# Non-standard linear recurring sequence subgroups and automorphisms of
irreducible cyclic codes
††thanks: This work was supported by the Estonian Research Council grant
PRG49.

Henk D. L. Hollmann  Institute of Computer Science
University of Tartu
51009 Tartu, Estonia
<EMAIL_ADDRESS>
###### Abstract

Let ${\cal U}$ be the multiplicative group of order $n$ in the splitting field
$\mathbb{F}_{q^{m}}$ of $x^{n}-1$ over the finite field $\mathbb{F}_{q}$. Any
map of the form $x\rightarrow cx^{t}$ with $c\in{\cal U}$ and $t=q^{i}$,
$0\leq i<m$, is $\mathbb{F}_{q}$-linear on $\mathbb{F}_{q^{m}}$ and fixes
${\cal U}$ set-wise; maps of this type will be called standard. Occasionally
there are other, non-standard $\mathbb{F}_{q}$-linear maps on
$\mathbb{F}_{q^{m}}$ fixing ${\cal U}$ set-wise, and in that case we say that
the pair $(n,q)$ is non-standard. We show that an irreducible cyclic code of
length $n$ over $\mathbb{F}_{q}$ has “extra” permutation automorphisms (others
than the standard permutations generated by the cyclic shift and the Frobenius
mapping that every such code has) precisely when the pair $(n,q)$ is non-
standard; we refer to such irreducible cyclic codes as non-standard or NSIC-
codes. In addition, we relate these concepts to that of a non-standard linear
recurring sequence subgroup as investigated in a sequence of papers by Brison
and Nogueira. We present several families of NSIC-codes, and two constructions
called “lifting” and “extension” to create new NSIC-codes from existing ones.
We show that all NSIC-codes of dimension two can be obtained in this way, thus
completing the classification for this case started by Brison and Nogueira.

###### Index Terms:

linear recurrence relation, linear recurring sequence, $f$-sequence,
$f$-subgroup, linear recurring sequence subgroup, non-standard sequence
subgroup, cyclic code, irreducible cyclic code, permutation automorphism

##  I Introduction

The general problem of determining the automorphism group of any cyclic code
is a difficult problem, and [1, Section 3.5] suggests to investigate special
cases such as irreducible cyclic codes. Usually, the only permutation
automorphisms of an irreducible cyclic code are those generated by the cyclic
shift and the Frobenius mapping; we refer to these codes as standard and to
the exceptional ones that possess “extra” permutation automorphisms as non-
standard. For briefness, we will refer to non-standard irreducible cyclic
codes as NSIC-codes. Examples include the $q$-ary simplex codes, certain even-
weight codes, and the duals of the binary and ternary Golay codes. The
ultimate goal is to obtain a full classification of the NSIC-codes. We present
several families of NSIC-codes, together with construction techniques called
lifting and extension to construct new NSIC-codes from existing ones. As one
of our main results, we show that every NSIC-code of dimension two can be
obtained in this way.

Interestingly, the notion of an NSIC-code can be related to various other
notions. In a series of papers, see, e.g., [2, 3, 4, 5], Brison and Nogueira
investigated the concept of a non-standard linear recurring sequence subgroup
or NSLRS-group, a multiplicative subgroup in an extension of a finite field
$\mathbb{F}_{q}$ with the property that the elements can be represented by a
non-cyclic linear recurring sequence with characteristic polynomial over
$\mathbb{F}_{q}$ (for precise definitions of these and other notions, we refer
to the next two sections). We show that the notion of an NSIC-codes coincides
with that of a NSLRS-group in the case where the characteristic polynomial of
the linear recurrence relation is required to be irreducible over
$\mathbb{F}_{q}$.

Let ${\cal U}$ be the multilicative subgroup of order $n$ in an extension
$\mathbb{F}_{q^{m}}$ of $\mathbb{F}_{q}$, with $m$ minimal. Usually, the
collection ${\cal L}(n,q)$ of $\mathbb{F}_{q}$-linear maps on
$\mathbb{F}_{q^{m}}$ that fix ${\cal U}$ set-wise consists of the maps
$x\rightarrow ux^{q^{i}}$ ($u\in{\cal U},0\leq i<m$) only. But occasonally,
${\cal L}(n,q)$ contains other, non-standard maps; in that case, we refer to
the pair $(n,q)$ as non-standard. We show that an irreducible cyclic code $C$
of length $n$ over $\mathbb{F}_{q}$ is NSIC-code precisely when $(n,q)$ is
non-standard; in fact, we show that ${\cal L}(n,q)$ is a group and ${\cal
L}(n,q)\cong{\rm PAut}(C)$, the permutation automorphism group of $C$.

The contents of this paper are the following. In Section II we introduce the
notation used in this paper. In Section III, we sketch the required background
on linear recurring sequences and provide definitions of some of the notions
used above. In Section IV we prove the equivalence of NSIC-codes and non-
standard pairs. Lifting and extension is discussed in Section V. In Section VI
we introduce equally-spaced polynomials and a related class of NSIC-codes
based on a new type of cyclic product codes. Section VII lists the NSIC-codes
known to us, and in Section VIII we classify the NSIC-codes of dimension 2.
Finally, in Section IX we discuss our results.

We have space to include just a few complete proofs, for all other proofs,
further background, and any unexplained notation, we refer to [6] and [7].
Together, these two papers form an improved and extended version of the arxiv
paper [8].

##  II Notation and preliminaries

Throughout this paper, $q$ denotes a power of a prime, $n$ is a positive
integer such that $\gcd(n,q)=1$, and $m={\rm ord}_{n}(q)$, the smallest
positive integer for which $n\mid q^{m}-1$. The finite field of size $q$ is
denoted by $\mathbb{F}_{q}$, and we use $\mathbb{F}_{q}^{*}$ to denote both
the non-zero elements of the field and the multiplicative subgroup of
$\mathbb{F}_{q}$. We let $\overline{\mathbb{F}}_{q}$ denote the algebraic
closure of $\mathbb{F}_{q}$. For a subset ${\cal H}$ of a group ${\cal G}$, we
write ${\cal H}\leq{\cal G}$ to denote that ${\cal H}$ is a subgroup of ${\cal
G}$. There is a unique multiplicative subgroup in $\overline{\mathbb{F}}_{q}$
of order $n$, consisting of the $n$-th roots of unity in
$\overline{\mathbb{F}}_{q}$, which we denote by ${\cal U}_{n,q}$. Note that
${\cal U}_{n,q}\leq\mathbb{F}_{q^{m}}^{*}$, and by the definition of $m$,
$\mathbb{F}_{q^{m}}$ is the smallest extension of $\mathbb{F}_{q}$ that
contains ${\cal U}_{n,q}$. We let $\xi$ denote a fixed element in
$\mathbb{F}_{q^{m}}^{*}$ of order $n$, and we mostly use $g$ to denote the
minimal polynomial of $\xi$ over $\mathbb{F}_{q}$; note that the degree
$\deg(g)$ of $g$ equals $m$. Then ${\cal U}_{n,q}=\mbox{$\langle\xi\rangle$}$,
the multiplicative group generated by $\xi$.

An important notion in this paper is the $q$-order. The $q$-order of $\xi$,
denoted by $\delta_{q}(\xi)$, is the smallest positive integer $d$ for which
$\xi^{d}\in\mathbb{F}_{q}$. It is not difficult to show that
$\delta_{q}(\xi)=\delta_{q}(n):=n/\gcd(n,q-1)$, so depends only on $n$.
Writing $d=\delta_{q}(n)$ and $e=(n,q-1)$, we have $n=de$ with $e\mid q-1$ and
$\gcd(d,(q-1)/e)=1$; as a consequence, the number $m={\rm ord}_{n}(q)$ depends
only on $d$ and is the smallest positive integer such that
$d\mid(q^{m}-1)/(q-1)$. For proofs and more information on the $q$-order, see
[6].

In this paper, we investigate the group ${\cal L}(n,q)$ of all
$\mathbb{F}_{q}$-linear maps $L$ on $\mathbb{F}_{q^{m}}$ that fix the group
${\cal U}_{n,q}$ of order $n$ in $\mathbb{F}_{q^{m}}^{*}$ set-wise, that is,
such that $L({\cal U}_{n,q})={\cal U}_{n,q}$. Note that ${\cal L}(n,q)$ is
indeed a group: since $m={\rm ord}_{n}(q)$, the $\mathbb{F}_{q}$-span of the
elements $1,\xi,\ldots,\xi^{m-1}$ of ${\cal
U}_{n,q}=\mbox{$\langle\xi\rangle$}$ is $\mathbb{F}_{q^{m}}$, so if $L\in{\cal
L}(n,q)$, then ${\rm Im}(L)=\mathbb{F}_{q^{m}}$, hence $L$ is invertible. We
refer to an $\mathbb{F}_{q}$-linear map on $\mathbb{F}_{q^{m}}$ of the form
$L:x\rightarrow cx^{q^{j}}$, for some $j\in[m]:=\\{0,1,\ldots,m-1\\}$ and
$c\in\mathbb{F}_{q^{m}}$, as standard. Note that such a standard map is
contained in ${\cal L}(n,q)$ if and only if $c\in{\cal U}_{n,q}$ . We write
${\cal L}_{\rm st}(n,q)$ to denote the subgroup of ${\cal L}(n,q)$ consisting
of the standard maps in ${\cal L}(n,q)$. In certain exceptional cases, ${\cal
L}(n,q)$ is stricktly larger than ${\cal L}_{\rm st}(n,q)$; in that case, we
refer to the pair $(n,q)$ as non-standard and to the group ${\cal U}_{n,q}$ as
non-standard (over $\mathbb{F}_{q}$); in the usual case where ${\cal
L}(n,q)={\cal L}_{\rm st}(n,q)$, we refer to both $(n,q)$ and ${\cal U}_{n,q}$
as standard.

Let ${\cal S}_{n}$ denote the symmetric group on $n$ symbols, the group of all
permutations on $[n]$. Define the map $\Psi:{\cal L}(n,q)\rightarrow{\cal
S}_{n}$ by $\Psi(L)=\pi$ if $L(\xi^{i})=\xi^{\pi(i)}$ for $i=0,\ldots,n-1$. It
is easily seen that $\Psi$ is a one-to-one group-homomorphism. We will write
${\cal S}(n,q)=\Psi({\cal L}(n,q))$ and ${\cal S}_{\rm st}(n,q)=\Psi({\cal
L}_{\rm st}(n,q))$ to denote the images in ${\cal S}_{n}$ under $\Psi$ of
${\cal L}(n,q)$ and ${\cal L}_{\rm st}(n,q)$. By the above, ${\cal
L}(n,q)\cong{\cal S}(n,q)$ and ${\cal L}_{\rm st}(n,q)\cong{\cal S}_{\rm
st}(n,q)=\\{x\rightarrow q^{i}x+a\pmod{n}\mid a\in\mathbb{Z}_{n},i\in[m]\\}$;
in particular, we note that ${\cal L}_{\rm st}(n,q)$ has size $nm$. For
missing proofs and more details, we refer to [6, 7].

##  III Linear recurring seqence subgroups

In this section, we establish the relation between non-standard pairs and non-
standard linear recurring relation subgroups [3]. For more background on
linear recurring sequence, see, e.g., [9] or [10].

A sequence ${\bf s}=s_{0},s_{1},\ldots$ in $\overline{\mathbb{F}}_{q}$ is
called an $m$th order linear recurring sequence if it satisfies a
(homogeneous) linear recurrence relation of the form

$s_{k}=\sigma_{m-1}s_{k-1}+\cdots+\sigma_{1}s_{k-m+1}+\sigma_{0}s_{k-m}$ (1)

for all integers $k\geq m$, where $m\geq 1$,
$\sigma_{0}\in\mathbb{F}_{q}^{*}$, and
$\sigma_{1},\ldots,\sigma_{m-1}\in\mathbb{F}_{q}$. The monic polynomial

$f(x)=x^{m}-\sigma_{m-1}x^{m-1}-\cdots-\sigma_{1}x-\sigma_{0}$ (2)

in $\mathbb{F}[x]$ with $f(0)=-\sigma_{0}\neq 0$ is called the characteristic
polynomial of the recurrence relation (1) and a sequence ${\bf s}$ that
satisfies (1) is called an $f$-sequence. Such a sequence is necessarily
periodic, and we denote the (smallest) period of an $f$-sequence ${\bf s}$ by
${\rm per}({\bf s})$. We say that an $f$-sequence ${\bf s}$ in
$\overline{\mathbb{F}}_{q}$ is cyclic if there exists
$\alpha\in\overline{\mathbb{F}}_{q}$ such that $s_{k+1}/s_{k}=\alpha$ for all
$k\geq 0$; note that then, necessarily, $f(\alpha)=0$. Recall that
$\xi\in\mathbb{F}_{q^{m}}$ denotes an element of order $n$, and ${\cal
U}_{n,q}=\mbox{$\langle\xi\rangle$}$ is the group of $n$-th roots of unity. If
there exists an $f$-sequence ${\bf s}$ in $\overline{\mathbb{F}}_{q}$ with
${\rm per}({\bf s})=n$ such that ${\cal
U}_{n,q}=\\{s_{0},s_{1},\ldots,s_{n-1}\\}$, then we say that ${\cal U}_{n,q}$
is an $f$-subgroup, and that ${\cal U}_{n,q}$ is represented by ${\bf s}$.
Sometimes, such a group is referred to as a linear recurring sequence subgroup
[2]. A rather uninteresting way for ${\cal U}_{n,q}$ to be an $f$-subgroup is
when $f$ is the minimal polynomial of $\xi$ over $\mathbb{F}_{q}$,
representing ${\cal U}_{n,q}$ by the cyclic $f$-sequence ${\bf s}$ with
$s_{k}=\xi^{k}$ ($k\geq 0$). But sometimes there exist non-cyclic
$f$-sequences that represent ${\cal U}_{n,q}$; in that case, if $f$ is a
polynomial over $\mathbb{F}_{q}$, we refer to ${\cal U}_{n,q}$ as a non-
standard $f$-subgroup or a non-standard linear recurring sequence subgroup
(NSLRS-group) over $\mathbb{F}_{q}$.

In this paper, we will be mainly interested in the case where the
characteristic polynomial of the linear recurrence relation is irreducible
over $\mathbb{F}_{q}$. Our approach is based on the following results. For
precise proofs, we refer to [6].

###### Theorem III.1

Let $g$ be irreducible over $\mathbb{F}_{q}$ of degree $m$, and let $\xi$ be a
zero of $g$ of order $n$ in $\mathbb{F}_{q^{m}}$, so that $m={\rm
ord}_{n}(q)$. Then a sequence ${\bf s}$ is a $g$-sequence in
$\mathbb{F}_{q^{m}}$ if and only if there are
$L_{0},\ldots,L_{m-1}\in\mathbb{F}_{q^{m}}$ such that

$s_{k}=L_{0}\xi^{k}+L_{1}\xi^{qk}+\cdots+L_{m-1}\xi^{q^{m-1}k}\qquad(k\geq
0).$ (3)

###### Proof:

(Sketch) Since $\xi,\xi^{q},\ldots,\xi^{q^{m-1}}$ are all zeros of $g$,
sequences of the form as in the theorem are indeed $g$-sequences. A dimension
argument shows that these are indeed all the $g$-sequences over
$\mathbb{F}_{q^{m}}$. ∎

###### Theorem III.2

With the assumptions as in Theorem III.1, a $g$-subgroup ${\cal U}$ in an
extension of $\mathbb{F}_{q}$ is unique, and has the form ${\cal
U}=\mbox{$\langle\xi\rangle$}={\cal U}_{n,q}$.

###### Proof:

(Sketch) All zeros $\xi^{q^{i}}$ ($i\in[m]$) of $g$ have the same period $n$,
and so every non-zero $g$-sequence ${\bf s}$ has minimal period ${\rm
per}({\bf s})=n$. ∎

For a generalization of this result to $f$-subgroups for general $f$, and for
additional background and references, see [9]. If we now combine the above
results, we obtain the following.

###### Theorem III.3

With the assumptions as in Theorem III.1, a group ${\cal U}$ is a $g$-subgroup
if and only if ${\cal U}={\cal U}_{n,q}=\mbox{$\langle\xi\rangle$}$. In
addition, there is a one-to-one correspondence between $g$-sequences
representing ${\cal U}_{n,q}$ and $\mathbb{F}_{q}$-linear maps $L\in{\cal
L}(n,q)$, where ${\bf s}$ corresponds to $L$ if $s_{k}=L(\xi^{k})$ ($k\geq
0)$; moreover, ${\bf s}$ is cyclic if and only if the corresponding map $L$ is
in ${\cal L}_{\rm st}(n,q)$.

###### Proof:

(Sketch) It is well-known that there is a one-to-one correspondence between
$\mathbb{F}_{q}$-linear maps on $\mathbb{F}_{q^{m}}$ and $q$-polynomials on
$\mathbb{F}_{q^{m}}$, maps of the form
$L(x)=L_{0}x+L_{1}x^{q}+\cdots+L_{m-1}x^{q^{m-1}}$ with
$L_{0},L_{1},\ldots,L_{m-1}\in\mathbb{F}_{q^{m}}$. Now the result essentially
is a direct consequence of the expression (3) for $g$-sequences in
$\mathbb{F}_{q^{m}}$. ∎

As a consequence, the notion of a non-standard pair $(n,q)$ coincides with
that of a non-standard $g$-subgroup over $\mathbb{F}_{q}$ with $g$ irreducible
of order $n$.

##  IV Automorphisms of cyclic codes

A linear $[n,k]_{q}$-code $C$ is an $\mathbb{F}_{q}$-linear $k$-dimensional
subspace of $\mathbb{F}_{q}^{n}$. We sometime refer to vectors in $C$ as code
words. The dual code $C^{\perp}$ of $C$ is the collection of all vectors ${\bf
a}\in\mathbb{F}_{q}^{n}$ for which $({\bf a},{\bf
c}):=a_{0}c_{0}+\cdots+a_{n-1}c_{n-1}=0$ for all ${\bf c}\in C$. A permutation
$\pi\in{\cal S}_{n}$ induces a permutation on $\mathbb{F}_{q}^{n}$ by mapping
a vector ${\bf c}=(c_{0},c_{1},\ldots,c_{n-1})$ to the vector

${\bf c}^{\pi}=(c_{\pi^{-1}(0)},c_{\pi^{-1}(1)},\ldots,c_{\pi^{-1}(n-1)}).$

The group of permutation automorphisms of $C$, denoted by ${\rm PAut}(C)$, is
the collection of all permutations $\pi\in{\cal S}_{n}$ with the property that
if ${\bf c}\in C$, then ${\bf c}^{\pi}\in C$. For later use, we remark that
${\rm PAut}(C^{\perp})={\rm PAut}(C)$ [11, Lemma 1.3, (i)]. The cyclic shift
is the permutation $\sigma=(0,1,\ldots,n-1)\in{\cal S}_{n}$, mapping $i$ to
$i+1\pmod{n}$ ($i\in[n])$. A linear code $C\subseteq\mathbb{F}_{q}^{n}$ is
cyclic if $\sigma\in{\rm PAut}(C)$. By identifying a vector ${\bf
c}=(c_{0},c_{1},\ldots,c_{n-1})\in\mathbb{F}_{q}^{n}$ with the polynomial
$c(x)=c_{0}+c_{1}x+\cdots+c_{n-1}x^{n-1}$ in ${\cal
R}_{n,q}:=\mathbb{F}_{q}[x]\bmod x^{n}-1$, a code of length $n$ over
$\mathbb{F}_{q}$ corresponds to a subset of ${\cal R}_{n,q}$. A linear code is
cyclic if and only if the corresponding subset is an ideal in ${\cal
R}_{n,q}$. Now any ideal of ${\cal R}_{n,q}$ is principal, that is, generated
by a unique monic polynomial. If $C=(g(x))$ is a cyclic code of length $n$,
then $g(x)$ is called the generator polynomial of $C$ and
$h(x)=(x^{n}-1)/g(x)$ is called the parity-check polynomial of $C$.

Let $\xi\in\mathbb{F}_{q^{m}}$ have order $n$ and degree $m={\rm ord}_{n}(q)$
over $\mathbb{F}_{q}$. Let ${\rm Tr}(x)=x+x^{q}+\cdots+x^{q^{m-1}}$ denote the
trace of $\mathbb{F}_{q^{m}}$over $\mathbb{F}_{q}$. The code $C_{n,q}$
consisting of all code words

$\mu(a)=({\rm Tr}(a\xi^{0}),{\rm Tr}(a\xi^{1}),\ldots,{\rm Tr}(a\xi^{n-1}))$

with $a\in\mathbb{F}_{q^{m}}$ is called the irreducible (or minimal) cyclic
code of length $n$ over $\mathbb{F}_{q}$. Note that $C_{n,q}$ is unique up to
a permutation, and has dimension $m$. The dual $C_{n,q}^{\perp}$ of $C_{n,q}$
has as generator polynomial the minimal polynomial $g(x)$ of $\xi$ over
$\mathbb{F}_{q}$, which has degree $m$; such a code is called maximal [12]. It
is well-known and easily verified that beside the cyclic shift $\sigma$, also
the Frobenius mapping $\phi:x\rightarrow qx\pmod{n}$ (considered as an element
of ${\cal S}_{n}$) is an automorphisms of $C_{n,q}$ (to see this, note that if
${\bf c}\in\mathbb{F}_{q}^{n}$, then $c(x)^{q}=c(x^{q})=c^{\phi}(x)$
corresponds to ${\bf c}^{\phi}$). We write ${\rm PAut}_{\rm st}(C_{n,q})$ to
denote the group $\langle\sigma,\phi\rangle$ generated by $\sigma$ and $\phi$;
note that $|{\rm PAut}_{\rm st}(C_{n,q})|=nm$. Recall that, by definition,
$C_{n,q}$ is a NSIC-code if and only if ${\rm PAut}(C_{n,q})\supsetneq{\rm
PAut}_{\rm st}(C_{n,q})$. The following result shows some unexpected
connections between the concept of a NSIC-code and various other notions of
non-standardness.

###### Theorem IV.1

An irreducible cyclic code $C_{n,q}$ is a NSIC-code if and only if $(n,q)$ is
a non-standard pair. More precise, we have that ${\rm PAut}(C_{n,q})={\rm
PAut}(C_{n,q}^{\perp})={\cal S}(n,q)$ and ${\rm PAut}_{\rm st}(C_{n,q})={\rm
PAut}_{\rm st}(C_{n,q}^{\perp})={\cal S}_{\rm st}(n,q)$.

###### Proof:

First, let $L\in{\cal L}(n,q)$, with $\Psi(L)=\pi\in{\cal S}_{n}$. Recall that
$C_{n,q}^{\perp}$ is the cyclic code with defining zero $\xi$. So if ${\bf
c}\in C_{n,q}^{\perp}$, then $c(\xi)=0$, hence

$0=L(0)=L(\sum_{i=0}^{n-1}c_{i}\xi^{i})=\sum_{i=0}^{n-1}c_{i}\xi^{\pi(i)}=c^{\pi}(\xi),$

and we conclude that ${\bf c}^{\pi}\in C_{n,q}^{\perp}$. Since ${\bf c}\in
C_{n,q}^{\perp}$ was arbitrary, $\pi\in{\rm PAut}(C_{n,q}^{\perp})$.

On the other hand, let $\pi\in{\rm PAut}(C_{n,q}^{\perp})$. Since
$(1,\xi,\ldots,\xi^{m-1})$ is a basis for $\mathbb{F}_{q^{m}}$ over
$\mathbb{F}_{q}$, we can define an $\mathbb{F}_{q}$-linear map $L$ on
$\mathbb{F}_{q^{m}}$ by setting $L(\xi^{i})=\xi^{\pi(i)}$ for
$i=0,\ldots,m-1$, and then extending $L$ by $\mathbb{F}_{q}$-linearity. We
claim that $L\in{\cal L}(n,q)$ with $\Psi(L)=\pi$. To see this, let
$j\in\\{m,\ldots,n-1\\}$. There are $a_{0},\ldots,a_{m-1}\in\mathbb{F}_{q}$
such that $\xi^{j}=a_{0}+a_{1}\xi+\cdots+a_{m-1}\xi^{m-1}$. This has two
consequences. First, by definition of $L$,

$L(\xi^{j})=a_{0}\xi^{\pi(0)}+\cdots+a_{m-1}\xi^{\pi(m-1)}.$ (4)

Next, the vector ${\bf c}=(a_{0},\ldots,a_{m-1},0,\ldots,0,-1,0,\ldots,0)$,
with the entry -1 in position $j$, has $c(\xi)=0$, so ${\bf c}\in
C_{n,q}^{\perp}$, and since $\pi\in{\rm PAut}(C_{n,q}^{\perp})$, we have ${\bf
c}^{\pi}\in{\rm PAut}(C_{n,q}^{\perp})$, that is,

$0=\sum_{i=0}^{n-1}c_{i}\xi^{\pi(i)}=a_{0}\xi^{\pi(0)}+\cdots+a_{m-1}\xi^{m-1}-\xi^{\pi(j)}.$
(5)

Combining (4) and (5), we conclude that $L(\xi)=\xi^{\pi(j)}$. Since $j$ was
arbitrary, we conclude that $L(\xi^{j})=\xi^{\pi(j)}$ for all $j\in[n]$, that
is, $\Psi(L)=\pi$. ∎

##  V Lifting and extension

In this section, we discuss two methods to create new non-standard pairs from
existing ones.

Let $\xi$ have order $n$ and degree $m$ over $\mathbb{F}_{q}$, and let $t$ be
a positive integer for which $\gcd(m,t)=1$. Since
$\gcd(n,q^{ti}-1)=\gcd(n,q^{ti}-1,q^{m}-1)=\gcd(n,q^{\gcd(m,ti)}-1)=\gcd(n,q^{\gcd(m,i)}-1)$,
we have $m={\rm ord}_{n}(q)={\rm ord}_{n}(q^{t})$, hence the minimal
polynomial $g(x)$ of $\xi$ over $\mathbb{F}_{q}$ is also the minimal
polynomial of $\xi$ over $\mathbb{F}_{q^{t}}$. So in view of the results in
Section III, the following result is not too surprising.

###### Theorem V.1

(Lifting) Let $m$ and $t$ be as above. Then ${\cal S}(n,q)={\cal S}(n,q^{t})$
and ${\cal S}_{\rm st}(n,q)={\cal S}_{\rm st}(n,q^{t})$. So $(n,q)$ is non-
standard if and only if $(n,q^{t})$ is non-standard.

###### Proof:

(Sketch) Let $L\in{\cal L}(n,q)$, so let $L(x)=\sum_{i=0}^{m-1}L_{i}x^{q^{i}}$
on $\mathbb{F}_{q^{m}}$ with $L_{i}\in\mathbb{F}_{q^{m}}$ ($i\in[m]$), where
we consider the indices modulo $m$. Define
$\tilde{L}(x)=\sum_{i=0}^{m-1}L_{it}x^{q^{ti}}$ ($x\in\mathbb{F}_{q^{mt}}$).
Then $L=\tilde{L}$ on $\mathbb{F}_{q^{m}}$, and $\tilde{L}$ is
$\mathbb{F}_{q^{t}}$-linear on $\mathbb{F}_{q^{tm}}$. Moreover, since
$\langle\xi\rangle\subseteq\mathbb{F}_{q^{m}}$, we have
$\tilde{L}(\xi^{i})=L(\xi^{i})=\xi^{\pi(i)}$ for all $i\in[n]$, where
$\pi=\Psi(L)$, so $\tilde{L}\in{\cal L}(n,q^{t})$ and
$\Psi(\tilde{L})=\Psi(L)$. Conversely, if $\tilde{L}\in{\cal L}(n,q^{t})$,
then $\tilde{L}$ is $\mathbb{F}_{q^{t}}$-linear, hence
$\mathbb{F}_{q}$-linear, on $\mathbb{F}_{q^{mt}}$, and since $\tilde{L}$ fixes
$\langle\xi\rangle$ set-wise, $\tilde{L}$ also fixes
$\mathbb{F}_{q}(\xi)=\mathbb{F}_{q^{m}}$ set-wise. So the restriction $L$ of
$\tilde{L}$ to $\mathbb{F}_{q^{m}}$ is in ${\cal L}(n,q)$. The claims are now
obvious. ∎

We will refer to the operation of passing from a non-standard pair $(n,q)$ to
a non-standard pair $(n,q^{t})$ with $\gcd(m,t)=1$ as lifting. In coding
terms, note that if $\tilde{C}$ is the $\mathbb{F}_{q^{t}}$-span of the code
$C_{n,q}$, then ${\rm PAut}(\tilde{C})={\rm PAut}(C_{n,q})$. And we have
$C_{n,q^{t}}\subseteq\tilde{C}$, with equality if and only if the parity-check
polynomial $g$ of $C_{n,q}$ (and of $\tilde{C}$) is also irreducible over
$\mathbb{F}_{q^{t}}$, which holds if and only if $\gcd(m,t)=1$.

The above operation on a non-standard pair $(n,q)$ changed the value of $q$.
Our next operation changes $n$. Recall that if $\xi\in\mathbb{F}_{q^{m}}^{*}$
has order $n$, then $d=\delta_{q}(n)=n/\gcd(n,q-1)$ is the smallest positive
integer such that $\xi^{d}\in\mathbb{F}_{q}$; in addition, $n=de$ with
$e=\gcd(n,q-1)$ and $\gcd(d,(q-1)/e)=1$. First, we prove a lemma.

###### Lemma V.2

Let $\xi\in\mathbb{F}_{q^{m}}$ have order $n$ and $q$-order
$\delta_{q}(\xi)=\delta_{q}(n)=d=n/\gcd(n,q-1)$, and let ${\cal
G}\leq\mathbb{F}_{q}^{*}$ with $\xi^{d}\in{\cal G}$. Then ${\cal
G}\langle\xi\rangle\leq\mathbb{F}_{q^{m}}^{*}$ and $|{\cal
G}\langle\xi\rangle|=d|{\cal G}|$.

###### Proof:

Obviously, ${\cal G}\mbox{$\langle\xi\rangle$}\leq\mathbb{F}_{q^{m}}^{*}$.
Since $\xi^{d}\in{\cal G}$, every element of ${\cal G}\langle\xi\rangle$ has
the form $g\xi^{i}$ with $g\in{\cal G}$ and $i\in[d]$. By definition of the
$q$-order, all these elements are distinct, hence ${\cal G}\langle\xi\rangle$
has order $d|{\cal G}|$. ∎

Now we have the following ‘extension” result.

###### Theorem V.3

(Extension) Let $n=de$ with $e=\gcd(n,q-1)$ and $d=\delta_{q}(n)$, so that
$\gcd(d,(q-1)/e)=1$. For a positive integer $f$, we have
$d=\delta_{q}(n)=\delta_{q}(nf)$ if and only if $f\mid(q-1)/e$. Let
$f\mid(q-1)/e$ and let $\theta$ have order $nf$ in an extension of
$\mathbb{F}_{q}$. Then $\theta\in\mathbb{F}_{q^{m}}^{*}$, and
$\langle\theta\rangle{\cal L}(n,q)\leq{\cal L}(nf,q)$; in particular, if
$(n,q)$ is non-standard and $f\mid d(q-1)/n$, then $(nf,q)$ is also non-
standard.

###### Proof:

With the assumptions in the theorem, we have that
$\delta_{q}(nf)=def/\gcd(def,q-1)=df/\gcd(df,(q-1)/e)=df/\gcd(f,(q-1)/e)=d$ if
and only if $f\mid(q-1)/e$. Next, assume that $f\mid(q-1)/e$. Let
$\xi\in\mathbb{F}_{q^{m}}^{*}$ have order $n$, and let $\theta$ have order
$nf$. Now ${\cal G}=\langle\theta^{d}\rangle\leq\mathbb{F}_{q}^{*}$ and
$|{\cal G}|=ef$, hence $\langle\theta\rangle={\cal G}\langle\xi\rangle$ by
Lemma V.2. Finally, if $g\in{\cal G}$ and $L\in{\cal L}(n,q)$, then
$gL(\langle\theta\rangle)=gL({\cal G}\mbox{$\langle\xi\rangle$})\subseteq
g{\cal G}\mbox{$\langle\xi\rangle$}\subseteq\mbox{$\langle\theta\rangle$}$,
hence $gL\in{\cal L}(n,q)$. Obviously, ${\cal L}_{\rm
st}(nf,q)=\langle\theta\rangle{\cal L}_{\rm st}(n,q)$, so the last conclusion
is immediate. ∎

Extension is related to the following code construction technique. Let $C$ be
a cyclic code of length $n$ over $\mathbb{F}_{q}$, with parity-check
polynomial $g(x)\in\mathbb{F}_{q}[x]$, so that $g(x)\mid x^{n}-1$, and let
$\nu\in\mathbb{F}_{q}^{*}$ with $\nu^{nf}=1$. Let
$\tilde{C}\subseteq\mathbb{F}_{q}^{nf}$ be the collection of words

${\tilde{{\bf c}}}=(c_{0},\nu
c_{1},\ldots,\nu^{n-1}c_{n-1},\nu^{n}c_{0},\ldots,\nu^{nf-1}c_{n-1})$

with ${\bf c}=(c_{0},\ldots,c_{n-1})\in C$. If $c(x)=a(x)(x^{n}-1)g(x)$ for
some $a(x)\in\mathbb{F}_{q}[x]$ with $\deg(a)<\deg(g)$, then
${\tilde{c}}(x)=c(\nu x)((\nu x)^{nf}-1)/((\nu x)^{n}-1)=a(\nu
x)(x^{nf}-1)/g(\nu x)$, hence $\tilde{C}$ is a cyclic code of length $nf$ over
${\bf F}_{q}$, with parity-check polynomial $(x^{nf}-1)/g(\nu x)$. Note that
if $g$ is irreducible with zero $\xi$, then $C=C_{n,q}$, and if the zero
$\xi/\nu$ of $g(\nu x)$ has order $nf$, then $\tilde{C}=C_{nf,q}$. Remark also
that if $\tilde{\nu}$ has order $f$, if $C_{0}$ is the cyclic code consisting
of all scalar multiples of the word
$(1,\tilde{\nu},\tilde{\nu}^{2},\ldots,\tilde{\nu}^{f-1})$, and if $C_{1}$ is
the code with code words $(c_{0},\nu c_{1},\ldots,\nu^{n-1}c_{n-1})$ with
$(c_{0},c_{1},\ldots,c_{n-1})\in C$, then $C_{1}$ is equivalent to $C$ and
$\tilde{C}$ is equivalent to the product code $C_{0}\times C_{1}$. For more
information on this product code construction and its relation with extension,
we refer to [6].

In [6] we show that multiple lifts and extensions can always be obtained by a
single lift, followed by a single extension.

##  VI Equally-spaced polynomials

A polynomial $f(x)\in\mathbb{F}_{q}[x]$ is called equally-spaced if it is of
the form $f(x)=g(x^{k})$ for some integer $k\geq 2$. Then
$\deg(f)=k\cdot\deg(g)$ and if $g(0)\neq 0$, then ${\rm ord}(f)=k\cdot{\rm
ord}(g)$ and ${\rm ord}_{q}(f)=k\cdot{\rm ord}_{q}(g)$ (see [6]). Equally-
spaced polynomials form a rich source of non-standard examples [5].

###### Theorem VI.1

Let $k\geq 2$ be a positive integer with $\gcd(k,q)=1$, let
$g(x)\in\mathbb{F}_{q}[x]$ be monic with $g(0)\neq 0$, of order $n\geq 1$ and
degree $m\geq 1$, and suppose that $f(x)=g(x^{k})$ is irreducible over
$\mathbb{F}_{q}$. Then the pair $(nk,q)$ is non-standard except when
$f(x)=x^{2}+1$.

For a proof, we refer to [5] or [6]. In [6] we also show that if $C$ is cyclic
of length $n$ over $\mathbb{F}_{q}$, with generator polynomial $g(x)$, then
the product code $\mathbb{F}_{q}^{k}\times C$ is again cyclic, with generator
polynomial $g(x^{k})$.

Note that under the conditions of Theorem VI.1, the polynomial $f(x)=g(x^{k})$
is irreducible over $\mathbb{F}_{q}$ if and only if ${\rm ord}_{nk}(q)=mk$.
The next theorem states when this occurs.

###### Theorem VI.2

Let $k$, $g(x)$, $n$, and $m$ be as in Theorem VI.1, so with $\gcd(nk,q)=1$
and with $g$ irreducible, so with $m={\rm ord}_{n}(q)$. Let $P(n,q)$ be the
set of numbers that have only prime factors $r$ for which $r|n$ and
$\gcd(r,(q^{m}-1)/n)=1$. Then the polynomial $f(x)=g(x^{k})$ is irreducible if
and only if $k\in P(n,q)$, with $4\not\,\mid k$ if $2\in P(n,q)$ and $n\equiv
2\bmod 4$.

The attraction of Theorem VI.2 is that it is constructive. For a proof, we
refer to [6].

##  VII Known non-standard pairs and their codes

The following non-standard pairs $(n,q)$ and corresponding irreducible (NSIC)
and maximal cyclic codes are known to us.
1) Pairs $(n,q)$ with $n\geq 5$ prime and $m={\rm ord}_{n}(q)=n-1$. The
corresponding codes are the repetition codes and their duals, the even-weight
codes, with corresponding polynomials $f(x)=(x^{n}-1)/(x-1)$. Here, ${\cal
L}(n,q)={\cal S}_{n}$.
2) Pairs $(n,q)$ with $n=q^{m}-1$, where $m>2$ or $m=2,q>2$. The corresponding
codes are the $q$-ary simplex codes and their duals, the primitive BCH codes
with designed distance 2, with as polynomials the primitive polynomials over
$\mathbb{F}_{q}$. Here, ${\cal L}(n,q)={\rm GL}(m,q)$.
3) The pair $(n,q)=(23,2)$, where $m={\rm ord}_{n}(q)=11$, with corresponding
code the (dual of the) binary Golay code, with group ${\cal L}(23,2)\cong
M_{23}$.
4) The pair $(n,q)=(11,3)$, where $m=5$, with corresponding codes the (dual of
the) ternary Golay code, with group ${\cal L}(11,3)\cong{\rm PSL}(2,11)$.
5) Pairs $(n,q)=(kn_{0},q)$ with $m={\rm ord}_{n}(q)=km_{0}$, where $n_{0}>1$
and $k\geq 2$ are integers with $k>2$ if $n_{0}=2$ and $m_{0}={\rm
ord}_{n_{0}}(q)$. (See Theorem VI.2 for the conditions under which this is the
case.) The corresponding codes are $\mathbb{F}_{q}^{k}\times C_{n_{0},q}$ and
their duals, with group ${\cal L}(n,q)\cong S_{k}\wr{\rm PAut}(C_{n_{0},q})$
(wreath product), and with corresponding polynomials the irreducible equally-
spaced polynomials $f(x)=g(x^{k})$ with $g(x)\in\mathbb{F}_{q}[x]$ and
$f(x)\neq x^{2}+1$.
6) In addition, for every non-standard pair $(n,q)$ with $m={\rm ord}_{n}(q)$,
we have non-standard pairs $(nf,q^{t})$ for every positive integer $t$ such
that $\gcd(t,m)=1$ and every positive integer $f$ such that
$f\mid(q^{t}-1)/\gcd(n,q^{t}-1)$ that can be obtained from $(n,q)$ by lifting
and extension as described in Section V.

Possibly, the above examples exhaust all possibilities. In the next section,
we show this to hold for the case where $m=2$.

##  VIII Classification for $m=2$

For degree $m=2$, we have the following result.

###### Theorem VIII.1

The non-standard pairs $(n,q)$ with $m={\rm ord}_{n}(q)=2$ and $q$-order
$d=n/\gcd(n,q-1)$ are the ones listed in Section VII for which $m=2$ and are
the following.
1) Pairs $(n,q)$ where $n=2e>4$ with $e\geq 1$ integer for which $e\mid q-1$,
with both $q$ and $(q-1)/e$ odd (the equally-spaced case); here $d=2$ and
${\cal L}(2e,q)\cong\mathbb{Z}_{e}\wr S_{2}$ (wreath product).
2) Pairs $(n,q)$ where $n=q^{2}-1$ and $q\geq 3$ (the primitive case); here
$d=q+1\geq 4$ and ${\cal L}(n,q)={\rm GL}(2,q)$. And in addition, all pairs
$(nf,q^{t})$ obtained from a non-standard pair $(n=q^{2}-1,q)$ as above by
lifting and extension, so with $t$ odd and $f\mid(q^{t}-1)/(q-1)$; here
$d=q+1$ and ${\cal L}(f(q^{2}-1),q^{t})=\langle\theta\rangle{\rm GL}(2,q)$,
where $\theta\in\mathbb{F}_{q^{t}}^{*}$ has order $f(q^{2}-1)$.

###### Proof:

(Sketch) If $d=2$, then the corresponding polynomial is of the form
$x^{2}-\sigma$ with $\sigma\in\mathbb{F}_{q}^{*}$ and is irreducible over
$\mathbb{F}_{q}$. A simple analysis, using Theorem VI.2, quickly leads to the
non-standard pairs as listed in case 1). Next, let $\xi\in\mathbb{F}_{q^{2}}$
be an element of degree $m=2$, order $n$, and $q$-order $d>2$, so with minimal
polynomial $g(x)=x^{2}-\sigma_{1}x-\sigma_{0}$, where
$\sigma_{0},\sigma_{1}\in\mathbb{F}_{q}^{*}$. Let $T\in{\cal L}(n,q)$ be the
“cyclic shift” map defined by $T(x)=\xi x$ on $\mathbb{F}_{q^{2}}$, and let
$L\in{\cal L}(n,q)$ with $L(1)=1$ and $L(\xi)=\omega+\nu\xi$, say. We proceed
in several steps.
1\. (Normalization) The element $\tilde{\xi}=\xi/\sigma_{1}$ has minimal
polynomial $\tilde{g}(x)=x^{2}-x-\lambda$, where
$\lambda=\sigma_{0}/\sigma_{1}^{2}$. Obviously, the $q$-order of $\tilde{\xi}$
again equals $d$. Writing $\tilde{\omega}=\omega/\sigma_{1}$, we have
$T(1)=\sigma_{1}\tilde{\xi},T(\tilde{\xi})=\sigma_{1}\tilde{\xi}^{2}=\sigma_{1}(\tilde{\xi}+\lambda)$
and $L(1)=1,L(\tilde{\xi})=\nu\tilde{\xi}+\tilde{\omega}$. Put $D={\rm
diag}(1,\sigma_{1}^{-1})$ and define

$\tilde{L}=L^{D}=\left[\begin{array}[]{cc}1&\tilde{\omega}\\\
0&\nu\end{array}\right],\qquad\tilde{T}=\sigma_{1}^{-1}T^{D}=\left[\begin{array}[]{cc}0&\lambda\\\
1&1\end{array}\right].$

Note that $\tilde{L}$ and $\tilde{T}$ are the matrices of the maps $L$ and
$\sigma_{1}^{-1}T$ on $\mathbb{F}_{q^{2}}$, with respect to the basis
$(1,\tilde{\xi})$.
2\. (Subgroup of ${\rm PGL}(2,q)$) Next, we identify the points of the
1-dimensional projective geometry ${\rm PG}(1,q)$ over $\mathbb{F}_{q}$ with
the sets $\mathbb{F}_{q}^{*}\alpha$ ($\alpha\in\mathbb{F}_{q^{2}}^{*}$), and
we consider the group ${\cal G}=\langle\tilde{T},\tilde{L}\rangle$ generated
by the matrices $\tilde{T}$ and $\tilde{L}$ as a subgroup of ${\rm PGL}(2,q)$.
Since $\tilde{\xi}^{d}\in\mathbb{F}_{q}^{*}$, every power $\tilde{\xi}^{j}$ of
$\tilde{\xi}$ is of the form $a\tilde{\xi}^{i}$ for some
$i\in\\{0,1,\ldots,d-1\\}$ and some $a\in\mathbb{F}_{q}^{*}$, hence ${\cal
G}\leq{\rm PGL}(2,q)$ fixes the set $O=\\{1,\xi,\ldots,\xi^{d-1}\\}$,
considered as a subset of ${\rm PG}(1,q)$, set-wise.
3\. (Subgroup structure of ${\rm PGL}(2,q)$) Now we use the known subgroup
structure of ${\rm PGL}(2,q)$, see, e.g., [14], [15], and a separate analysis
for the cases $3\leq d\leq 5$, to conclude that either $L$ is standard, or
${\cal G}$ is conjugate in ${\rm PGL}(2,q)$ to one of its subgroups ${\rm
PGL}(2,q_{0})$ or ${\rm PSL}(2,q_{0})$. Assuming the last case, we can then
show that $d=q_{0}+1$, where $q=q_{0}^{t}$ with $t$ odd, and
$\lambda,\tilde{\omega},\nu\in\mathbb{F}_{q_{0}}$ with
$\mathbb{F}_{q_{0}}=\mathbb{F}_{p}(\lambda)$, hence
$\tilde{\xi}\in\mathbb{F}_{q_{0}}^{2}$.
4\. As a consequence, $L$ fixes $\mathbb{F}_{q_{0}^{2}}$, and a simple
analysis reveals that the restriction of $L$ to $\mathbb{F}_{q_{0}^{2}}$ is
non-standard if $L$ itself is non-standard. Moreover, since $L$ fixes both
$\mathbb{F}_{q_{0}^{2}}^{*}$ and $\langle\xi\rangle$ set-wise, it also fixes
the intersection
$\mathbb{F}_{q_{0}^{2}}^{*}\cap\langle\xi\rangle=\langle\xi^{\delta}\rangle$,
where $\delta$ is the $q_{0}$-order of $\xi$. So, writing
$\theta=\xi^{\delta}$, we conclude that the group $\langle\theta\rangle$ is
non-standard over $\mathbb{F}_{q_{0}}$, as witnessed by the non-standard map
$L$ on $\mathbb{F}_{q_{0}^{2}}$.
5\. We have shown that $\theta$ is non-standard over $\mathbb{F}_{q_{0}}$,
with (maximal) $q_{0}$-order $q_{0}+1$. Now by [4, Theorem 2.4], $\theta$ is
primitive in $\mathbb{F}_{q_{0}^{2}}$.
6\. Finally, it can be shown that, as expected, the non-standard group
$\langle\xi\rangle$ over $\mathbb{F}_{q}$ can be obtained for the non-standard
group $\langle\theta\rangle$ over $\mathbb{F}_{q_{0}}$ by lifting and
extension. ∎

In [7] we shown that a nonstandard element $\xi$ of order $n$ and degree
$m={\rm ord}_{n}(q)$ over $\mathbb{F}_{q}$ with maximal $q$-order
$d=(q^{m}-1)/(q-1)$ is necessarily primitive, so has order $n=q^{m}-1$, thus
generalizing [4, Theorem 2.4] for $m=2$ to all $m$. The proof of the
generalization is completely different and uses the recently completed
classification of 1/2-transitive linear groups [13]. We take this opportunity
to remark that this result does not follow from [16, Theorem 2.3] since the
given proof is only valid when the subgroup $G$ in that theorem is proper.

##  IX Discussion and conclusions

Let $n$ and $q$ be positive integers with $\gcd(n,q)=1$, and let $m$ be the
multiplicative order of $q$ molulo $n$, the smallest positive integer such
that $n\mid q^{m}-1$. We say that the pair $(n,q)$ is non-standard if the
collection ${\cal L}(n,q)$ of $\mathbb{F}_{q}$-linear maps on
$\mathbb{F}_{q^{m}}$ that fixes the group ${\cal U}_{n,q}$ of $n$-th roots of
unity set-wise contains maps not of the form $x\rightarrow cx^{q^{j}}$ with
$c\in\mathbb{F}_{q^{m}}$ and $j\in[m]$. In this paper, we have first linked
this notion to that of non-standard linear recurring sequence subgroups [2, 3,
4, 5]. Then we showed that a pair $(n,q)$ is non-standard precisely when an
irreducible cyclic code of length $n$ over $\mathbb{F}_{q}$ has “extra”
permutation automorphisms (others than those generated by the cyclic shift and
some Frobenius mapping); in this paper, we refer to such codes as NSIC-codes.
A result by Brison and Nogueira from [4] for the case $m=2$ states that a non-
standard pair $(n,q)$ with (maximal) $q$-order $d=\delta_{q}(n)=q+1$ is
necessarily primitive, so has $n=q^{2}-1$. Using the known subgroup structure
of ${\rm PG}(2,q)$ in combination with this result, we have finished the
classification of the non-standard pairs and NSIC-codes for the case of
dimension $m=2$ that was initiated by Brison and Nogueira in [2, 3, 4].

Substantial information is available on the subgroup structure of ${\rm
PGL}(m,q)$ for $m=3$, and to a lesser extend also for $m=4$, see, e.g., [17].
By using similar methods, in combination with our generalization of [4,
Theorem 2.4] to all $m$ in [7], we expect that classification is also possible
for the case $m=3$, and perhaps even for the case $m=4$.

## Acknowledgment

I want to thank Karan Khathuria, Vitaly Skachek, and Ludo Tolhuizen for
proofreading and discussions.

## References

  * [1] P. Charpin, Open problems on cyclic codes, in: Handbook of Coding Theory I, V.S. Pless, W.C. Huffman, and R.A. Brualdi, Eds. Amsterdam: Elsevier,1998, 963–1063.
  * [2] O.J. Brison, J.E. Nogueira, Linear recurring sequence subgroups in finite fields, Finite Fields Appl. 9 (2003), 413–422.
  * [3] O.J. Brison, J.E. Nogueira, Second order linear sequence subgroups in finite fields, Finite Fields Appl. 14 (2008), 277–290.
  * [4] O.J. Brison, J.E. Nogueira, Second order linear sequence subgroups in finite fields-II, Finite Fields Appl. 15 (2009), 40–53.
  * [5] O.J. Brison, J.E. Nogueira, Non-standard sequence subgroups in finite fields, Finite Fields Appl. 16 (2010), 187–203.
  * [6] Henk D. L. Hollmann, Non-standard linear recurring sequence subgroups in finite fields and automorphisms of cyclic codes I,” submitted to Finite Fields Appl.
  * [7] Henk D. L. Hollmann, Non-standard linear recurring sequence subgroups in finite fields and automorphisms of cyclic codes II, in preparation.
  * [8] Henk D.L. Hollmann, Non-standard linear recurring sequence subgroups in finite fields and automorphisms of cyclic codes, http://arxiv.org/abs/0807.0595v1, July 3, 2008.
  * [9] Henk D.L. Hollmann, Medet Zhanbulatuly, Some basic results on finite linear recurring sequence subgroups, Finite Fields Appl. 73: 101844, 2021.
  * [10] R. Lidl, H. Niederreiter, Finite fields (2ed), Cambridge University Press, 1997.
  * [11] W.C. Huffman, Codes and groups, in: Handbook of Coding Theory II, V.S. Pless, W.C. Huffman, and R.A. Brualdi, eds., Elsevier, Amsterdam: Elsevier, 1998, 1345–1440.
  * [12] F.J. MacWilliams, N.J.A. Sloane, The Theory of Error-Correcting Codes, North Holland, 1983.
  * [13] M.W. Liebeck, C.E. Praeger, J. Saxl, The classification of 3/2-transitive permutation groups and 1/2-transitive linear groups, Proc. Amer. Math. Soc. 147(12):1, 2019, 5023–5037.
  * [14] L.E. Dickson, Linear groups, Dover, New York, 1958.
  * [15] X. Faber, Finite p-Irregular Subgroups of ${\rm PGL}(2,k)$, 2012, available at https://arxiv.org/abs/1112.1999v3.
  * [16] J.-P. Zanotti, Automorphism Groups of BWD-codes, J. Comb. Theory, Series A, 78, 1997, 303–308.
  * [17] O.H. King, The subgroup structure of finite classical groups in terms of geometric configurations, in: Surveys in Combinatorics 2005, Webb, ed., Cambridge University Press 2005, 29–56.