the photon density.

Figure 7: The collinear vector boson density at $\mu=100$ GeV and $\mu=500$
GeV as a function of $x$.

The heavy vector-boson density vanished (in the approach applied here) for
scales $\mu<m_{{\rm W}}$ therefore the densities are only shown for larger
scales. One can see very nicely, that for higher scales, the photon and W
densities approach each other, as it should.

Figure 8: The TMD vector boson density at $\mu=100$ GeV and $\mu=500$ GeV as a
function of $k_{\rm T}$.

In the transverse momentum distribution, one can observe the similarity of
photon and W densities at large $k_{\rm T}$, while significant differences are
visible at smaller $k_{\rm T}$.

###  2.6 Application: Predictions using PB collinear and TMD distributions

The evolution of the TMD gluon density has been used to perform fits to the
DIS precision data [61], as described in detail in Ref. [21].

The CASCADE3 MC generator [62], engineered specifically to conform to the PB
TMD approach, is the only generator based on TMDs. Special strength of this
approach is that CASCADE3 allows to simulate parton showers fully consistent
with TMD parton distribution functions (PDFs). The PB method and CASCADE3
provide successful predictions for Deep-Inelastic Scattering (DIS) [21, 63],
inclusive Drell-Yan (DY) at different $\sqrt{s}$ and mass ranges [64, 65, 66,
67], Z+jets [68, 69] and dijets [70, 71]. Furthermore, the PB method has also
been included in various publications of CMS [72, 70, 73] and H1 [63] showing
a great potential for becoming commonly used in the experimental community.

##  3 Description of the program components

###  3.1 Subroutines and functions

The source code of uPDFevolv2 and this manual can be found under:
`https://updfevolv.hepforge.org/`

  * sminit

to initialise

  * sminfn

to generate starting distributions in $x$ and $k_{t}$

  * smbran

to simulate perturbative branchings

  * szval

to calculate $z$ values for the splitting

  * smqtem

to generate $t$ from the corresponding Sudakov factor

  * updfgrid

to build, fill and normalise the updf grid.

  * asbmy(q)

to calculate $\frac{\alpha_{s}(q)}{2\pi}$ in LO, NLO or NNLO using the utility
from QCDnum.

  * Utility routines:
  * evolve tmd

Main routine to perform parton evolution

  * updfread

example program to read and plot the results

  * gadap

1-dimensional Gauss integration routine

  * gadap2

2-dimensional Gauss integration routine

  * divdif

linear interpolation routine (CERNLIB)

  * ranlux

Random number generator `RANLUX` (CERNLIB)

###  3.2 Parameter in steering files

  *   * Ipdf = 60500

LHApdf set name for collinear valence quark starting distribution

  * Qg = 1.3

starting value $q_{0}$ for perturbative evolution

  * Qs = 1.0

Gaussian width for the intrinsic $k_{T}$ distribution

  * Iordas = 2

order in perturbation theory for splitting functions and $\alpha_{s}$ (iordas
= 1 is LO, iordas=2 is NLO, iordas=3 is NNLO)

  * Iqord = 2

ordering definition to calculate scale in $\alpha_{s}$, the transverse
momentum $q_{\rm t}$ (and $z_{\rm dyn}$).

  * Iqord=0 : Angular ordering with the scale of $(1-z)^{2}q^{\prime\,2}$ in $\alpha_{s}$ and $z_{\rm dyn}$

  * Iqord=1 : Virtuality ordering with the scale of $(1-z)q^{\prime\,2}$ in $\alpha_{s}$ and $z_{\rm dyn}$

  * Iqord=2: Angular ordering with the scale of $q^{\prime\,2}$ in $\alpha_{s}$ and fixed $z_{M}$

  * zmaxfixed = 0.999999

active only for Iqord=2

  * Nev = 1000000

Number of generated events

  * Q0ord=0.01

active only for Iqord=0 and 1
$q_{0}$ value for dynamical $z_{max}=1-\frac{q_{0}}{q^{\prime}}$

  * Ikern=0
  * Ikern=0 : full evolution with starting distribution (run ends up with two grid files for TMD and collinear distribution)

  * Ikern=1 : only kernel to be used in xFitter (run ends up with four grid files, for gluon and quark TMD and collinear kernels)

  * mc=1.47

charm mass

  * mb=4.5

bottom mass

  * mt=173.

top mass

  * asZ=0.118

$\alpha_{s}(m_{{\rm Z}})$

###  3.3 Storing of the outputs

  * updf-grid.dat

name of the grid file for TMD outcome.

  * updf-grid_int.dat

name of the grid file for collinear outcome.

  * test.root

name of the root file contains histograms with collinear pdfs.

In order to have enough statistics, usually 900 jobs with $10^{6}$ events are
needed. The results of each job are then added to produce the final grid
files. The code for doing this can be obtained from the authors.

##  4 Program Installation

uPDFevolv2 follows the standard AUTOMAKE convention. To install the program,
do the following

    1) Get the source

    tar xvfz uPDFevolv2-XXXX.tar.gz
    cd uPDFevolv2-XXXX

    2) Generate the Makefiles (do not use shared libraries)
    ./configure

    3) Compile the binary
    make

    4) Install the executable
    make install

    4) The executable is in bin

    run it with:
    bin/updf_evolve < steering.txt

    plot the result with:

    bin/updfread

##  5 Acknowledgments

We are grateful to R. Zlebcik and L. Keersmaekers for many discussions during
the evolution of the uPDFevolv2 code. A. Lelek acknowledges funding by
Research Foundation-Flanders (FWO) (application number: 1272421N).

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