# Commuting Involutions and Elementary Abelian Subgroups of Simple Groups

Robert M. Guralnick  R.M. Guralnick, Department of Mathematics, University of
Southern California, Los Angeles, CA 90089-2532, USA<EMAIL_ADDRESS>and
Geoffrey R. Robinson  G. R. Robinson, Department of Mathematics, King’s
College, Aberdeen, AB24 3FX, UK<EMAIL_ADDRESS>Dedicated to the
memory of our good friend Jan Saxl

###### Abstract.

Motivated in part by representation theoretic questions, we prove that if $G$
is a finite quasi-simple group, then there exists an elementary abelian
subgroup of $G$ that intersects every conjugacy class of involutions of $G$.

The first author was partially supported by the NSF grant DMS-1901595 and a
Simons Foundation Fellowship 609771

We thank Thomas Breuer for computations with sporadic groups and Ross Lawther
for comments on the exceptional groups in characteristic $2$.

##  1\. Introduction

Let $G$ be a finite group. An elementary abelian subgroup $E$ of $G$ is a
called a broad subgroup if every involution of $G$ is conjugate to an element
of $E$. This definition is motivated in part by the fact that (using a Theorem
of R.Knörr [5]) an irreducible character which vanishes identically on non-
identity elements of $E$ lies in a $2$-block of defect zero. This in turn
allows us to prove that when $G$ contains a broad subgroup, the number of
irreducible characters of $G$ which lie in $2$-blocks of positive defect of
$G$ is at most $|C_{G}(t)|$ for some (non-trivial) involution $t\in G$. Our
main result (which depends upon the classification of finite simple groups)
is:

######  Theorem 1.

Let $G$ be a finite quasi-simple group. Then $G$ contains a broad subgroup.

Notice that it is immediate that a direct product of finite groups has a broad
subgroup if each direct factor has a broad subgroup.

We have the following corollary, which may be of independent interest:

######  Corollary 2.

Let $G$ be a finite quasi-simple group. If $x$ and $y$ are involutions of $G$,
then $x$ commutes with some conjugate of $y$.

The promised application to character theory is provided by:

######  Corollary 3.

Let $G$ be a finite group with no non-trivial normal subgroup of odd order.
Then there is a non-trivial involution $t\in F^{\ast}(G)$ such that the total
number of irreducible characters of $G$ which do not lie over $2$-blocks of
defect zero of $F^{\ast}(G)$ is at most $|C_{G}(t)|$.

This applies in particular when $G$ is a non-Abelian simple group, and since
$G={\rm SL}(2,2^{n})$ (for $n\geqslant 2)$ has exactly $2^{n}$ irreducible
characters which lie in $2$-blocks of positive defect, while
$|C_{G}(t)|=2^{n}$ for each non-trivial involution $t\in G$, the inequality of
Corollary 3 may be sharp. If $G$ is quasisimple and the center has even order,
then there is a central element of order $2$ and so the corollary is not
useful.

Note that there is no direct analog of the theorem for odd primes. For example
if $p\geq 5$, let $S=\operatorname{SL}_{n}(p)$ with $4\leq n\leq p$. Since
$Z(S)$ has order prime to $p$, it suffices to produce an example in $S$. Let
$x$ be a regular unipotent element (i.e. $x$ has a single Jordan block of size
$n$). Note that $x$ has order $p$. Then the centralizer of $x$ is an
elementary abelian $p$-group of rank $n-1$. It is straightforward to check
that the only elements $y\in C_{S}(x)$ which have two Jordan blocks have
Jordan blocks of sizes $(n\pm 1)/2$ if $n$ is odd and two Jordan blocks of
size $n/2$ if $n$ is even. Thus, $x$ does not commute with an element of order
$p$ that has Jordan blocks of size $n-1$ and $1$.

For $p=3$, we give an example where elements of order $3$ are semisimple. Let
$S=\operatorname{SL}_{9}(q)$ with $q\equiv 4\allowbreak\mkern
4.0mu({\operator@font mod}\,\,9)$. Then $Z=Z(S)=\langle z\rangle$ has order
$3$. Let $x\in S$ satisfy $x^{3}=z$. So in $G=S/Z$, $xZ$ is an element of
order $3$. Let $y\in S$ be an element of order $3$ with a $7$ dimensional
eigenspace with eigenvalue $1$. We claim that no conjugate of $y$ commutes
with $x$ modulo $Z$. Suppose there is a conjugate $y^{\prime}$ of $y$
commuting with $x$ modulo $Z$. Then replacing $y^{\prime}$ by $y$, we see that
$[x,y]=u\in Z$. Then $y$ is conjugate to $yu$ whence by considering
eigenvalues, $u=1$. So $x$ and $y$ commute in $S$. Since $y$ has a
$1$-dimensional eigenspace, $x$ must preserve this space. However, $x$ has no
eigenvalues in the base field, a contradiction.

The result also fails in general for almost simple groups for $p=2$. Let
$G=\operatorname{O}^{+}_{4m}(2^{a})$. There are two conjugacy classes of
involutions in $\operatorname{SO}_{4m}^{+}(2^{a})=\Omega_{4m}^{+}(2^{a})$
interchanged by an involution in $G$. Clearly such an outer involution cannot
commute with any element of either of the two conjugacy classes interchanged.

Note that all three families of counterexamples above are also counterexamples
to the corollary. We also remark that it is easy to check that the quaternion
group $Q_{8}$ is the only extra-special $2$-group which has a broad subgroup,
while it is clear that for $p$ odd, a non-Abelian $p$-group of exponent $p$
never has a broad elementary $p$-subgroup.

In the next three sections, we prove Theorem 1 for the various families of
quasi-simple groups. In the last two sections, we prove Corollary 3 and
discuss some other character theoretic results.

##  2\. Alternating and Sporadic Groups

We make a few preliminary remarks before starting the proof. We may assume
that $G$ has no odd order normal subgroups and in particular if $G$ is
quasisimple, we may assume that the center is a $2$-group.

If $G$ has a unique class of noncentral involutions, then the result holds by
considering the abelian subgroup $E=\langle Z(G),x\rangle$ where $x$ is a
noncentral involution. If $G$ has $3$ conjugacy classes of involutions,
$C_{1},C_{2}$ and $C_{3}$ and there exist $x_{i}\in C_{i}$ with
$x_{1}x_{2}x_{3}=1$, then the elementary abelian subgroup $\langle
x_{1},x_{2}\rangle$ intersects each $C_{i}$. If $H$ is a subgroup of $G$ and
$H$ intersects every conjugacy class of involutions of $G$ and $H$ has a broad
subgroup, then so does $G$.

The case of alternating groups is trivial.

######  Lemma 2.1.

Let $G=A_{n},n\geq 5$. Let $E$ be the elementary abelian subgroup of $S_{n}$
with all orbits of size $2$ if $n$ is even and all orbits but one of size $2$
if $n$ is odd. Then every involution in $S_{n}$ is conjugate to an element of
$E$ by an element of $A_{n}$.

######  Lemma 2.2.

Let $G=2A_{n},n\geq 5$. Then $G$ contains a broad subgroup.

###### Proof.

If $x\in A_{n}$ is an involution, then $x$ lifts to an involution in $G$ if
and only if the number of points moved by $x$ is a multiple of $8$. Now choose
a partition of $n$ into subsets $X_{0},X_{1},\ldots,X_{d}$ where $|X_{i}|=8$
for $i>0$ and $|X_{0}|<8$. Let $E$ be the elementary abelian $2$-group
$E_{1}\times\ldots\times E_{d}$ where $E_{i}$ acts as a regular elementary
abelian group of order $8$ on $X_{i}$. Note that if $x\in E$, then $x$ moves
$8e$ points for some $0\leq e\leq d$ and so every element in $E$ lifts to an
involution in $G$, whence the lift of $E$ to $G$ is also elementary abelian. ∎

All but five of the sproadic groups have at most $2$ noncentral conjugacy
classes of involutions and so the observations above yield the result. The
remaining cases are $Co_{2},Co_{1},Fi_{22},Fi_{23}$ and $BM$. The first four
have three nontrivial classes of involutions and $BM$ has four such classes.
In these cases, we just use GAP [2] to produce $E$. One deals with the quasi-
simple sporadic groups in a similar manner [2].

######  Lemma 2.3.

Let $G$ be a quasi-simple sporadic group. Then there exists an elementary
abelian subgroup $E$ intersecting every conjugacy class of involutions in $G$.

##  3\. Group of Lie type in characteristic $2$

In this section $q$ is a power of $2$. We consider the finite simple groups of
Lie type over $\mathbb{F}_{q}$. We will typically work with the simply
connected group (which will have center of odd order) which is sufficient. We
note the result that centralizer of an involution is connected. It is not true
that all centralizers of unipotent elements are connected.

######  Lemma 3.1.

Let $G$ be a connected reductive algebraic group over any algebraically closed
field of characteristic $2$. Let $g\in G$ be an involution. Then $C_{G}(g)$ is
connected.

###### Proof.

Clearly, we can assume that $G$ is semisimple. Then $Z(G)$ is finite of odd
order and so the result does not depend upon the isogeny class of $G$. So we
may assume that $G$ is a direct product and so it suffices to assume that $G$
is simple.

If $G$ is exceptional, we just quote [8, Chapter 22]. So assume that $G$ is
classical. By Lang’s theorem, it suffices to show that if two involutions in
$G(q)$ (untwisted) are conjugate in the algebraic group, they are already
conjugate in $G(q)$ (and indeed we may assume that $q$ is large). If
$G=\operatorname{SL}_{2n}(q)$, then any involution is conjugate to one in the
unipotent radical $Q$ of the stabilizer of $n$-dimensional space and two
elements in $Q$ are conjugate if and only if they have the same rank (identify
$Q$ with $n\times n$ matrices), whence the result. If
$G=\operatorname{SL}_{n}(q)$, we can reduce to the case that $n$ is even (or
use a slightly different unipotent radical). Similarly if
$G=\operatorname{Sp}_{2n}(q)$, then any involution acts trivially on a totally
singular $n$-dimensional subspace [3] and so is contained in the unipotent
radical $Q$ of the stabilizer of such a space [3]. We can identify $Q$ with
the space of symmetric bilinear forms on an $n$-dimensional space and the
orbits of the Levi subgroup are the congruence classes of such forms. Two
forms are congruent if and only they have the same rank and either both are
alternating or not and again the result holds.

Next consider $G=\operatorname{SO}^{+}_{2n}(q)=\operatorname{SO}(V)$. Let
${\bar{G}}$ be the corresponding algebraic group. Let $C$ be a $G$-conjugacy
class of involutions and let ${\bar{C}}$ be the conjugacy class of $\bar{G}$
containing $C$. Recall that $G\leq H=\operatorname{Sp}_{2n}(q)$. It follows by
[1] or [3] that if $D$ is the $H$-class containing $C$, then $D\cap G=C$ or
$C\cup\tau(C)$ for $\tau$ any element of $\operatorname{O}_{2n}^{+}(q)$ not in
$\operatorname{SO}_{2n}^{+}(q)$. By the result for symplectic groups, $D$ is
precisely the set of $\mathbb{F}_{q}$ points of the corresponding conjugacy
class in the algebraic group.

Let $x\in C$. Let $(\ ,\ )$ denote the alternating form preserved by $G$. If
$(xv,v)\neq 0$ for some $v\in V$, then $x$ preserves a nondegenerate $2$-space
and so $x$ commutes with a transvection $\tau$ (over the base field). In
particular, $\tau(C)=C$ and so $C=D\cap G=\bar{C}\cap H$ as required. Suppose
that $(xv,v)=0$ for all $v\in V$. It follows just as for the symplectic case
that $x$ acts trivially on a totally singular $n$-dimensional space and so $x$
is contained in the unipotent radical of the stabilizer of such a space and
the orbits of the Levi subgroup on the unipotent radical correspond to the
orbits of $\operatorname{GL}_{n}$ on skew symmetric matrices. The only
invariant is the rank and so there is no fusion after field extensions. Thus
$\bar{C}\cap H=C$ as required.

∎

###  3.1. Linear, Unitary and Symplectic Groups

The case of $\operatorname{SL}_{n}(q)$ is obvious.

######  Lemma 3.2.

Let $G=\operatorname{SL}_{n}(q),n\geq 2$. Let $m=n/2$ if $n$ is even and let
$m=(n+1)/2$ if $n$ is odd. Let $P$ be the stabilizer of an $m$-dimensional
subspace $W$. Then any involution in $G$ is conjugate to an element acting
trivially on $W$. In particular, if $Q$ is the unipotent radical of the
stabilizer of $W$, then $Q$ is elementary abelian and every involution of $G$
is conjugate to an element of $Q$.

For symplectic and unitary groups, we use the following elementary
observation.

######  Lemma 3.3.

Let $G=\operatorname{SU}_{2n}(q),n\geq 2$ or $\operatorname{Sp}_{2n}(q),n>2$.
Then any involution $g$ acts trivially on a totally isotropic subspace of
dimension $n$ and so the elementary abelian subgroup $Q$ that is the unipotent
radical of the stabilizer of a totally singular subspace of dimension $n$
intersects every conjugacy class of involutions.

###### Proof.

For $G=\operatorname{Sp}_{2n}(q)$, the argument is easier (and is given in
[3]). So we assume that $G=\operatorname{SU}_{2n}(q)$. If $n=1$, this is
clear.

Let $W$ be the fixed space of the involution of $g$. Note that $\dim W\geq n$.
If $W$ is totally singular, the result follows. Otherwise $g$ is trivial on a
nonsingular $1$-space $L$. Then $g$ acts on $L^{\perp}$ and since $\dim
L^{\perp}=2n-1$, the fixed space of $g$ has dimension at least $n$ and so is
not totally singular. Thus $g$ leaves invariant some nondegenerate $1$-space
orthogonal to $L$ and so acts trivially on a nondegenerate $2$-space. The
result then follows by induction. ∎

If $G=\operatorname{SU}_{2n+1}(q)$, then any involution is conjugate to an
element of $\operatorname{SU}_{2n}(q)$. So to prove the theorem it suffices to
consider only even dimensional unitary groups. Thus, the previous lemma gives
the result for $\operatorname{SU}$ and $\operatorname{Sp}$.

###  3.2. Orthogonal Groups

We now consider $G=\operatorname{SO}_{2n}^{\epsilon}(q)=\operatorname{SO}(V)$
with $n\geq 4$. This case is different than the other families of classical
groups and we have to examine the conjugacy classes of involutions more
closely. We should note that (in characteristic $2$) we denote the simple
group by $\operatorname{SO}$ (so in particular $\operatorname{SO}$ is not the
full orthogonal group of determinant $1$ isometries preserving the quadratic
form – often $SO^{\epsilon}_{n}(q)$ is denoted by $\Omega^{\epsilon}_{n}(q)$).

Suppose that $n=2m$ is even. Decompose $V=V_{1}\perp\ldots\perp V_{m}$ as the
orthogonal sum of $m$ four dimensional spaces. If $V$ has $+$ type, choose all
the summands to be of $+$ type. Otherwise choose $V_{m}$ to be of $-$ type.
Let $E$ be a Sylow $2$-subgroup of
$\operatorname{SO}(V_{1})\times\ldots\times\operatorname{SO}(V_{m})$. Note
that $E$ is elementary abelian and that the normalizer of $E$ contains
elements in $\operatorname{O}(V)$ not in $\operatorname{SO}(V)$. Thus it
suffices to show that $E$ intersects every $\operatorname{O}(V)$ conjugacy
class of involutions. Such involutions are described in [8, Chap. 6]. They
possible conjugacy classes are labelled $W(2)^{e}\oplus W(1)^{f}$ or
$V(2)^{2}\oplus W(2)^{e}\oplus W(1)^{f}$ (see [8, Chapter 6]). They are
distinguished by their Jordan form and whether $(gv,v)$ vanishes identically
for $v\in V$ with respect to the alternating form left invariant by $G$ (the
classes involving $V(2)^{2}$ are the ones where this function does not vanish
everywhere).

Note that involutions in $\operatorname{SO}_{4}^{+}(V)$ are either trivial,
$W(2)$ or $V(2)^{2}$. Involutions in $\operatorname{SO}_{4}^{-}(V)$ are either
trivial or are in the class $V(2)^{2}$. If $V$ has $+$ type, then it is clear
any class of involutions intersects $E$. Similarly if $V$ has $-$ type, it is
clear that any involution other than $W(2)^{m}$ is conjugate to an element of
$E$. However, this class of involutions in the algebraic group is not
invariant under the graph automorphism (i.e. there are two $\operatorname{SO}$
classes which are fused in $\operatorname{O}$). Since the classes are stable
under any field automorphism, it follows that this class is not present in
$\operatorname{SO}^{-}_{n}(q)$.

If $n$ is odd, decompose $V=W\perp U$ with $\dim U=2$ and $W$ of $+$ type (and
so $U$ as the same type as $V$). By [8, Chap. 6], any involution in $g$ is
conjugate to an element in $\operatorname{SO}(W)$ and so the result follows by
the case that $n$ is even. This completes the proof of the theorem in this
case.

###  3.3. Exceptional Groups

Note that for the groups ${{}^{2}}B_{2}(2^{2a+1})$,
${{}^{2}}F_{4}(2^{2a+1})^{\prime}$ or $G_{2}(q)$, there are at most $2$
noncentral conjugacy classes of involutions and the theorem holds – see [8,
Chap. 22]. Also since centralizers of involutions are connected, all classes
of involutions intersect a subfield group and so one can work over the base
field (and so use the character tables). We do not use this.

In $F_{4}(q)$, the center of a Sylow $2$-subgroup has order $q^{2}$ and
intersects of three of the four nontrivial classes of involutions and so the
result holds in the case as well. It is easy to see that each conjugacy class
of involutions in ${{}^{2}}E_{6}(q)$ or $E_{6}(q)$ intersects $F_{4}(q)$. This
can be seen by the tables in [8] – indeed they exhibit the classes of
unipotent elements in the exceptional groups by first working in $E_{8}$ and
then determining which classes intersect the smaller groups and if they split.
Alternatively, using the Jordan block structure of involutions of $F_{4}$
acting on the Lie algebra and the $26$ dimensional module [6] and noting that
the Lie algebra of $E_{6}$ as an $F_{4}$-module is the direct sum of the Lie
algebra of $F_{4}$ and the $26$-dimensional module, one can see there three of
the nontrivial classes in $F_{4}$ remain distinct in $E_{6}$. Thus, the result
for these cases follows from the result for $F_{4}$.

In $E_{7}$, using the standard labelling, we see that we can choose $4$
commuting simple root subgroups and the group generate by meets all four
classes (see the labelling given in [1] or [6]). Each class of involutions in
$E_{8}$ intersects $E_{7}$ (arguing as in the case of $F_{4}<E_{6}$). Thus,
the theorem holds for $E_{7}$ and $E_{8}$.

###  3.4. Exceptional Multipliers

If $G$ is a simple group of Lie type in characteristic $2$, then almost always
its Schur mulitplier has odd order and so there is nothing more to do. There
are a handful of cases where this fails. See [4, Table 6.1.3]. In each of the
cases, one produces the required broad subgroup using GAP [2].

##  4\. Groups of Lie type in odd characteristic

In this section, we consider finite simple groups of Lie type over a field
$\mathbb{F}_{q}$ with $q$ odd. We prove a slightly stronger result which makes
the proof easier. So let $X$ be a simple algebraic group over an algebraically
closed field of odd characteristic $p$. Let $\sigma$ be a Lang-Steinberg
endomorphism of $X$ and let $H=X_{\sigma}$ the fixed points of $\sigma$. Then
$H$ is a finite group of Lie type over a finite field $\mathbb{F}_{q}$ for
some $q$ a power of $p$. Let $G=O^{p^{\prime}}(H)$. Then aside from the cases
that $H={{}^{2}}G_{2}(3)$ or $H$ is $\operatorname{SL}_{2}(3)$ or
$\operatorname{PGL}_{2}(3)$, we have $G$ is quasi-simple. Note that in the
last cases $G$ is solvable and in the first case
$[H,H]\cong\operatorname{PSL}_{2}(8)$. We will exclude these cases (clearly
the main theorem holds in these cases).

We say that a subgroup $S$ of $H$ is toral if $S$ is contained in a torus $T$
in $X$. It follows that $S$ is contained in a maximal torus of $H$ (which is
defined to be $C_{T}(\sigma)$ for $T$ a $\sigma$ invariant maximal torus of
$X$). Note that if $\phi:X\rightarrow Y$ is an isogeny of algebraic groups,
then tori map to tori and the inverse image of a maximal torus is a maximal
torus.

We shall prove the following:

######  Theorem 4.1.

Let $G$ be a finite quasi-simple group of Lie type over a field of odd
characteristic. Then there exists a toral subgroup $E$ of $G$ which intersects
every conjugacy class of involutions of $G$.

If $G$ is as in the theorem and $Z$ is a central subgroup, then the result for
$G/Z$ implies the result for $G$ (since as noted above toral subgroups lift to
toral subgroups and any involution in $G$ maps to an involution in $G$). On
the other hand, if we prove that there exists a toral subgroup $S$ of $G$ such
that $S$ intersects every conjugacy class of $2$-elements $g\in G$ with
$g^{2}\in Z$, then $SZ/Z$ is toral and intersects each class of involutions in
$SZ/Z$. Thus, we can choose a particular form of the group and prove the
result needed for that form.

If $G$ is simply connected and split, the result is quite easy.

######  Lemma 4.2.

Let $G$ be a finite quasi-simple simply connected group of Lie type over the
field of $q$-elements. Let $S$ be a maximal torus of $G$ contained in a Borel
subgroup of $G$. Then $S$ intersects every conjugacy class of involutions of
$G$.

###### Proof.

Let $T$ be a maximal torus of $X$ containing $S$. Every involution of $X$ is
conjugate to an element of $T$ (indeed every semisimple element of $X$ is
conjugate to an element of $T$). Note that $S$ is a direct product of $r$
copies of a cyclic group of order $q-1$ and so that $S$ contains the
$2$-torsion subgroup of $T$. Let $g\in G$ be an involution, then $g$ is
conjugate in $X$ to some element of $S$. Since $X$ is simply connected,
$g^{X}\cap G=g^{G}$ and so $g$ is conjugate in $G$ to an element of $S$. ∎

We can now complete the proof for the exceptional groups. The previous lemma
implies the result for the simply connected groups of type $G_{2}(q)$,
$F_{4}(q)$, $E_{6}(q)$, $E_{7}(q)$ and $E_{8}(q)$. Aside from the case of
$E_{7}(q)$, the centers are either trivial or have order $3$ and so the result
holds.

We note that in the simply connected group $E_{7}(q)$, there are maximal tori
that are direct products of seven cyclic groups of order $q-1$ and also of
order $q+1$ [7]. Arguing as above, one of these maximal tori contain a
conjugate of any element of order $4$ in $G$ and so any element whose square
is central.

This leaves only the cases of ${{}^{2}}G_{2}(3^{2k+1}),k\geq 1$ and
${{}^{2}}E_{6}(q)$. In the first case, there is only one class of involutions
and the result follows since any semisimple element is contained in some
maximal torus. In the case of ${{}^{2}}E_{6}(q)$ (and since the center has odd
order, it suffices to consider the simply connected case), there is a maximal
torus that is the direct product of six copies of the cyclic group of order
$q+1$ [7]. This contains the $2$-torsion subgroup of a maximal torus of the
algebraic group and we argue as above.

We next consider the classical groups. We will work with the form of the group
that acts on the natural module $V$.

We first consider $\operatorname{SL}_{n}(q)$. If $n$ is odd, the center has
odd order and so the result follows by Lemma 4.2. We next handle the case that
$n$ is even.

######  Lemma 4.3.

Let $G=\operatorname{SL}_{2m}(q)\leq GL_{2m}(q)=L$. Then there exists a toral
subgroup $A$ of $L$ such that $A$ intersects every conjugacy class of elements
of $G$ containing elements whose square is central.

###### Proof.

Note that semisimple elements in $G$ which are conjugate in $L$ are already
conjugate in $G$. Let $Z$ be the Sylow $2$-subgroup of the center of $G$ with
$z$ a generator. Consider the subgroup $B=B_{1}\times\ldots\times B_{m}$ of
$L$ with each $B_{i}\cong\operatorname{GL}_{2}(q)$ acting on a direct sum of
$m$ two dimensional spaces. Let $A_{i}$ be a cyclic subgroup of $B_{i}$ of
order $q^{2}-1$ and set $A=A_{1}\times\ldots\times A_{m}$. Note that $A$
contains an element $w$ with $w^{2}=z$ (by considering the two dimensional
case). Suppose that $y^{2}=z^{i}$ with $y\in G$. If $i$ is odd, then there is
an odd power of $y$ with square $z$ and so $y$ is conjugate to a power of $w$.
If $i=2j$, then $yz^{-j}$ is an involution in $G$. It is easy to see that an
involution in $G$ is conjugate to an element of $A$ (since each eigenspace is
even dimensional). Thus, $A$ contains a conjugate of every element of $G$
whose square is central. Clearly $A$ is toral and so the result holds. ∎

As we noted above, this implies the result for any quotient of
$\operatorname{SL}$.

The same proof holds for $G=\operatorname{SU}_{2m}(q),m\geq 2$ and so the main
result for any quotient of $\operatorname{SU}_{2m}(q)$. If
$G=\operatorname{SU}_{n}(q)$ with $n$ odd, then $Z(G)$ has odd order and we
see that there is a toral subgroup that is a direct product of $n-1$ copies of
a cyclic group of order $q+1$ which contains the $2$-torsion subgroup of a
maximal torus of the algebraic group and the result follows as above.

If $G=\operatorname{Sp}_{2n}(q)$ with $q\equiv 1\allowbreak\mkern
4.0mu({\operator@font mod}\,\,4)$, then the split torus $T$ contains the
$4$-torsion subgroup of the maximal torus containing it in the algebraic
group. Thus every element of order $4$ in $G$ is conjugate to an element of
$T$ whence the results holds for $\operatorname{Sp}_{2n}(q)$ and its central
quotient. If $q\equiv 3\allowbreak\mkern 4.0mu({\operator@font mod}\,\,4)$,
there is a maximal torus of $G$ that is the direct product of $n$ copies of
the cyclic group of order $q+1$ which again contains the $4$-torsion subgroup
of a maximal subgroup of the algebraic group and the result follows.

Finally we consider the quasi-simple groups related to orthogonal groups in
dimension at least $7$. We will prove the result for the groups
$G=\Omega^{\pm}_{n}(q)\leq L=\operatorname{SO}^{\pm}_{n}(q)$. As noted above,
this suffices to prove the result in general.

First suppose that $n=2m+1$ is odd. Then
$\operatorname{SO}_{n}(q)=\operatorname{SO}(V)$ has no center and the simple
group $\Omega_{n}(q)$ consists of the elements with spinor norm $1$ and has
index $2$. Decompose $V=W\perp L$ with $W$ a hyperplane such that the central
element in $\operatorname{SO}(W)$ has spinor norm $1$. Now decompose
$W=W_{1}\perp\ldots\perp W_{m}$ as an orthogonal sum of $m$ nondegenerate
$2$-paces with at least one summand of each type. If $g$ is an involution in
$\Omega_{n}(q)$, then $g$ is conjugate to an element of the maximal torus
$T=\operatorname{SO}(W_{1})\times\ldots\times\operatorname{SO}(W_{m})$. If the
multiplicity of the trivial eigenspace of $g$ is greater than $1$, this is
clear since the class of $g$ (for $g$ an involution even in
$\operatorname{SO}(V)$) is determined by the dimension and type of its fixed
space. If the fixed space of $g$ is $1$-dimensional and $g$ has spinor norm
$1$, then the $-1$ eigenspace of $g$ has the same type as $W$ and so $g$ is
conjugate to the element acting as $-1$ on $W$. Thus, $T\cap\Omega_{n}(q)$ is
a toral subgroup intersecting each conjugacy class of involutions and the
result holds.

Next consider the case that $n=4m+2,m\geq 2$. First consider the case that
$L=\operatorname{SO}_{n}^{\epsilon}(q)=\operatorname{SO}(V)$ with
$q\equiv\epsilon 1\allowbreak\mkern 4.0mu({\operator@font mod}\,\,4)$, Note
that $Z(L)=Z(G)$ has order $2$ in this case. Decompose
$V=V_{1}\perp\ldots\perp V_{2m+1}$ where the $V_{i}$ are $2$-dimensional
spaces of the same type as $V$. Let
$T=\operatorname{SO}(V_{1})\times\ldots\times\operatorname{SO}(V_{2m+1})$.
Then $T$ is a maximal torus of $\operatorname{SO}(V)$. It is easy to see that
any involution of spinor norm $1$ is conjugate to an element of $T$. Moreover,
if $x\in G$ and $x^{2}=-I$, then $x$ is conjugate to an element of $T$ (reduce
to the two dimensional case).

Thus, $T\cap\Omega(V)$ intersects any conjugacy of elements of $G$ whose
square is central. This proves the theorem for $\Omega(V)$ and it simple
quotient and so the result holds.

Suppose that $q\equiv\epsilon 3\allowbreak\mkern 4.0mu({\operator@font
mod}\,\,4)$. In this case the nontrivial involution in $Z(L)$ has spinor norm
$-1$ and so $\Omega(V)$ has trivial center and is the simple group. Decompose
$V=V_{1}\perp\ldots\perp V_{2m+1}$ so that there are at least $2$ summands of
each type. Let
$T=\operatorname{SO}(V_{1})\times\ldots\times\operatorname{SO}(V_{2m+1})$.
Then $T$ intersects every conjugacy class of involutions of
$\operatorname{SO}(V)$ and the result holds.

Finally consider the case that $n=4m,m\geq 2$. Suppose that
$L=\operatorname{SO}_{n}^{-}(q)=\operatorname{SO}(V)$. Note that in this case
$-I$ has spinor norm $-1$ and so $\Omega_{n}^{-}(q)$ is simple. Decompose
$V=V_{1}\perp\ldots\perp V_{2m}$ into an orthogonal sum of $2m$ nondegenerate
$2$-dimensional spaces (note both types must occur in this decomposition). It
is clear that any involution in $\Omega(V)$ is conjugate to an element of
$T=\operatorname{SO}(V_{1})\times\ldots\times SO(V_{2m})$ and the result
holds.

Finally suppose that $L=\operatorname{SO}_{n}^{+}(q)$. Decompose
$V=V_{1}\perp\ldots\perp V_{2m}$ with each $V_{i}$ of dimension $2$ and all of
the same type. If $q\equiv 1\allowbreak\mkern 4.0mu({\operator@font
mod}\,\,4)$, take the type of $V_{i}$ to be $+$ and $-$ otherwise. In the case
the central involution in $L$ has spinor norm $1$ and indeed the involution in
$\operatorname{SO}(V_{i})$ has spinor norm $1$. Set
$T=SO(V_{1})\times\ldots\times SO(V_{2m})$, a direct product of $2m$ cyclic
groups of order $q\pm 1$ (and each cyclic group has order a multiple of $4$).
If $x\in G$ and $x^{2}=-I$, then $x$ is conjugate to an element of $T$ and any
involution of spinor norm $1$ is also conjugate to an element of $T$ and the
result follows.

##  5\. Proof of Corollary 3

As mentioned in the introduction, a theorem of R. Knörr [5], tells us that
when $p$ is a prime and $G$ is a finite group of order divisible by $p$, then
the irreducible character $\chi$ of $G$ lies in a $p$-block of defect zero of
$G$ (hence vanishes on all elements of $G$ of order divisible by $p$) if and
only if $\chi$ vanishes on all elements of order $p$ in $G$. Hence when $G$
has an elementary Abelian subgroup $E$ which meets every conjugacy class of
elements of order $p$ of $G$, the irreducible character $\chi$ lies in a
$p$-block of defect zero of $G$ if and only if $\chi$ vanishes identically on
the non-identity elements of $E$.

We may conclude, using the results of [9] that if $G$ has such an elementary
Abelian $p$-subgroup $E$, then $\sum_{a\in
E^{\\#}}|\chi(a)|^{2}\geqslant|E|-1$ whenever $\chi$ lies in a $p$-block of
positive defect of $G$.

Using the orthogonality relations, we deduce that if $G$ has $k_{+}$
irreducible characters lying in $p$-blocks of positive defect, we have

$\sum_{a\in E^{\\#}}|C_{G}(a)|\geqslant k_{+}(|E|-1).$

In particular, we have $k_{+}\leqslant|C_{G}(a)|$ for some $a\in E^{\\#}.$

More generally, let $G$ be a finite group, and $N\lhd G$. We will first prove
that if $N$ has an elementary Abelian $p$-subgroup $E$ which meets every
conjugacy class of elements of order $p$ of $N$, then there an element $x$ of
order $p$ in $N$ such that at most $|C_{G}(x)|$ irreducible characters of $G$
lie in $p$-blocks which do not cover $p$-blocks of defect zero of $N$.

We first need a slight extension of the above result of Knörr.

######  Lemma 5.1.

Let $G$ be a finite group and $N$ be a normal subgroup of $G$. Let $B$ be a
$p$-block of $G$ with defect group $D$ such that $D\cap N\neq 1.$ Then for any
irreducible character $\chi\in B$, there is an element $x\in N$ of order $p$
with $\chi(x)\neq 0$.

###### Proof.

Let $(R,K,F)$ be a $p$-modular system for $G$. We note that $H=N_{G}(D\cap
N)\geqslant N_{G}(D)$ so by Brauer’s First Main Theorem, there is a unique
Brauer correspondent block for $B$ in $N_{G}(D\cap N)$, say $B^{\prime}.$ Let
$\sigma=\sum_{\\{n\in N:n^{p}=1\\}}n$, which is an element of $Z(RG)$.

We claim that $\omega_{\chi}(\sigma)\in J(R)$, so that
$\omega_{\chi}(\sigma-1_{G})\not\in J(R)$, and in particular, there must be
some non-identity element $x\in N$ of order $p$ with $\chi(x)\neq 0$.

Now, using the Brauer homomorphism, there is an irreducible character $\mu\in
B^{\prime}$ (which may be assumed to have $D\cap N$ in its kernel) such that
$\omega_{\chi}(\sigma)\equiv\omega_{\mu}(\sigma^{\ast})$ ( mod $J(R)$ ), where
$\sigma^{\ast}=\sum_{\\{n\in C_{G}(D\cap N):n^{p}=1\\}}n.$

However, let $Z=\Omega_{1}(Z(D\cap N))$, and let $Z^{+}$ be the sum of its
elements in $RG$. It is clear that $\sigma^{\ast}=Z^{+}T$ for some element $T$
of $RC_{G}(D\cap N)$ which is itself a sum of certain elements of order
dividing $p$. Notice that $T$ commutes with $Z^{+}.$ It follows that
$\sigma^{\ast}$ has nilpotent image in $Z(FN_{G}(D\cap N))$, so that
$\omega_{\mu}(\sigma^{\ast})\in J(R)$, which suffices to complete the proof.

∎

We conclude from the Lemma (and Clifford’s Theorem) that the irreducible
character $\chi$ of $G$ lies over characters in $p$-blocks of positive defect
of $N$ if and only if $\chi(x)\neq 0$ for some element $x$ of order $p$ in
$N$.

In particular, if $E$ is the above broad elementary Abelian $p$-subgroup of
$N$, then no irreducible character $\chi$ of $G$ which lies over irreducible
characters in $p$-blocks of positive defect zero of $N$ can vanish identically
on $E^{\\#}.$ We may then conclude as above that there is an element $x$ of
order $p$ in $N$ such that the total number of irreducible characters of $G$
which do not vanish identically on $p$-singular elements of $N$ is at most
$|C_{G}(x)|$.

Finally, we turn our attention to the case $p=2$, $O(G)=1$ and
$N=F^{\ast}(G)$. If $O_{2}(G)\neq 1$, let $Z=\Omega_{1}(Z(O_{2}(G)))\neq 1.$
Then by Clifford’s Theorem, whenever $\chi$ is an irreducible character of
$G$, ${\rm Res}^{G}_{Z}(\chi)$ is not a multiple of the regular character, so
that $\chi$ does not vanish identically on $Z^{\\#}.$ Hence, as before, we
have $\sum_{z\in Z^{\\#}}|\chi(z)|^{2}\geqslant|Z|-1$, and we have
$k(G)\leqslant|C_{G}(z)|$ for some $z$ of order $2$ in $Z$, where (as usual),
$k(G)$ denotes the number of complex irreducible characters of $G$.

Suppose then that $O_{2}(G)=1$. Then $F^{\ast}(G)$ is a direct product of non-
Abelian simple groups, and $F^{\ast}(G)$ has a broad elementary $2$-subgroup,
say $E$, by Theorem 1 and the remarks following. Now the arguments above tell
us that there is some element $t\in E^{\\#}$ such that the number of
irreducible characters of $G$ which do not vanish identically on $2$-singular
elements of $F^{\ast}(G)$ is at most $|C_{G}(t)|$, and the proof of Corollary
3 is complete.

We remark that if equality holds in Corollary 3, then every irreducible
character of $G$ which does not vanish at $t$ takes value $\pm 1$ at $t$, and
that each such character has odd degree, so lies in a $2$-block of full defect
of $G$ (and has height zero in that block).

##  6\. Further character-theoretic applications

One of our motivations for asking the question answered by Corollary 2 is the
following:

######  Lemma 6.1.

Let $G$ be a finite group, and let $t,u$ be non-trivial involutions of $G$
such that $t$ does not commute with any $G$-conjugate of $u$. Let $C=C_{G}(t)$
and $D=C_{G}(u)$. Then there is a non-trivial ‘irreducible character $\mu$ in
the principal $2$-block of $G$ such that $\mu(t)\mu(u)\neq 0$ and
$\mu(1)\leqslant([C:O(C)]-1)(\sqrt{[D:O(D)]-1})$.

###### Proof.

We first note that $u$ is not expressible as the product of two conjugates of
$t$. More generally, the product of two conjugates of $t$ never lies in the
$2$-section of $u$, for if $t^{x}t^{y}$ has $2$-part $u^{z}$, then $t^{x}$ and
$t^{y}$ both invert the involution $u^{z}$, contrary to hypothesis.

By the usual character-theoretic formula for the coefficient of $g$ in the
product of two class sums, we see that the class function

$\sum_{\chi\in{\rm Irr}(G)}\frac{\chi(t)^{2}\chi}{\chi(1)}$

vanishes identically on the $2$-section of $u$ in $G$.

By a well-known consequence of Brauer’s Second Main Theorem, the class
function

$\sum_{\chi\in B}\frac{\chi(t)^{2}\chi}{\chi(1)}$

also vanishes identically on the $2$-section of $u$ in $G$, where $B$ is the
principal $2$-block of $G$. It follows easily that

$\sum_{1\neq\chi\in B}\frac{\chi(t)^{2}|\chi(u)|}{\chi(1)}\geqslant 1.$

Choose a non-trivial irreducible character $\mu\in B$ with $\mu(t)\mu(u)\neq
0$ and with $r(u)=\frac{|\mu(u)|}{\mu(1)}$ maximal. Then

$\sum_{1\neq\chi\in B}\chi(t)^{2}\geqslant\frac{1}{r(u)}.$

By Brauer’s Second and Third Main Theorem, we have

$\sum_{\chi\in B}\chi(t)^{2}=\sum_{\theta\in b}\theta(1)^{2},$

where $b$ is the principal $2$-block of $C=C_{G}(t)$ . Since each irreducible
character $\theta$ of $b$ has $O(C)$ in its kernel, we deduce that

$\sum_{1\neq\chi\in B}\chi(t)^{2}\leqslant[C:O(C)]-1$

.

A similar argument with $u$ certainly allows us to conclude that
$|\mu(u)|\leqslant\sqrt{[D:O(D)]-1}.$ Hence we may conclude that

$\mu(1)\leqslant([C:O(C)]-1)(\sqrt{[D:O(D)]-1}),$

and the result follows.

∎

In view of Corollary 2, this result may not be applied as it stands to finite
simple groups, but we have seen that the hypotheses may be satisfied within
almost simple groups, so we mention:

######  Corollary 6.2.

Let $G$ be an almost simple group with $F^{\ast}(G)=S$, and let $t$ be an
involution of $G\backslash S$ and $u$ be an involution of $S$ which commutes
with no conjugate of $t$. Then the principal $2$-block of $S$ contains a
$t$-stable non-trivial irreducible character of degree at most
$([C:O(C)]-1)(\sqrt{[D:O(D)]-1}),$ where $C=C_{S}(t)$ and $D=C_{S}(u)$.

###### Proof.

We might as well work within $\langle t\rangle S$, so we suppose that
$G=\langle t\rangle S$. We may argue in a similar fashion to the previous
result. Now the principal $2$-block of $G$ contains two linear characters, $1$
and $\lambda$, say, and we also have $\lambda(t)^{2}\lambda(u)=1.$ This time,
we find that

$\sum_{\chi\in B:\chi(1)>1}\frac{\chi(t)^{2}|\chi(u)|}{\chi(1)}\geqslant 2.$

where $B$ is again the principal $2$-block of $G$. However, the irreducible
characters $\chi\in B$ with $\chi(t)\chi(u)\neq 0$ come in pairs, both members
of which lie over the same $t$-stable irreducible character of the principal
$2$-block of $S$ (if $\chi$ is one such, so is $\lambda\chi\neq\chi$, and note
that $\chi(t)^{2}\chi(u)$ is unchanged on replacing $\chi$ by $\lambda\chi$).
Also, $C_{G}(t)=\langle t\rangle\times C_{S}(t)$, so the result follows in a
fashion similar to the previous Corollary.

∎

## References

  * [1] M. Aschbacher and G. Seitz, Involutions in Chevalley groups over fields of even order, Nagoya Math. J. 63 (1976), 1–91.
  * [2] T. Breuer,

          https://urldefense.com/v3/__http://www.math.rwth-aachen.de/*Thomas.Breuer/ctbllib
      /doc/chap2_mj.html*X7AD5E36B7D44C507__;fiM!!LIr3w8kk_Xxm
      !8LRHz5pIJnyEnPG58yg-O5Ke8LeSLCifFpGruohaZkTEe-om4n5NN-wMHsqCqzk$

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  * [4] D. Gorenstein, R. Lyons and R. Solomon, The classification of the finite simple groups. Number 3. Part I. Chapter A. Almost simple K-groups. Mathematical Surveys and Monographs, 40.3. American Mathematical Society, Providence, RI, 1998.
  * [5] R. Knörr, Virtually irreducible lattices, Proc. London Math. Soc. (3),59, (1989), 99-132.
  * [6] R. Lawther, Jordan block sizes of unipotent elements in exceptional algebraic groups, Comm. Algebra 23 (1995), 4125–4156.
  * [7] M. Liebeck, J. Saxl and G. Seitz, Subgroups of maximal rank in finite exceptional groups of Lie type, Proc. London Math. Soc. (3) 65 (1992), 297–325.
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#  $(n,\sigma)-$equivalence relation between skew constacyclic codes

Hassan Ou-azzou<EMAIL_ADDRESS>Mustapha Najmeddine
<EMAIL_ADDRESS>Nuh Aydin<EMAIL_ADDRESS>Department of Mathematics,
ENSAM–Meknes, Moulay Ismail University, Morocco  Department of Mathematics,
Kenyon College, Gambier, OH 43022, USA

###### Abstract

In this paper we generalize the notion of $n$-equivalence relation introduced
by Chen et al. in [12] to classify constacyclic codes of length $n$ over a
finite field $\mathbb{F}_{{}_{q}},$ where $q=p^{r}$ is a prime power, to the
case of skew constacyclic codes without derivation. We call this relation
$(n,\sigma)$-equivalence relation, where $n$ is the length of the code and
$\sigma$ is an automorphism of the finite field. We compute the number of
$(n,\sigma)$-equivalence classes, and we give conditions on $\lambda$ and
$\mu$ for which $(\sigma,\lambda)$-constacyclic codes and
$(\sigma,\lambda)$-constacyclic codes are equivalent with respect to our
$(n,\sigma)$-equivalence relation. Under some conditions on $n$ and $q$ we
prove that skew constacyclic codes are equivalent to cyclic codes. We also
prove that when $q$ is even and $\sigma$ is the Frobenius autmorphism, skew
constacyclic codes of length $n$ are equivalent to cyclic codes when
$\gcd(n,r)=1$. Finally we give some examples as applications of the theory
developed here.

###### keywords:

Contacyclic codes, cyclic codes, skew constatcyclic codes, skew polynomials,
equivalent codes.

##  1 Introduction

Coding theory plays a fundamental role in various applications, such as error
detection and correction, data transmission, data storage and reliable
communication. It involves the study of efficient encoding and decoding
methods for transmitting data reliably over noisy channels. Cyclic codes are
one of the most important families of linear codes for both theoretical and
practical reasons. They establish a key link between coding theory and
algebra, and their structure makes them convenient for implementation. Many of
the best-known codes are either cyclic or related to cyclic codes. Cyclic
codes were introduced in the 1950s by Prange in [21] as linear codes with the
property that the cyclic shift of any codeword is another codeword. One of the
important generalizations of cyclic codes is the class of constacyclic codes,
first introduced in [8]. Like cyclic codes, constacyclic codes also are also
conveniently implemented using simple shift registers. Their rich algebraic
structure makes them a major subject of study for theoretical reasons. There
are useful generalizations of cyclic codes such as quasi-cyclic (QC) and
quasi-twisted (QT) codes that have been used to obtain a large number of new
linear codes with better parameters e.g., [2, 1, 3, 6, 19]. One of the
generalizations of cyclic codes is skew-cyclic codes, which were introduced in
[9]. Their study necessitates the use of the Ore ring
$R:=\mathbb{F}_{{}_{q}}[x,\sigma]$, where $\sigma$ is an automorphism on
$\mathbb{F}_{{}_{q}}$, and the multiplication is defined by the extension of
the rule $x\cdot a=\sigma(a)x$ for all $a\in\mathbb{F}_{{}_{q}}$.

For a nonzero element $\lambda$ of $\mathbb{F}_{{}_{q}}$,
$\lambda$-constacyclic codes of length $n$ over $\mathbb{F}_{{}_{q}}$ are
characterized as the ideals $\langle g(x)\rangle$ of the quotient ring
$\mathbb{F}_{q}[x]/\langle x^{n}-\lambda\rangle$ where the generator
polynomial $g(x)$ is the unique monic polynomial of minimum degree in the
code, and it is a divisor of $x^{n}-\lambda$. Cyclic codes ($\lambda=1$) are a
particular case of constacyclic codes. Although the quotient space
$R/R\cdot\langle x^{n}-\lambda\rangle$ is not always a ring, hence skew cyclic
codes are not necessarily ideals, they can be viewed as left $R$-submodules
$\langle g(x)\rangle$ of the left $R$-module $R/R\cdot\langle
x^{n}-\lambda\rangle$ where $g(x)$ is a right divisor of $x^{n}-\lambda$.
Other characterizations of skew cyclic and skew constacyclic codes can be
found in [16, 6, 9, 10].

In a previous work [12], Chen et al. introduced an equivalence relation
“$\sim_{n}$” called $n$-equivalence for the nonzero elements of
$\mathbb{F}_{q}$ to classify constacyclic codes of length $n$ over
$\mathbb{F}_{q}$ such that the constacyclic codes belonging to the same
equivalence class have the same minimum distance. For
$\lambda,\mu\in\mathbb{F}_{{}_{q}}^{*}$, $\lambda\sim_{n}\mu$ means the
existence of a nonzero scalar $a\in\mathbb{F}_{{}_{q}}^{*}$ such that the map
$\psi_{a}$ from the ring $\mathbb{F}_{q}[x]/\langle x^{n}-\mu\rangle$ to
$\mathbb{F}_{q}[x]/\langle x^{n}-\lambda\rangle$, defined by
$\psi_{a}(f(x))=f(ax)$ is an isometry with respect to the Hamming distance.
Equivalently, $\lambda\sim_{n}\mu$ if the polynomial $\lambda x^{n}-\mu$ has
at least one root in $\mathbb{F}_{{}_{q}}[x].$ So, it is easy to relate the
generator polynomial of a $\lambda$-constacyclic code $C=\langle g(x)\rangle$
with that of $\psi_{a}(C)=\langle g(ax)\rangle$. Some papers of interest about
the equivalence of constacyclic codes can be found in [4, 5, 11, 13, 15].

In this paper we generalize the notion of $n$-equivalence relation introduced
by Chen et al. in [12] to classify constacyclic codes of length $n$ over
finite field $\mathbb{F}_{{}_{q}}$ to the case of skew constacyclic codes
without derivation. We call this relation $(n,\sigma)$-equivalence relation,
where $n$ is the length of the code and $\sigma$ is an automorphism of the
finite field. We compute the number of $(n,\sigma)$-equivalence classes, and
we give conditions on $\lambda$ and $\mu$ for which
$(\sigma,\lambda)$-constacyclic codes and $(\sigma,\lambda)$-constacyclic
codes are equivalent with respect to the $(n,\sigma)$-equivalence relation.
Under some conditions on $n$ and $q$ we prove that skew constacyclic codes are
equivalent to cyclic codes. We also prove that when $q$ is even and $\sigma$
is the Frobenuis autmorphism, skew constacyclic codes of length $n$ are
equivalent to cyclic codes if $\gcd(n,r)=1$. Finally we give some examples as
applications of the theory developed here.

The rest of this paper is organized as follows: Section 2 provides a review of
the basic background on skew polynomial rings and skew constacyclic codes. In
section 3, we study the properties of the equivalence relation
$(n,\sigma)$-equivalence. Section 4 investigates the equivalence between skew
constacyclic codes over $\mathbb{F}_{2^{r}}$ using the properties of the
$(n,\sigma)$-equivalence relation. Finally, in section 5 we present some
examples as applications of the theory developed here.

##  2 Skew $(\sigma,\lambda)-$constacyclic codes

Let $\mathbb{F}_{{}_{q}}$ be a finite field with $q$ elements, where $q=p^{r}$
for some prime number $p$ and $r\in\mathbb{Z}_{\geq 1}$, and let
$\sigma:\mathbb{F}_{{}_{q}}\rightarrow\mathbb{F}_{{}_{q}}$ be an automorphism
of $\mathbb{F}_{{}_{q}}$. Recall that the Frobenius automorphism of
$\mathbb{F}_{{}_{q}}$ is defined as the automorphism
$\theta:\mathbb{F}_{{}_{q}}\rightarrow\mathbb{F}_{{}_{q}}$ that maps $a$ to
$a^{p}$ and generates the Galois group
$\operatorname{Aut}(\mathbb{F}_{{}_{q}}).$ Then each automorphism
$\sigma\in\operatorname{Aut}(\mathbb{F}_{{}_{q}})$ can be expressed as a power
of $\theta$, which means there exists an integer $s$, $0\leq s<r,$ such that
$\sigma(a)=a^{p^{s}}$ for all $a\in\mathbb{F}_{{}_{q}}.$ It is important to
note also that $\theta$ keeps $\mathbb{F}_{p}$ fixed i.e.,
$\theta(\mathbb{F}_{p})=\mathbb{F}_{p}$, and generates the cyclic group of
automorphisms of $\mathbb{F}_{{}_{q}}$.

######  Definition 1 (Skew $(\sigma,\lambda)$-constacyclic code).

Let $\lambda\in\mathbb{F}_{{}_{q}}^{*}$ and
$\sigma\in\operatorname{Aut}(\mathbb{F}_{{}_{q}})$ be an automorphism of
$\mathbb{F}_{{}_{q}}.$ A linear code $C\subseteq\mathbb{F}_{{}_{q}}^{n}$ is
called a skew $(\sigma,\lambda)$-constacyclic code of length $n$ over
$\mathbb{F}_{{}_{q}}$ if for each codeword $(c_{0},c_{1},\ldots,c_{n-1})$ of
$C,$ we have

$\left(\lambda\sigma(c_{n-1}),\sigma(c_{0}),\sigma(c_{1}),\ldots,\sigma(c_{n-2})\right)\in
C.$

For $\lambda=\pm 1$ we obtain skew $\sigma$-cyclic codes and
$\sigma$-negacyclic codes as special cases.

Let us recall that a map
$T:\mathbb{F}_{{}_{q}}^{{}^{n}}\rightarrow\mathbb{F}_{{}_{q}}^{{}^{n}}$ is
called $\sigma$-semi linear transformation if it is an additive map satisfying
$T(\alpha u)=\sigma(\alpha)T(u)$ for $\alpha\in\mathbb{F}_{{}_{q}}$ and
$u\in\mathbb{F}_{{}_{q}}^{n}.$ In the particular case
$\sigma=\operatorname{Id},\ T$ is a linear transformation. We can obtain some
important classes of codes from semi linear transformations.

######  Remark 1.

Let $\lambda\in\mathbb{F}_{{}_{q}}^{*}$ and
$\sigma\in\operatorname{Aut}(\mathbb{F}_{{}_{q}})$ be an automorphism of
$\mathbb{F}_{{}_{q}}.$

  1. 1.

Skew $(\sigma,\lambda)$-constacyclic codes are invariant under the
$\sigma-$semi linear transformation $T_{\sigma,\lambda},$ called the
$(\sigma,\lambda)-$constacyclic shift, defined by:

$T_{\sigma,\lambda}(v_{0},v_{1},\ldots,v_{n-1})=\left(\lambda\sigma(v_{n-1}),\sigma(v_{0}),\sigma(v_{1}),\ldots,\sigma(v_{n-2})\right),\text{for
all }\ v=(v_{0},v_{1},\ldots,v_{n-1})\in\mathbb{F}_{{}_{q}}^{n}.$

  2. 2.

$\lambda$-constacyclic codes are invariant under the linear transformation
$T_{\lambda}$ defined by:

$T_{\lambda}(v_{0},v_{1},\ldots,v_{n-1})=\left(\lambda
v_{n-1},v_{0},v_{1},\ldots,v_{n-2}\right),\text{for all }\
v=(v_{0},v_{1},\ldots,v_{n-1})\in\mathbb{F}_{{}_{q}}^{n}.$

  3. 3.

When $\sigma=\operatorname{Id},$ we get
$T_{\lambda}=T_{\sigma,\operatorname{Id}}.$

To investigate the structure of skew $(\sigma,\lambda)-$constacyclic codes we
need to recall the notion of skew polynomials. A $\sigma$-skew polynomial ring
over $\mathbb{F}_{{}_{q}}$ is the set

$\mathbb{F}_{{}_{q}}[x,\sigma]=\left\\{a_{0}+a_{1}x+\ldots+a_{n-1}x^{n-1}:a_{i}\in\mathbb{F}_{{}_{q}},n\in\mathbb{Z}_{\geq
0}\right\\}$

endowed with the usual polynomial addition and the multiplication is defined
by

$xa=\sigma(a)x.$

In the following proposition, we collect some properties of the skew
polynomial ring $\mathbb{F}_{{}_{q}}[x,\sigma]$.

######  Proposition 1 ([20], p. 483-486).

Let $s$ be the order of the automorphism $\sigma,$ and denote by
$\operatorname{Fix}_{\sigma}(\mathbb{F}_{{}_{q}})$ the subfield of
$\mathbb{F}_{{}_{q}}$ fixed by $\sigma$.

  1. 1.

The ring $\mathbb{F}_{{}_{q}}[x,\sigma]$ is, in general, a non-commutative
ring unless $\sigma$ is the identity automorphism of $\mathbb{F}_{{}_{q}}.$

  2. 2.

An element $f\in\mathbb{F}_{{}_{q}}[x,\sigma]$ is central if and only if $f$
belongs to $\operatorname{Fix}_{\sigma}(\mathbb{F}_{{}_{q}})[x^{s}]$ i.e., the
centre of $\mathbb{F}_{{}_{q}}[x,\sigma]$ is
$\operatorname{Z}(R)=\operatorname{Fix}_{\sigma}(\mathbb{F}_{{}_{q}})[x^{s}].$

  3. 3.

Two-sided ideals of $\mathbb{F}_{{}_{q}}[x,\sigma]$ are generated by elements
of the form $h(x^{s})x^{m}$, where $m$ is an integer and
$h(x)\in\operatorname{Fix}_{\sigma}(\mathbb{F}_{{}_{q}})[x].$

  4. 4.

The ring $\mathbb{F}_{{}_{q}}[x,\sigma]$ is a right (resp. left) Euclidean
domain.

It is important to note that in the polynomial ring $R$, the evaluation of
skew polynomials is different from the evaluation of polynomials in the usual
polynomial rings over commutative rings.

######  Definition 2 ([18] ).

For $a\in\mathbb{F}_{{}_{q}},$ and $f(x)\in R$, the right evaluation of $f(x)$
at $a$ is the element $f(a):=r$ such that $f(x)=q(x)(x-a)+r.$

In [18, Proposition 2.9], Lam and Leroy proved that, for each
$\alpha\in\mathbb{F}_{{}_{q}}$ we have

$f(\alpha)=a_{0}+a_{1}N_{1}(\alpha)+\ldots+a_{n-1}N_{n-1}(\alpha)=\displaystyle{\sum_{i=0}^{n}}a_{i}N_{i}(\alpha)$
(1)

where $N_{0}(\alpha)=1,$ and for each $i$ in $\mathbb{N}^{*}$

$N_{i}(\alpha)=\sigma^{i-1}(\alpha)\ldots\sigma(\alpha)\alpha.$

By taking $\sigma(a)=a^{p^{s}}$ for all $a\in\mathbb{F}_{{}_{q}},$ we obtain
for each $i\in\mathbb{N}^{*}$ that

$N_{i}(\alpha)=\alpha^{p^{s(i-1)}}\alpha^{p^{s(i-2)}}\ldots\alpha^{p^{s}}\alpha=\alpha^{\sum_{j=0}^{i-1}p^{sj}}=\alpha^{{\frac{p^{si}-1}{p^{s}-1}}}.$
(2)

Set $[i]_{s}=\dfrac{p^{si}-1}{p^{s}-1}$. Then the right evaluation of
$f(x)=\displaystyle{\sum_{i=0}^{n}}a_{i}x^{i}$ at
$\alpha\in\mathbb{F}_{{}_{q}}$ is given by

$f(\alpha)=\displaystyle{\sum_{i=0}^{n}}a_{i}\alpha^{[i]_{s}}.$ (3)

Now we recall the following characterization of skew
$(\sigma,\lambda)-$constacyclic codes.

######  Theorem 1 ( Theorem 3, Theorem 4 in [6]).

A linear code $C\subseteq\mathbb{F}_{{}_{q}}^{{}^{n}}$ is a skew
$(\sigma,\lambda)-$constacyclic code if and only if it is a left ideal of
$\mathbb{F}_{{}_{q}}[x,\sigma]/\langle x^{n}-\lambda\rangle.$ Moreover,

  1. 1.

There is a monic polynomial of least degree
$g(x)\in\mathbb{F}_{{}_{q}}[x,\sigma]$ such that $g(x)$ right divides
$x^{n}-\lambda$ and $C=\langle g(x)\rangle.$

  2. 2.

The set $\\{g(x),xg(x),\ldots,x^{k-1}g(x)\\}$ forms a basis of $C$ and the
dimension of $C$ is $k=n-\deg(g).$

  3. 3.

A generator matrix $G$ of $C$ is given by:

$G=\left(\begin{array}[]{cccccccc}g_{{}_{0}}&g_{{}_{1}}&\cdots&g_{{}_{n-k}}&0&\cdots&\cdots&0\\\
0&\sigma(g_{{}_{0}})&\sigma(g_{{}_{1}})&\cdots&\sigma(g_{{}_{n-k}})&0&\cdots&0\\\
\vdots&\ddots&\ddots&\ddots&&\ddots&&\vdots\\\
\vdots&&\ddots&\ddots&\ddots&&\ddots&\vdots\\\
0&\ldots&&0&\sigma^{k-1}(g_{{}_{0}})&\sigma^{k-1}(g_{{}_{1}})&\ldots&\sigma^{k-1}(g_{{}_{n-k}})\\\
\end{array}\right)$ (4)

where $k=n-\deg(g)$ and $g(x)=\displaystyle\sum_{i=0}^{n-k}g_{i}x^{i}.$

  4. 4.

There is a one-to-one correspondence between right divisors of $x^{n}-\lambda$
and skew $(\sigma,\lambda)-$constacyclic codes of length $n$ over
$\mathbb{F}_{{}_{q}}$.

######  Definition 3 ( Section 1.6, and 1.7 in [17]).

Let $C_{1}$ and $C_{2}$ be two linear codes of length $n$ over
$\mathbb{F}_{{}_{q}}$. Then $C_{1}$ and $C_{2}$ are called  equivalent if
there exists an $n\times n$ monomial matrix matrix $P$ such that
$C_{1}=C_{2}P$.

######  Remark 2.

Two equivalent codes have the same minimum Hamming distance and in general
same parameters.

In the following theorem we show that in some cases skew
$(\sigma,\lambda)-$constacyclic codes are the same as ordinary
$\lambda$-constacyclic codes.

######  Theorem 2.

Let $\sigma$ be an automorphism of $\mathbb{F}_{{}_{q}}$ of order $m$ and $C$
be a skew $(\sigma,\lambda)$-constacyclic code of length $n$ over
$\mathbb{F}_{{}_{q}}$. If $\gcd(m,n)=1$ and
$\lambda\in\operatorname{Fix}_{\sigma}(\mathbb{F}_{{}_{q}})$ then $C$ is a
$\lambda$-constacyclic code of length $n$.

######  Proof.

Let $C=\langle g(x)\rangle$ be a skew $(\sigma,\lambda)$-constacyclic code of
length $n$ such that $\gcd(m,n)=1$. As shown in [23, Theorem 16], we know that
there exist integers $\alpha_{1},\alpha_{2}$ such that
$\alpha_{1}m=1+\alpha_{2}n$, where $\alpha_{2}>0$.

Let $c(x)=c_{0}+c_{1}x+\ldots+c_{n-1}x^{n-1}\in C$. We need to show that

$\lambda c_{n-1}+c_{0}x+\ldots+c_{n-2}x^{n-1}\in C.$

Since $C$ is a skew $(\sigma,\lambda)$-constacyclic code,
$x^{\alpha_{1}m}c(x)\in C$ and we have

$\displaystyle x^{\alpha_{1}m}c(x)$
$\displaystyle=x^{1+\alpha_{2}n}(c_{0}+c_{1}x+\ldots+c_{n-1}x^{n-1})$
$\displaystyle=\sigma^{1+\alpha_{2}n}(c_{0})x^{1+\alpha_{2}n}+\sigma^{1+\alpha_{2}n}(c_{1})x^{2+\alpha_{2}n}+...+\sigma^{1+\alpha_{2}n}(c_{n-1})x^{n+\alpha_{2}n}$
$\displaystyle=\sigma^{\alpha_{1}m}(c_{0})x\lambda^{\alpha_{2}}+\sigma^{m\alpha_{1}}(c_{1})x^{2}\lambda^{\alpha_{2}}+...+\sigma^{\alpha_{1}m}(c_{n-1})\lambda^{1+\alpha_{2}}$
$\displaystyle=(\lambda
c_{n-1}+c_{0}x+c_{1}x^{2}+...+c_{n-2}x^{n-1})\lambda^{\alpha_{2}}$
$\displaystyle=\lambda^{\alpha_{2}}(\lambda
c_{n-1}+c_{0}x+c_{1}x^{2}+...+c_{n-2}x^{n-1})\in C\ \ \text{since}\ \
\lambda\in\operatorname{Fix}_{\sigma}(\mathbb{F}_{{}_{q}}).$

Therefore, $\lambda c_{n-1}+c_{0}x+c_{1}x^{2}+...+c_{n-2}x^{n-1}\in C,$ and
hence $C$ is a $\lambda$-constacyclic code.

######  Theorem 3.

Let $\sigma$ be an automorphism of $\mathbb{F}_{{}_{q}}$ of order $m$ and $C$
be a skew $(\sigma,\lambda)$-constacyclic code of length $n$ over
$\mathbb{F}_{{}_{q}}$. If $\gcd(m,n)=d$ and
$\lambda\in\operatorname{Fix}_{\sigma}(\mathbb{F}_{{}_{q}})$ then $C$ is a
quasi-twisted code of length $n$ and index $d$ with shift constant $\lambda$.

######  Proof.

Let $C=\langle g(x)\rangle$ be a skew $(\sigma,\lambda)-$constacyclic code of
length $n$ over $\mathbb{F}_{{}_{q}}$ such that $\gcd(m,n)=d$. We know that
there exist integers $\alpha_{1},\alpha_{2}$ such that
$\alpha_{1}m=d+\alpha_{2}n$, where $\alpha_{2}>0$.

Let $n=ds$ and

$c=(c_{{}_{0,0}},c_{{}_{0,1}},\ldots,c_{{}_{0,d-1}},c_{{}_{1,0}},c_{{}_{1,1}},\ldots,c_{{}_{1,d-1}},\ldots,c_{{}_{s-1,0}},c_{{}_{s-1,1}},\ldots,c_{{}_{s-1,d-1}})\in
C.$

We will show that

$T_{\lambda}^{d}(c)=(\lambda c_{{}_{s-1,0}},\lambda
c_{{}_{s-1,1}},\ldots,\lambda
c_{{}_{s-1,d-1}},c_{{}_{0,0}},c_{{}_{0,1}},\ldots,c_{{}_{1,d-1}},\ldots,c_{{}_{s-2,0}},c_{{}_{s-1,1}},\ldots,c_{{}_{s-2,d-1}})$

where $T_{\lambda}$ is the $\lambda$-constacyclic shift defined in Remark 1.
Let $T_{\lambda,\sigma}$ be the $\sigma$-semi-linear transformation defined by

$T_{\lambda,\sigma}(c_{0},c_{1},\ldots,c_{n-1})=(\lambda\sigma(c_{n-1}),\sigma(c_{0}),\ldots,\sigma(c_{n-2}))=T_{\lambda}(\sigma(c_{n-1}),\sigma(c_{0}),\ldots,\sigma(c_{n-2})).$

We know that $C$ is a skew $\lambda$-constacyclic code if
$T_{\lambda,\sigma}(c)\in C$ for each $c\in C$, and for each positive integer
$k$ we have $T_{\lambda,\sigma}^{k}(c)=T_{\lambda}^{k}(\sigma^{k}(c)).$ It
follows that

$T_{\lambda,\sigma}^{d+\alpha_{2}n}(c)=T_{\lambda}^{d+\alpha_{2}n}(\sigma^{d+\alpha_{2}n}(c))=T_{\lambda}^{d+\alpha_{2}n}(\sigma^{\alpha_{1}m}(c))=T_{\lambda}^{d+\alpha_{2}n}(c)=\lambda^{\alpha_{2}}T_{\lambda}^{d}(c)\in
C.$ (5)

Hence, $T_{\lambda}^{d}(c)\in C$ and so we obtain the desired result.

##  3  Equivalence between skew constacyclic codes

In [12] Chen et al. introduced the notion of $n$-equivalence relation on
$\mathbb{F}_{{}_{q}}^{*}$ for the purpose of classifying ordinary constacyclic
codes of length $n$ over $\mathbb{F}_{{}_{q}}.$ In this section we generalize
this notion to the case of skew constacyclic codes by defining an equivalence
relation on $\mathbb{F}_{{}_{q}}^{*}$ called $(n,\sigma)$-equivalence
relation, where $\sigma\in\operatorname{Aut}(\mathbb{F}_{{}_{q}})$ is an
automorphism of $\mathbb{F}_{{}_{q}}.$

######  Definition 4 ($(\sigma,n)-$equivalence relation).

Let $n$ be a positive integer. We say that $\lambda$ and $\mu$ are
$(n,\sigma)$-equivalent in $\mathbb{F}_{q}^{*}$, denoted by
$\lambda\sim_{n,\sigma}\mu$, if the skew polynomial $\lambda
x^{n}-\mu\in\mathbb{F}_{{}_{q}}[x,\sigma]$ has at least one root in
$\mathbb{F}_{{}_{q}}^{*}$, i.e., there is $\alpha\in\mathbb{F}_{{}_{q}}^{*}$
such that $\lambda N_{n}(\alpha)=\mu.$

######  Proposition 2.

The relation $\sim_{n,\sigma}$ is an equivalence relation on
$\mathbb{F}_{{}_{q}}^{*}$.

######  Proof.

  1. 1.

$\lambda\sim_{n,\sigma}\lambda$ since $1$ is a root of $\lambda x^{n}-\lambda$
in $\mathbb{F}_{{}_{q}}$.

  2. 2.

$\lambda\sim_{n,\sigma}\mu\iff$ there exists $\alpha\in\mathbb{F}_{{}_{q}}$
such that $\alpha$ is a root of the polynomial $f(x)=\lambda x^{n}-\mu$. By
equation (1), this is equivalent to

$\lambda N_{n}(\alpha)=\mu\iff\lambda=\mu
N_{n}(\alpha^{-1})\iff\mu\sim_{n,\sigma}\lambda.$

  3. 3.

If $\lambda\sim_{n,\sigma}\mu$ and $\mu\sim_{n,\sigma}\xi$ then there are
$\alpha,\beta$ in $\mathbb{F}_{{}_{q}}^{*}$ such that

$\lambda N_{n}(\alpha)=\mu\ \ \text{and}\ \ \mu N_{n}(\beta)=\xi$

Therefore

$\lambda N_{n}(\alpha)=N_{n}(\beta^{-1})\xi$

and so

$\lambda N_{n}(\alpha)N_{n}(\beta)=\lambda N_{n}(\alpha\beta)=\xi.$

Hence $\lambda\sim_{n,\sigma}\xi.$

The following theorem gives characterizations of $(n,\sigma)$-equivalence
between two elements $\lambda$ and $\mu$ of $\mathbb{F}_{{}_{q}}^{*}.$

######  Theorem 4.

Let $\sigma\in\operatorname{Aut}(\mathbb{F}_{{}_{q}})$ be an automorphism of
$\mathbb{F}_{{}_{q}}$ such that $\sigma(a)=a^{p^{s}}$ for all
$a\in\mathbb{F}_{{}_{q}}$, and let $\lambda,\mu\in\mathbb{F}_{q}^{*}$. The
following four statements are equivalent:

  1. 1.

$\lambda\sim_{n,\sigma}\mu$

  2. 2.

There is an element $\alpha$ in $\mathbb{F}_{q}^{*}$ such that $\lambda
N_{n}(\alpha)=\mu$ and the following map

$\begin{array}[]{cccc}\varphi_{\alpha}:&\mathbb{F}_{q}[x,\sigma]/\langle
x^{n}-\mu\rangle&\longrightarrow&\mathbb{F}_{q}[x,\sigma]/\langle
x^{n}-\lambda\rangle,\\\ &&&\\\ &f(x)&\longmapsto&f(\alpha x)\end{array}$

is an $\mathbb{F}_{q}$-algebra isometry with respect to the Hamming distance.

  3. 3.

$\lambda^{-1}\mu\in\langle N_{n}(\xi)\rangle$, where $\xi$ is a primitive
element of $\mathbb{F}_{{}_{q}}.$

  4. 4.

$\lambda^{-1}\mu\in\langle\xi^{[n]_{s}}\rangle$, where $\xi$ is a primitive
element of $\mathbb{F}_{{}_{q}}.$

  5. 5.

$\left(\lambda^{-1}\mu\right)^{d}=1$, where
$d=\dfrac{q-1}{\operatorname{gcd}([n]_{s},q-1)}$.

In particular, the number of $(n,\sigma)$-equivalence classes in
$\mathbb{F}_{q}^{*}$ is equal to $\operatorname{gcd}([n]_{s},q-1)$.

######  Proof.

  1. 1.

(1) $\Rightarrow$ (2). Suppose that $\lambda\sim_{n,\sigma}\mu$ then there is
$\alpha\in\mathbb{F}_{{}_{q}}$ such that $\lambda N_{n}(\alpha)=\mu$. We will
prove that $\varphi_{\alpha}$ is an $\mathbb{F}_{{}_{q}}$-algebra isometry.
Let us first show that $\varphi_{\alpha}$ is an $\mathbb{F}_{{}_{q}}$-algebra
isomorphism. Let $f(x)=\displaystyle{\sum_{i=0}^{n-1}}a_{i}x^{i}$ and
$g(x)=\displaystyle{\sum_{i=0}^{n-1}}b_{i}x^{i}$ in
$\mathbb{F}_{{}_{q}}[x,\sigma]/\langle x^{n}-\mu\rangle.$ On the one hand we
have

$\begin{array}[]{rl}f(x)g(x)&=\displaystyle{\sum_{j=0}^{n-1}}\displaystyle{\sum_{i=0}^{j}}a_{i}\sigma^{i}\left(b_{j-i}\right)x^{j}+\displaystyle{\sum_{j=0}^{n-1}}\displaystyle{\sum_{i=j+1}^{n-1}}a_{i}\sigma^{i}\left(b_{n-i+j}\right)x^{j+n}\\\
&=\displaystyle{\sum_{j=0}^{n-1}}\left(\displaystyle{\sum_{i=0}^{j}}a_{i}\sigma^{i}\left(b_{j-i}\right)+\displaystyle{\sum_{i=j+1}^{n-1}}a_{i}\sigma^{i}(b_{n-i+j})\sigma^{j}(\mu)\right)x^{j},\\\
&(\text{ since $x^{j+n}=\sigma^{j}(\mu)x^{j}(\mod x^{n}-\mu).$ })\par\\\
\end{array}$

As $\varphi_{\alpha}\left(x^{j}\right)=N_{j}(\alpha)x^{j}$, we obtain

$\begin{array}[]{rl}\varphi_{\alpha}(f(x)g(x))&=\displaystyle{\sum_{j=0}^{n-1}}\left(\displaystyle{\sum_{i=0}^{j}}a_{i}\sigma^{i}\left(b_{j-i}\right)+\displaystyle{\sum_{i=j+1}^{n-1}}a_{i}\sigma^{i}(b_{n-i+j})\sigma^{j}(\mu)\right)N_{j}(\alpha)x^{j}\\\
&\\\ \end{array}$

On the other hand we have:

$\begin{array}[]{rl}\varphi_{\alpha}(f(x))\varphi_{\alpha}(g(x))=&\displaystyle{\sum_{j=0}^{n-1}}\left(\displaystyle{\sum_{i=0}^{j}}a_{i}N_{i}(\alpha)\sigma^{i}\left(b_{j-i}N_{j-i}(\alpha)\right)\right)x^{j}+\\\
&\displaystyle{\sum_{j=0}^{n-1}}\left(\displaystyle{\sum_{i=j+1}^{n-1}}a_{i}N_{i}(\alpha)\sigma^{i}\left(b_{n-i+j}N_{n+j-i}(\alpha)\right)\right)x^{j+n}.\\\
\end{array}$

As $x^{j+n}=\sigma^{j}(\lambda)x^{j}(\mod x^{n}-\lambda)$, we have

$\begin{array}[]{rl}\varphi_{\alpha}(f(x))\varphi_{\alpha}(g(x))&=\displaystyle{\sum_{j=0}^{n-1}}\left(\displaystyle{\sum_{i=0}^{j}}a_{i}\sigma^{i}\left(b_{j-i}\right)N_{i}(\alpha)\sigma^{i}\left(N_{j-i}(\alpha)\right)\right)x^{j}+\\\
&\displaystyle{\sum_{j=0}^{n-1}}\left(\displaystyle{\sum_{i=j+1}^{n-1}}a_{i}\sigma^{i}\left(b_{n-i+j}\right)N_{i}(\alpha)\sigma^{i}\left(N_{n+j-i}(\alpha)\right)\sigma^{j}(\lambda)\right)x^{j}.\\\
\end{array}$

According to [14, Proposition 2.1] we have
$N_{i}(\alpha)\sigma^{i}\left(N_{j-i}(\alpha)\right)=N_{j}(\alpha),$ and so

$N_{i}(\alpha)\sigma^{i}\left(N_{n+j-i}(\alpha)\right)\sigma^{j}(\lambda)=N_{j+n}(\alpha)\frac{\sigma^{j}(\mu)}{\left(\sigma^{j}\left(N_{n}(\alpha)\right)\right)}=\sigma^{j}(\mu)N_{j}(\alpha),\
\text{since }\lambda N_{n}(\alpha)=\mu.$

It follows that

$\begin{array}[]{rl}\varphi_{\alpha}(f(x))\varphi_{\alpha}(g(x))&=\displaystyle{\sum_{j=0}^{n-1}}\left(\displaystyle{\sum_{i=0}^{j}}a_{i}\sigma^{i}\left(b_{j-i}\right)+\displaystyle{\sum_{i=j+1}^{n-1}}a_{i}\sigma^{i}\left(b_{n-i+j}\right)\sigma^{j}(\mu)\right)N_{j}(\alpha)x^{j}\\\
&=\varphi_{\alpha}(f(x)g(x)),\end{array}$

which means that $\varphi_{\alpha}$ is a morphism of an
$\mathbb{F}_{{}_{q}}$-algebra.

Finally we observe that for any $f(x)\in\mathbb{F}_{q}[x,\sigma]/\langle
x^{n}-\mu\rangle$, $f(x)$ and $f(\alpha x)$ have the same weight. Therefore,
$\varphi_{\alpha}$ preserves the Hamming weight and so it is an
$\mathbb{F}_{{}_{q}}$-algebra isometry with respect to Hamming distance.

  2. 2.

(2) $\Rightarrow$ (3). Suppose there exists an $\alpha\in\mathbb{F}_{q}^{*}$
such that the map

$\varphi:\mathbb{F}_{q}[x,\sigma]/\langle
x^{n}-\mu\rangle\to\mathbb{F}_{q}[x,\sigma]/\langle x^{n}-\lambda\rangle,\ \
f(x)\mapsto f(\alpha x)$

is an $\mathbb{F}_{q}$-algebra isometry with respect to the Hamming distance.
From the fact that $\mu=\varphi_{\alpha}(\mu)$ and
$x^{n}=\lambda\mod(x^{n}-\lambda)$ we obtain

$\mu=\varphi_{\alpha}(\mu)=\varphi_{\alpha}\left(x^{n}\right)=\varphi_{\alpha}(x)^{n}=(\alpha
x)^{n}=N_{n}(\alpha)x^{n}=N_{n}(\alpha)\lambda.$

This gives $\lambda^{-1}\mu=N_{n}(\alpha).$ Let $\xi$ be a primitive element
of $\mathbb{F}_{{}_{q}}$ then $\alpha=\xi^{i}$ for some $0\leq i\leq q-1.$ It
follows that

$\lambda^{-1}\mu=N_{n}(\xi^{i})=N_{n}(\xi)^{i},\ \text{ which means that }\
\lambda^{-1}\mu\in\langle N_{n}(\xi)\rangle.$

  3. 3.

(3) $\Rightarrow$ (4). Since $N_{n}(\xi)=\xi^{[n]_{s}}$ we have
$\langle\xi^{[n]_{s}}\rangle=\langle N_{n}(\xi)\rangle$ and so
$\lambda^{-1}\mu\in\langle\xi^{[n]_{s}}\rangle.$

  4. 4.

(4) $\Rightarrow$ (5). Since $\mathbb{F}_{q}^{*}=\langle\xi\rangle$ is a
cyclic group, $\langle\xi^{[n]_{s}}\rangle$ is the unique subgroup of order
$d=\dfrac{q-1}{\operatorname{gcd}([n]_{s},q-1)}$ and any subgroup with order
dividing $d$ is contained in the subgroup $\langle\xi^{[n]_{s}}\rangle$. It
follows that $(\lambda\mu^{-1})^{d}=1.$

  5. 5.

(5) $\Rightarrow$ (1). Let $\lambda$ and $\mu$ in $\mathbb{F}_{q}^{*}$ such
that $(\lambda\mu^{-1})^{d}=1.$ That means $\lambda\mu^{-1}$ belongs to the
cyclic subgroup of $\mathbb{F}_{{}_{q}}^{*}$ of order $d.$ However, the unique
subgroup of $\mathbb{F}_{{}_{q}}^{*}$ of order $d$ is
$\langle\xi^{[n]_{s}}\rangle$. It follows that
$\lambda^{-1}\mu=\xi^{i[n]_{s}}\in\langle\xi^{[n]_{s}}\rangle,$ for some
integer $0\leq i\leq q-1.$ Let $\alpha=\xi^{i},$ then
$\lambda^{-1}\mu=\alpha^{[n]_{s}}=N_{n}(\alpha),$ and so $\lambda
N_{n}(\alpha)=\mu$ which means that $\lambda\sim_{n,\sigma}\mu.$

Finally, by the equivalence of (1) and (5), the number of
$(n,\sigma)$-equivalence classes in $\mathbb{F}_{q}^{*}$ is equal to the index
of $\langle\xi^{[n]_{s}}\rangle$ in $\mathbb{F}_{q}^{*}$, which is given by

$\left|\mathbb{F}_{q}^{*}\ :\
\langle\xi^{[n]_{s}}\rangle\right|=\frac{q-1}{\left|\langle\xi^{[n]_{s}}\rangle\right|}=\dfrac{q-1}{\frac{q-1}{\operatorname{gcd}([n]_{s},q-1)}}=\operatorname{gcd}([n]_{s},q-1).$

######  Corollary 1.

Let $\lambda,\mu\in\mathbb{F}_{{}_{q}}^{*}$ be such that
$\lambda\sim_{n,\sigma}\mu$ i.e., $\lambda N_{n}(\alpha)=\mu$ for some
$\alpha\in\mathbb{F}_{{}_{q}}^{*}.$ Then the skew
$(\sigma,\lambda)$-constacyclic codes of length $n$ over $\mathbb{F}_{{}_{q}}$
are equivalent to skew $(\sigma,\mu)$-constacyclic codes of length $n$ over
$\mathbb{F}_{{}_{q}}$ via the following $\mathbb{F}_{{}_{q}}$-algebra
isomorphism $\varphi_{\alpha},$ which preserves the Hamming weight.

$\begin{array}[]{cccc}\varphi_{\alpha}:&\mathbb{F}_{q}[x,\sigma]/\langle
x^{n}-\mu\rangle&\longrightarrow&\mathbb{F}_{q}[x,\sigma]/\langle
x^{n}-\lambda\rangle,\\\ &&&\\\ &f(x)&\longmapsto&f(\alpha x)\end{array}$ (6)

######  Remark 3 ( Theorem 1, [7]).

The result of the above corollary generalizes the result [7, Theorem 1], where
the authors proved that the skew $(\sigma,\lambda)$-constacyclic codes are
equivalent to skew cyclic codes if there is $\alpha\in\mathbb{F}_{{}_{q}}^{*}$
such that $\lambda=N_{n}(\alpha^{-1})$, and to skew negacyclic codes if
$\lambda=-N_{n}(\alpha^{-1})$. This corresponds to the special case $\mu=1$ (
or $\mu=-1$ ) and $\lambda\sim_{n,\sigma}1$ ( or $\lambda\sim_{n,\sigma}-1$)
in our result.

######  Corollary 2.

Let $\sigma\in\operatorname{Aut}(\mathbb{F}_{{}_{q}})$ be an automorphism of
$\mathbb{F}_{{}_{q}}$ such that $\sigma(a)=a^{p^{s}}$ for all
$a\in\mathbb{F}_{{}_{q}}$, and let $n$ be a positive integer such that
$\gcd([n]_{s},q-1)=1$. Then any two nonzero elements $\lambda$ and $\mu$ of
$\mathbb{F}_{q}^{*}$ are $(n,\sigma)$-equivalent to each other, i.e., $\lambda
N_{n}(\alpha)=\mu$ for an $\alpha\in\mathbb{F}_{q}^{*}$, and the map
$\varphi_{{}_{\alpha}}:\mathbb{F}_{q}[x,\sigma]/\langle
x^{n}-\mu\rangle\rightarrow\mathbb{F}_{q}[x,\sigma]/\langle
x^{n}-\lambda\rangle$ which maps $f(x)$ to $f(\alpha x)$ is an
$\mathbb{F}_{q}$-algebra isometry with respect to the Hamming distance.

######  Proof.

Since $\operatorname{gcd}([n]_{s},q-1)=1$, there is only one
$(n,\sigma)$-equivalence class in $\mathbb{F}_{q}^{*}.$ In other words, for
any $\lambda,\mu\in\mathbb{F}_{q}^{*}$, we have $\lambda\sim_{n,\sigma}\mu$.

The following result was proved in [7] by using a basic result from field
theory (see [22]) which says that the equation $x^{n}=\xi^{i}$ has a solution
in $\mathbb{F}_{{}_{q}}^{*}$ if and only if $\gcd(n,q-1)$ divides $i.$ In our
case we obtain a similar result by using the fact that we have only one
$(n,\sigma)$-equivalence class in $\mathbb{F}_{q}^{*}$ when
$\gcd([n]_{s},q-1)=1.$

######  Corollary 3 (Proposition 1, [7]).

Let $\sigma\in\operatorname{Aut}(\mathbb{F}_{{}_{q}})$ be an automorphism of
$\mathbb{F}_{{}_{q}}$ such that $\sigma(a)=a^{p^{s}}$ for all
$a\in\mathbb{F}_{{}_{q}},$ and let $n$ be a positive integer such that
$\gcd([n]_{s},q-1)=1$. Then all skew $(\sigma,\lambda)$-constacyclic codes are
equivalent to skew cyclic codes.

######  Proposition 3.

Let $\sigma\in\operatorname{Aut}(\mathbb{F}_{{}_{q}})$ be an automorphism of
$\mathbb{F}_{{}_{q}}$ such that $\sigma(a)=a^{p^{s}}$ for all
$a\in\mathbb{F}_{{}_{q}},$ and let $n$ be a positive integer
$d=\gcd([n]_{s},q-1).$ For each $\lambda\in\mathbb{F}_{{}_{q}}^{*}$ such that
$\lambda^{d}=1,$ all skew $(\sigma,\lambda)$-constacyclic codes are equivalent
to skew cyclic codes.

######  Proof.

In the statement (5) of Theorem 4, we take $\mu=1.$ By the fact that
$\lambda^{d}=(\lambda^{-1})^{d}=1,$ the result holds.

##  4 Skew constacyclic codes over $\mathbb{F}_{2^{r}}$

Let $q=2^{r}$ and $\sigma\in\operatorname{Aut}(\mathbb{F}_{{}_{q}})$ be the
Frobenius automorphism of $\mathbb{F}_{q},$ i.e., $\sigma(a)=a^{2}$ for all
$a\in\mathbb{F}_{q}.$ Note that the order of $\sigma$ is $r.$ In the following
theorem we investigate equivalence among $(\sigma,\delta)$-constacyclic codes
of length $n$ over $\mathbb{F}_{q}.$

######  Theorem 5.

Let $q=2^{r}$ and $\sigma\in\operatorname{Aut}(\mathbb{F}_{{}_{q}})$ be the
Frobenuis automorphism of $\mathbb{F}_{{}_{q}}.$ Let $C$ be a
$(\sigma,\lambda)$-constacyclic code of length $n$ over $\mathbb{F}_{{}_{q}}.$
The number of $(n,\sigma)$-equivalence classes is equal $2^{\gcd(n,r)}-1$, and

  1. 1.

If $\gcd(n,r)=1,$ then $C$ is equivalent to a cyclic code of length $n$ over
$\mathbb{F}_{{}_{q}}.$

  2. 2.

If $\gcd(n,r)=d\neq 1$, then there is a unique $0\leq j\leq 2^{d}-2$ such that
$C$ is equivalent to a skew $(\sigma,\xi^{j})$-constacyclic code, where $\xi$
is a primitive element of $\mathbb{F}_{{}_{q}}.$

######  Proof.

We have $\gcd([n]_{1},q-1)=\gcd(2^{n}-1,2^{r}-1)=2^{\gcd(n,r)}-1.$ Then
according to Theorem 4, the number of $(n,\sigma)$-equivalence classes is
$2^{\gcd(n,r)}-1.$

  1. 1.

If $n=1\mod r$, then $\gcd([n]_{1},q-1)=1$ and so we have only one equivalence
class which means that all $(\sigma,\lambda)$-constacylic codes of length $n$
over $\mathbb{F}_{{}_{q}}$ are equivalent to skew cyclic codes of length $n$
over $\mathbb{F}_{{}_{q}}.$ On the other hand the order of $\sigma$ is $r$ and
$\gcd(n,2k)=1,$ then by Theorem 2, $C$ is equivalent to a cyclic code of
length $n$ over $\mathbb{F}_{{}_{q}}.$

  2. 2.

If $\gcd(n,r)=d\neq 1,$ and $\gcd([n]_{1},q-1)=2^{d}-1,$ we can decompose
$\mathbb{F}_{{}_{q}}^{*}$ into a disjoint union of cosets of the subgroup
$\langle\xi^{2^{d}-1}\rangle$ as follows:

$\mathbb{F}_{{}_{q}}^{*}=\langle\xi\rangle=\langle
N_{n}(\xi)\rangle\cup\xi\langle
N_{n}(\xi)\rangle\cup\ldots\cup\xi^{2^{d}-2}\langle N_{n}(\xi)\rangle.$

So the element $\lambda\in\mathbb{F}_{{}_{q}}^{*}$ belongs to exactly one of
the cosets, i.e., there is a unique integer $0\leq j\leq 2^{d}-2,$ such that
$\lambda\in\xi^{j}\langle N_{n}(\xi)\rangle$ and so $\lambda$ is
$(\sigma,n)$-equivalent to $\xi^{j}$.

When $q=2^{r}$ and $r$ is an even number we have the following result.

######  Corollary 4.

Let $q=2^{2k}$ and let $\sigma$ be the Frobenuis automorphism of
$\mathbb{F}_{{}_{q}}.$ Then all $(\sigma,\lambda)$-constacyclic codes of odd
length $n$ are equivalent to cyclic codes of length $n$ over
$\mathbb{F}_{{}_{q}}.$

######  Proof.

Let $C$ be a $(\sigma,\lambda)$-constacyclic code of odd length $n$ over
$\mathbb{F}_{{}_{q}}.$ On the one hand we have
$\gcd([n]_{1},2k)=2^{\gcd(n,2k)}-1=2^{1}-1=1,$ hence $C$ is equivalent to a
skew cyclic code of length $n$ over $\mathbb{F}_{{}_{q}}.$ On the other hand
the order of $\sigma$ is $2k$ and $\gcd(n,2k)=1$. Then by Theorem 2, $C$ is
equivalent to a cyclic code of length $n$ over $\mathbb{F}_{{}_{q}}.$

######  Corollary 5.

All skew $(\sigma,\lambda)$-constacylic codes of odd length $n$ over
$\mathbb{F}_{4}$ are equivalent to cyclic codes.

##  5 Examples

In this section, we give some examples of applications of the theory developed
here.

######  Example 1 (Skew constacyclic codes over $\mathbb{F}_{4}$).

Let $q=4$, and $\xi$ be a primitive element of $\mathbb{F}_{4}$. In this
example, we are interested in skew constacyclic codes of length $n$ over
$\mathbb{F}_{4}$. Since the degree of the extension field
$\mathbb{F}_{4}/\mathbb{F}_{2}$ is $2$, we have
$\text{Aut}(\mathbb{F}_{4})=\\{\text{id},\theta\\}$, where $\theta$ is the
Frobenius automorphism of $\mathbb{F}_{4}$, i.e., $\theta(a)=a^{2}$ for all
$a\in\mathbb{F}_{4}$. According to Theorem 4, the number of
$(n,\theta)$-equivalence classes is equal to
$\gcd([n]_{1},q-1)=\gcd(2^{n}-1,3)$. By Theorem 5, we have two cases:

  1. 1.

If $n$ is odd, we have $\gcd(2^{n}-1,2^{2}-1)=2^{\gcd(n,2)}-1=2-1=1$, and so
there is only one $(n,\theta)$-equivalence class. Since $n$ is coprime with
the order of $\theta$, all skew $(\theta,\lambda)$-constacyclic codes of odd
length over $\mathbb{F}_{4}$ are $(n,\theta)$-equivalent to cyclic codes. This
means that there exists an element $\alpha$ in $\mathbb{F}_{4}^{*}$ such that
$\alpha^{[n]_{1}}\lambda=1$. Studying skew $(\lambda,\theta)$-constacyclic
codes, in this case, is reduced to studying cyclic codes of length $n$ over
$\mathbb{F}_{4}$.

  2. 2.

If $n$ is even, we have $\gcd(2^{n}-1,2^{2}-1)=2^{\gcd(n,2)}-1=3$, then we
have $3$ $(n,\theta)$-equivalence classes. Let $\lambda\in\mathbb{F}_{4}^{*}$,
then by Theorem 5, $\lambda$ is $(n,\theta)$-equivalent to $1,\xi,$ or
$\xi^{2}$. To construct skew $(\lambda,\theta)$-constacyclic codes, in this
case, we need to study three classes, which are skew $\theta$-cyclic codes,
skew $(\xi,\theta)$-constacyclic, and $(\xi^{2},\theta)$-constacyclic codes.

######  Example 2 (Skew constacyclic codes over $\mathbb{F}_{8}$).

Let $q=8$, and $\xi$ be a primitive element of $\mathbb{F}_{8}$. In this
example, we are interested in skew constacyclic codes of length $n$ over
$\mathbb{F}_{8}$. Since the degree of the extension field
$\mathbb{F}_{8}/\mathbb{F}_{2}$ is $3$, we have
$\text{Aut}(\mathbb{F}_{8})=\\{\text{id},\theta,\theta^{2}\\}$, where $\theta$
is the Frobenius automorphism of $\mathbb{F}_{8}$, i.e., $\theta(a)=a^{2}$ for
all $a\in\mathbb{F}_{8}$. In this case, we need to study skew
$(\theta,\lambda)$-constacyclic and skew $(\theta^{2},\lambda)$-constacyclic
codes of length $n$ over $\mathbb{F}_{8}$.

  1. 1.

Case 1: Skew $(\theta,\lambda)$-constacyclic codes: According to Theorem 4,
the number of $(n,\theta)$-equivalence classes is equal to
$\gcd([n]_{1},q-1)=\gcd(2^{n}-1,7)$. By Theorem 5, we have two cases:

    1. (a)

If $\gcd(n,3)=1$, we have $\gcd(2^{n}-1,2^{3}-1)=2^{\gcd(n,3)}-1=2-1=1$, and
so there is only one $(n,\theta)$-equivalence class. Since $n$ is coprime to
the order of $\theta$, the order of $\theta$ is $3$, then all skew
$(\theta,\lambda)$-constacyclic codes of odd length over $\mathbb{F}_{8}$ are
$(n,\theta)$-equivalent to cyclic codes. This means that there exists an
element $\alpha$ in $\mathbb{F}_{8}^{*}$ such that
$\alpha^{[n]_{1}}\lambda=1$. Studying skew $(\lambda,\theta)$-constacyclic
codes, in this case, is reduced to studying cyclic codes of length $n$ over
$\mathbb{F}_{8}$.

    2. (b)

If $\gcd(n,3)=3$, we have $\gcd(2^{n}-1,2^{3}-1)=2^{\gcd(n,3)}-1=7$. Then we
have $7$ $(n,\theta)$-equivalence classes. So, in this case, we need to
consider all possible skew $(\theta,\lambda)$-constacyclic codes for any
$\lambda\in\mathbb{F}_{8}^{*}$.

  2. 2.

Case 2: Skew $(\theta^{2},\lambda)$-constacyclic codes: According to Theorem
4, the number of $(n,\theta^{2})$-equivalence classes is equal to
$\gcd([n]_{2},q-1)=\gcd\left(\frac{2^{2n}-1}{2^{2}-1},7\right)$. In this case,
we performed computations using Magma software, where we calculated
$\operatorname{gcd}([n]_{2},q-1)$ for $n$ ranging from $1$ to $100,000,000$.
We observed two distinct cases.

    1. (a)

If $\gcd(n,3)=1$, we have $\gcd([n]_{2},7)=1$, and so there is only one
$(n,\theta^{2})$-equivalence class. Since $n$ is coprime to the order of
$\theta^{2}$, the order of $\theta^{2}$ is $3$, then all skew
$(\theta^{2},\lambda)$-constacyclic codes of odd length over $\mathbb{F}_{8}$
are $(n,\theta^{2})$-equivalent to cyclic codes. This means that there exists
an element $\alpha$ in $\mathbb{F}_{8}^{*}$ such that
$\alpha^{[n]_{2}}\lambda=1$. Studying skew $(\lambda,\theta^{2})$-constacyclic
codes, in this case, is reduced to studying cyclic codes of length $n$ over
$\mathbb{F}_{8}$.

    2. (b)

If $\gcd(n,3)=3$, we have $\gcd([n]_{2},7)=7$. Then we have $7$
$(n,\theta^{2})$-equivalence classes. So, in this case, we need to consider
all possible skew $(\theta^{2},\lambda)$-constacyclic codes for any
$\lambda\in\mathbb{F}_{8}^{*}$.

##  Conclusion

In this paper, we introduced the $(n,\sigma)$-equivalence relation, where $n$
is the length of the code and $\sigma$ is an automorphism of the finite field.
We computed the number of $(n,\sigma)$-equivalence classes, and we give
conditions on $\lambda$ and $\mu$ for which $(\sigma,\lambda)$-constacyclic
codes and $(\sigma,\lambda)$-constacyclic codes to be equivalent using the
properties of our $(n,\sigma)$-equivalent relation. Under some conditions on
$n$ and $q$, we proved that skew constacyclic codes are equivalent to cyclic
codes. For future work, it would be interesting to introduce the concept of
“skew cyclotomic cosets” to facilitate the factorization of $x^{n}-1$ and
$x^{n}-a$ in $\mathbb{F}_{q}[x,\sigma].$ This may give an efficient way to
classify skew constacyclic codes of length $n$ over a finite field
$\mathbb{F}_{{}_{q}}$.

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