# Critical and geometric properties of magnetic polymers across the globule-
coil transition

Kamilla Faizullina  Ilya Pchelintsev  Evgeni Burovski  HSE University, 101000
Moscow, Russia

###### Abstract

We study a lattice model of a single magnetic polymer chain, where Ising spins
are located on the sites of a lattice self-avoiding walk in $d=2$. We consider
the regime where both conformations and magnetic degrees of freedom are
dynamic, thus the Ising model is defined on a dynamic lattice and
conformations generate an annealed disorder. Using Monte Carlo simulations, we
characterize the globule-coil and ferromaget-to-paramagnet transitions, which
occur simultaneously at a critical value of the spin-spin coupling. We argue
that the transition is continuous—in contrast to $d=3$ where it is first-
order. Our results suggest that at the transition the metric exponent takes
the theta-polymer value $\nu=4/7$ but the crossover exponent $\phi\approx
0.7$, which differs from the expected value for a $\theta$-polymer.

##  I Introduction

A linear polymer in thermal equilibrium in a solvent can be either extended
(“swollen”), or collapsed into a dense globule, depending on the interplay
between the excluded volume effects, van der Waals attraction between monomers
and its screening by the solvent [1]. The physics of the phase transition
between these two states, the so-called globule-coil transition or
$\theta$-transition, is well captured by a simple lattice model of an
interacting self-avoiding walk (ISAW), with an attractive interaction between
monomers on the nearest neighboring sites of the lattice [2].

For magnetic polymers, where monomers carry magnetic moments (“spins”), the
key parameter is the ratio of the relaxation times of magnetic and
conformational degrees of freedom [3]: if spins are fast, conformations
generate a quenched disorder for the magnetic subsystem [3, 4, 5, 6]; in the
opposite limit, the chain with quenched spins is qualitatively equivalent to a
disordered copolymer; several models of this kind have been discussed in the
literature [7, 8, 9, 10].

The regime where both spins and conformations have comparable relaxation times
has so far received much less attention. In this regime, spins are defined on
a dynamic lattice, whose thermal fluctuations need to be taken into account
self-consistently, on an equal footing with spin fluctuations. In this
direction, Ref. [11] introduced a model where monomers of a SAW carry Ising
spins, which interact via a short-range ferromagnetic interaction. The model
is investigated on a three-dimensional (3D) cubic lattice using a mean-field
approximation and Monte-Carlo (MC) simulations. In the absence of external
magnetic field, Ref. [11] finds a first-order magnetic induced collapse
transition—from a swollen paramagnetic phase to a ferromagnetic globular
phase. (Upon increasing the magnetic field, the transition is reported to
become continuous.) In Ref. [12] we considered a dynamic Hydrophobic-polar
(HP) model in two dimensions (2D). The collapse transition was found to be
consistent with a (continuous) $\theta$-transition of a nonmagnetic ISAW.

In this paper, we consider a ferromagnetic Ising model with spins placed on a
self-avoiding walk (SAW) on a 2D square lattice. Using MC simulations, we also
find a joint ferromagnetic and globule-coil transition, however our results
indicate that it is continuous—unlike the 3D model, where it is first order
[11]. We argue that the transition is characterized by the theta-point metric
exponent $\nu$, but the crossover exponent $\theta$ is markedly different. We
also explore geometric properties of the model, and stress the role of the
surface terms.

##  II Model and method

We consider the model of Ref. [11]: Let $\mathcal{U}_{N}$ be a set of all SAW
conformations of $N$ monomers joined by $N-1$ links on a 2D square lattice.
Each monomer $i$ in a conformation $u\in\mathcal{U}_{N}$ carries an Ising
spin, $s_{i}=\pm 1$, see Fig. 1. The spin-spin interaction is short-ranged:
two spins interact if they are nearest neighbors on the lattice. Given a SAW
conformation $u\in\mathcal{U}_{N}$ and a sequence of $N$ spins, $\\{s\\}$, the
Hamiltonian is

$E(\\{s\\},u)=-J\sum_{\langle i,j\rangle\in u}s_{i}s_{j}-h\sum_{j\in
u}s_{j}\;.$ (1)

Here the summation in the first term runs over pairs of spins, $i,j\in u$,
which are nearest neighbors on the 2D lattice, and $J>0$ is the ferromagnetic
exchange coupling. In the second term, $h$ is the magnetic field.

The partition function corresponding to Eq. (1) reads

$Z=\sum_{u\in\mathcal{U}_{N}}\sum_{\\{s\\}}e^{-\beta E(\\{s\\},u)},$ (2)

where $\beta=1/kT$ is the inverse temperature. To set the energy units, we
take $\beta=1$ without loss of generality. Note that the summations in Eq. (2)
run over both conformations and spin configurations.

For $h=J=0$, spins decouple from conformations, and the model (1)-(2) reduces
to a non-interacting SAW. In the limit $h\gg J$, all spins are aligned, and
Eq. (1)-(2) reduces to the ISAW model. In this work we only consider the case
$h=0$. In the limit $J\ll 1$, the model (1)–(2) describes Ising spins located
on a non-interacting SAW—for the spins, the geometry is effectively one-
dimensional and spontaneous magnetization is absent in the thermodynamic limit
[3, 4, 5]. For $J\gg 1$, it is natural to expect a dense ferromagnetically
ordered globule.

We note that since Eq. (1) only involves a single coupling constant, it is
natural to expect that the ferromagnetic ordering sets in simultaneously with
the globule-coil transition. In the next sections we verify this expectation
and characterize the corresponding transition.

Method.— Most popular methods for Monte Carlo (MC) simulations of SAW-like
model are based on chain growth techniques with pruning and enrichment [13],
and their flat-histogram generalizations [14]. We use a different strategy: we
work directly with fixed-length configurations and employ a variant of the
worm algorithm [15] for interacting SAW-like models [16]. Specifically, the
method uses two sets of MC updates. First is a bilocal reptation update, where
we simultaneously remove a monomer from one end of a chain and add a monomer
to the other end— the direction of the new edge and the value of the new spin
are selected at random, see Fig. 1(a)-(b). This is nothing but the BEE move of
Ref. [17]. Second, to render the reptation dynamics ergodic and improve
convergence for dense configurations, we also use the “reconnect” update,
where we rotate a single edge in the middle of the chain and attach it to the
end of the chain—which needs to be adjacent to an internal monomer, see Fig.
1(a)-(c). The reconnect update is non-local since it reverses directions of
$O(N)$ links of the SAW. However the Metropolis acceptance probability [18]
equals unity since the update does not change the energy, Eq. (1). The
reconnect update allows the simulation to escape from conformations where the
end of the chain is trapped inside a dense configuration [16]. Furthermore, to
improve convergence of magnetic observables, we also use standard Wolff
cluster updates [19] for spins which keep the conformation fixed.

Figure 1:  Spin/SAW configurations and MC updates. Straight lines show a
sample SAW, open circles denote spins-up, $s_{j}=+1$, and closed circles
denote spins-down, $s_{j}=-1$. The BEE move is changing (a) to (b), where the
edge shown in dashed red line in (a) is removed and the edge shown in dashed
red line in (b) is added. The reconnect update is changing the configuration
between (a) and (c). Note that configurations (a) and (c) have the same energy
Eq. (1).

##  III Numerical simulations

We simulate our model on a square 2D lattice for chains of up to $N=10^{4}$
monomers. We typically use up to $10^{9}$ MC updates for thermalization and
collect statistics for $10^{10}$ to $10^{11}$ MC steps. Here in a single MC
step we select an update (a BEE move, a reconnect or a spin cluster update) at
random.

We perform simulations for $h=0$ and $0<J<2$. We collect statistics for the
mean energy, Eq. (1), per spin, $\epsilon=\langle E\rangle/N$, the mean
magnetization per spin, $\langle m\rangle\equiv\langle\sum_{j\in
u}s_{j}\rangle/N$ and its powers, $\langle m^{2}\rangle$ and $\langle
m^{4}\rangle$. To characterize the structural properties of the model, we
measure the mean end-to-end distance of the SAW, $\langle R^{2}_{N}\rangle$.
111In the literature, the gyration radius is often considered instead; however
the asymptotic properties of the radius of gyration and end-to-end distance
are expected to be the same (see, e.g. [17]), and the latter is simpler to
work with numerically.  Here and elsewhere in the text, angular brackets
denote the MC average approximating the average over the Gibbs distribution
(2).

Figure 2:  (top) mean squared magnetization as a function of $J$ for several
values of $N$. Solid squares with errorbars are MC results, lines are to guide
an eye only. Errorbars are estimated via binning analysis. In these
simulations we use at least $7\times 10^{9}$ MC steps per data point. (bottom)
Mean energy as a function of $J$ for several values of $N$. Squares are MC
data with errorbars, and lines are to guide an eye. See text for discussion.

Fig. 2(top) shows simulation results for mean square magnetization, $\langle
m^{2}\rangle$, as a function of $J$ for several representative values of the
SAW lengths $N$. At small values of $J$, $\langle m^{2}\rangle\to 0$ at
increasing $N$, which is consistent with the spontaneous magnetization being
zero in the thermodynamic limit [3, 4, 5]. For larger values of the coupling
constant, magnetization increases with increasing $J$ and starts saturating
for $J\gtrsim 0.88$, which suggests a ferromagnetic ordering for large $J$.

Fig. 2(bottom) illustrates the behavior of the mean energy, which approaches
the asymptotic $N\to\infty$ value of $-2J$ for a densely packed fully
magnetized walk. Finite-size corrections are clearly visible for both $\langle
m^{2}\rangle$ and $\langle\epsilon\rangle$, and we note that corrections are
more pronounced for $J\gtrsim 0.82$, especially in Fig. 2(bottom).

Figure 3:  Mean squared end-to-end distance as a function of $N$ from $N=100$
to $N=3000$ for several values of $J$. Stars are MC data with errorbars,
dashed lines are to guide an eye, and solid lines are $R^{2}\sim N^{2\nu}$
with $\nu=3/4$ (the solid red line) and $\nu=4/7$ (the solid black line). See
text for discussion.

Fig. 3 shows the dependence of the mean end-to-end distance, $\langle
R^{2}_{N}\rangle$, on $N$ for several values of the coupling constant $J$. For
$N\gg 1$ the scaling is visually consistent with a power-law,

$\langle R^{2}_{N}\rangle\sim N^{2\nu}(1+\cdots)\;,$ (3)

where dots represent corrections-to-scaling. For comparison, Fig. 3 also shows
the asymptotic power laws $N^{2\nu}$ with $\nu=3/4$—which is a non-interacting
SAW value (see e.g.,[21]),— and $\nu=4/7$—which is the exact value for the 2D
ISAW at the $\theta$-point [22].

Numerical data in Fig. 3 seem to indicate that the scaling of the end-to-end
distance for our model crosses over from a non-interacting SAW limit for small
$J$ to a $\theta$-point scaling for $J\sim 0.83$, and further on towards
$\nu=1/2$, which is expected for a dense globular phase. 222 Following Ref.
[39], we also fit the data shown in Fig. 3 with a four-parameter model,
$\log(R_{N}^{2}+k_{1})=2\nu\log(N+k_{2})+b$, with fit parameters $\nu$, $b$,
$k_{1}$ and $k_{2}$. Here $b$, $k_{1}$ and $k_{2}$ are phenomenological
parameters meant to mimic corrections to scaling. While this model is not
expected to be fully accurate—it misrepresents corrections-to-scaling
exponents and thus produces wrong results for $\nu$ close to the globular
phase—it does support the expectation from a visual inspection of Fig. 3 that
the metric exponent $\nu$ agrees with the $\theta$-point value $\nu=4/7$
around $J\approx 0.83$.  Taken together, our numerical results shown in Figs.
2 and 3, indicate that both magnetic and structural properties of the model
undergo a change at around $J\sim 0.83$.

Figure 4:  (top) Binder cumulants (4) as a function of $J$ for several values
of $N$. Solid squares with errorbars are MC results, lines are to guide an eye
only. Errorbars are estimated via a Gaussian resampling from errorbars of
$\langle m^{4}\rangle$ and $\langle m^{2}\rangle$. (bottom) Scaled mean end-
to-end distance (3) with $\nu=4/7$, which is the exact value for the 2D ISAW
at the $\theta$-point [22]. Squares are MC data with errorbars, and lines are
to guide an eye. See text for discussion.

The joint transition.—To locate the magnetic transition between paramagnetic
and ferromagnetic phases, we compute the fourth-order Binder cumulant,

$U_{4}=1-\frac{\langle m^{4}\rangle}{3\langle m^{2}\rangle^{2}}\;,$ (4)

which is expected to become scale-independent at the transition [24].

Fig. 4(top) shows the dependence of the Binder cumulant (4) on interaction $J$
for several values of $N$. For large values of the coupling constant (not
shown in Fig. 4), $U_{4}$ tends to the value $2/3$ from below, as expected for
a ferromagnetic state [24]. Curves of the cumulant $U_{4}$ for varying $N$
cross around $J\approx 0.834$, indicative of the paramagnetic-to-ferromagnetic
phase transition. Finite-size corrections are clearly visible in Fig. 4(top),
thus to get a more precise estimate for the transition temperature, we analyze
the pairwise crossings of the $U_{4}$ vs $N$ curves for a series of $N$ values
from $N=2000$ to $N=9000$. The final estimate for the critical values is

$J_{c}=0.8340(5)\,,\qquad U_{4}^{(c)}=0.308(8)\;.$ (5)

This result (5) is close to, but distinct from the estimate
$J_{c}=1/1.18\approx 0.847$, stated as preliminary without much discussion in
Ref. [11].

Fig. 4(bottom) shows the dependence of the mean squared end-to-end distance
(3). Here we rescale the values of $R^{2}_{N}$ by $N^{2\nu}$ with $\nu=4/7$,
as suggested by the analysis in the previous section. With this rescaling,
$\langle R^{2}\rangle/N^{2\nu}$ becomes $N$-independent (modulo corrections-
to-scaling) at $J_{\theta}=0.833(1)$ which is consistent with Eq. (5) within
the combined errorbars.

We also checked that the existence of the crossing is sensitive to the value
of the metric exponent $\nu$: if $\nu$ is changed by more then 0.07, the
crossing disappears.

We thus conclude that our numerical data suggest that (i) the ferromagnetic
and globule-coil transition occur simultaneously at the critical coupling
constant given by Eq. (5), and (ii) the scaling of the end-to-end distance at
the transition is consistent with the $\theta$-point metric exponent
$\nu=4/7$.

The crossover exponent.—We turn our attention to estimating the crossover
exponent $\phi$ which quantifies the deviation from criticality via the scaled
coupling $x=(J-J_{c})/N^{-\phi}$ [21]. Specifically, the end-to-end distance
is expected to follow $\langle R^{2}_{N}\rangle=N^{2\nu}f(x)$ where $f(\cdot)$
is a dimensionless function of a dimensionless variable. To probe this Ansatz,
we perform data collapse of the end-to-end distance, where we keep $\nu=4/7$
fixed at its theta-point value [22], and vary $J_{c}$ and $\phi$. This
procedure is illustrated in Fig. 5.

Figure 5:  Data collapse for the scaled end-to-end distance, $\langle
R^{2}_{N}\rangle/N^{2\nu}$, vs the scaled coupling $x=(J-J_{c})N^{\phi}$. We
fix $\nu=4/7$ and vary $J_{c}$ and $\phi$. On this plot, $J_{c}=0.832$ and
$\phi=0.7$. From visual inspection of the quality of the collapse, we estimate
$J_{c}=0.833(1)$ and $\phi=0.7(1)$. See text for discussion.

We find that our MC data are consistent with $J_{c}=0.833(1)$ and
$\phi=0.7(1)$, where the errorbars are conservative estimates from visual
inspection of the quality of the data collapse. We note that the value of
$J_{c}$ is consistent with Eq. (5). The crossover exponent clearly differs
from the $\theta$-point value for the ISAW model, where the Coulomb gas
prediction is $\phi=3/7$ [22] and numerical estimates are somewhat larger (see
Ref. [25] and the discussion therein).

We also perform a similar data collapse analysis for the magnetization, where
the scaling Ansatz is $\langle m^{2}\rangle=N^{-2\beta\phi}g(x)$, where $g(x)$
is a scaling function and $\beta$ is the order parameter exponent. Fig. 6
illustrates the procedure where we take $\beta=1/8$—which is the value for the
2D Ising universality class. While the quality of our numerical data does not
allow for estimating critical exponents with accuracy of any less then, say,
50%, we find that our data are consistent with the order parameter exponent
taking the 2D Ising value, and the crossover exponent $\phi\approx 0.7$.

Figure 6:  Data collapse for the second moment of magnetization $\langle
m^{2}\rangle$. In this plot we use $\phi=0.71$, $J_{c}=0.832$ and $\beta=1/8$.
See text for discussion.

We stipulate that a high-precision estimate of the crossover exponent and/or
the order parameter exponent should take into account two sources of
corrections. First, for a disordered Ising model, logarithmic corrections
[26], are known to lead to apparently varying exponents [27]. Second, non-
universal corrections due to the surface tension are strong for 2D SAWs [28]
because the surface-to-volume ratio in 2D scales as $\sim N^{-1/2}$ which is
close to the universal $\theta$-point values $\nu=4/7$ and $\phi=3/7$.

Bulk to surface ratio.— Strictly speaking, the very notions of bulk and
surface are not well defined for $J<J_{c}$, where typical conformations are
coil-like. To come up with a quantitative characteristic which is meaningful
across the globule-coil transition and can be interpreted as a bulk-to-surface
ratio in the globular phase, we consider a local neighborhood of a monomer. We
note that each monomer (apart from two endpoints of the chain) can be
classified according to the number of its neighbor monomers as being either
1D-like (two neighbors), 2D-like (four neighbors) or surface-like (3
neighbors).

For a length-$N$ conformation, we count the numbers of monomers of each kind;
dividing by $N$ we obtain the fractions, $n_{\alpha}$ ($\alpha=2,3,4$), so
that $n_{2}+n_{3}+n_{4}=1-2/N$. Qualitatively, the ratio $n_{2}/(n_{3}+n_{4})$
characterizes a blob-and-link structure of a coil-type conformation, and
$n_{4}/n_{3}$ can be interpreted as a proxy for a bulk-to-surface ratio.

Fig. 7 shows the fractions of each kind of monomers as a function of $J$ for
chains of length $N=1000$ to $4900$. For comparison, we also compute the
corresponding fractions for an ISAW model (i.e., Eqs. (1)-(2) with $h\gg J$).

Figure 7:  Fractions of monomers with two neighbors, $n_{2}$, (left), three
neighbors, $n_{3}$, (center) and four neighbors, $n_{4}$, (right). Solid
circles are the MC data for the Ising model (1)-(2), stars are the MC data for
the ISAW model, and dotted lines are to guide an eye. The vertical solid black
line is the theta-point for the ISAW, taken from Ref.[25]. The vertical dashed
brown line is Eq. (5).

Several features stand out in Fig. 7. First, even in the non-interacting SAW
limit, $J\to 0$, conformations are not fully 1D-like, as $n_{2}\approx 0.75$
only (the finite-size corrections become negligible for $N\gtrsim 100$). The
“bulk” fraction, $n_{4}$, is vanishingly small in the $J\to 0$ regime, and the
fraction of the “surface” monomers, $n_{3}$, tends to 0.25 for $J\to 0$. In
the opposite limit of large $J$, the 1D-like fraction tends to zero and the
“bulk” fraction grows. Most surprisingly, the “surface” fraction, $n_{3}$,
develops a peak for both ISAW and Ising models in the vicinity of their
respective collapse transitions.

While the relation between these results to a bulk/surface ratio of real
polymer chains is qualitative at best, and that more work is needed to
understand the nature of the peaks of $n_{3}(J)$, these results do illustrate
the importance of surface effects and stress the qualitative difference
between the magnetic SAW models and spin networks with mixed 1D / 2D local
connectivity [29].

Relation to the Ising model on rectangular lattices.—

It is instructive to compare the critical value of the Binder cumulant,
$U_{4}^{(c)}$, Eq. (5), to the values for a usual Ising model on a regular
grid. For the Ising model on a rectangular $L\times W$ lattice, the critical
value of $U_{4}$ depends on the boundary conditions and on the aspect ratio of
the lattice, $L/W$ [30, 31]. The dependence on the boundary conditions is
strong: on an $L\times L$ lattice with periodic boundary conditions,
$U^{(c)}\approx 0.61$, while open boundary conditions lead to $U^{(c)}\approx
0.4$. Furthermore, on the lattice with open boundary conditions, $U^{(c)}$
decreases continuously for increasing aspect ratio $L/W$ down to $\approx
0.35$ for $L/W=2$ [31] and further down for larger aspect ratios.

The critical value $U_{4}^{(c)}$, Eq. (5), is approximately compatible with
the result for the Ising model on a rectangular lattice with open boundary
conditions and the aspect ratio given by the ratio of the eigenvalues of the
gyration tensor of an interacting SAW at the $\theta$-point [25]. More work is
needed to accurately trace this connection.

The nature of the transition.— In 3D, the transition is clearly first-order
[11]. Our simulations indicate that the transition is continuous in 2D. First
of all, the Binder cumulant (4) is a monotonic function of $J$ for fixed $N$,
cf Fig. 4(top). This is consistent with a continuous transition, and is in
contrast to the expected behavior for a first-order transition, where the
cumulant is non-monotonic and develops a dip at $J_{c}$ as $N$ increases [32].

We then perform simulations for the specific heat capacity per monomer, which
is given by the second moment of the energy, Eq. (1)–(2),

$C=\frac{1}{N}\left(\langle E^{2}\rangle-\langle E\rangle^{2}\right)$ (6)

For finite values of $N$, the heat capacity is expected to have a peak in the
critical region. The peak can be rounded and shifted by finite-size
corrections, and the evolution of the peak height and shape is expected to be
very different for first-order and continuous transitions: For a first order
transition, the height of the peak of $C(J)$ is expected to be linear in $N$,
while the width is expected to shrink as $\sim N^{-1}$ [32]. For continuous
transitions, the structure of $C(J)$ in the vicinity of $J_{c}$ is controlled
by the heat capacity exponent $\alpha$, which is typically different from
unity.

Figure 8: The specific heat capacity per monomer, Eq. (6), as a function of
the coupling constant $J$. Errorbars are estimated via statistical resampling
from MC data for the first and second moments of the energy. See text for
discussion.

Fig. 8 shows our numerical results for the specific heat capacity. We note
that numerical cancellations in Eq. (6) magnify statistical errors of MC
simulations, thus limiting the values of $N$ accessible in these simulations
to be about an order of magnitude smaller then those in Figs. 2-6—which is
comparable to the values reported in Ref. [11]. At these values of $N\leqslant
500$, shown in Fig. 8, finite-corrections are very strong. Nevertheless, the
available numerical data suggest that the peak height dependence on $N$ is
sublinear and the peak widths shrinks slower then $1/N$. The overall shape of
$C(J)$ curves in Fig. 8 is drastically different from those observed for a
first-order transition in 3D in Ref. [11]. We interpret these observations,
however limited, as an additional indication of a transition being continuous,
with the heat capacity exponent $\alpha<1$.

We also note that we observe a single peak of $C(J)$, not a two-peak structure
reported for a site-diluted Ising model [33] and a network of Ising spins with
mixed 1D/2D local connectivity [29]. The difference with the latter is not
surprising given the role of the surface-like spins, cf Fig. 7.

To further check the nature of the transition, we compute distributions of
observables. Fig. 9 shows the distribution of the magnetization for $N=10^{4}$
in the vicinity of the transition, Eq. (5). The distribution is Gaussian-like
on the paramagnetic side, $J<J_{c}$, broadens on approach to the critical
coupling, and develops a clear ferromagnetic structure ($m=\pm 1$) for
$J>J_{c}$. In the critical region, we see no signs of a phase coexistence
which would signal a first-order transition.

Figure 9:  Distribution of the magnetization $m=\sum_{j\in u}s_{j}$/N for
$N=10000$. The coupling constants are $J=0.830<J_{c}$ (blue points),
$J=0.833\approx J_{c}$ (orange), $J=0.836$ (just above the $J_{c}$, green),
and $J=0.840>J_{c}$ (red). Each simulation uses $\sim 7\times 10^{9}$ MC
steps.

##  IV Conclusions and outlook

Concluding, we study a 2D model of a magnetic polymer chain where monomers of
a self-avoiding walk on a lattice carry Ising spins [11]. We use a variant of
the worm algorithm to simulate fixed-length chains of up to $10^{4}$ monomers.
We find a joint transition—where both spins order ferromagnetically and the
SAW collapses into a globular phase—at $J/T=0.8340(5)$. The very fact that the
transitions occur simultaneously can be traced to the specifics of the model,
which only has a single coupling constant, the exchange integral for the
short-range spin-spin interaction. What is less clear a priori, is the nature
of the transition. Our results suggest that the transition is continuous, in
contrast to a similar 3D model, where it is reported to be first-order [11].
Our numerical results suggest that some critical exponents (but not all of
them) are inherited from the “parent” models, namely the $\theta$-polymer ISAW
model, and the Ising model. Specifically, we present numerical evidence that
the metric exponent $\nu$ at the transition takes the $\theta$-point value
$\nu=4/7$, but the crossover exponent $\phi\approx 0.7$, which is clearly
different from the $\theta$-polymer value of $3/7$. We also present
indications that the magnetic order parameter exponent $\beta$ is consistent
with the 2D Ising universality class value $\beta=1/8$, however the accuracy
of this observation given our simulation results is relatively weak.

We study geometric properties of the model and classify the local connectivity
of monomers of the chain into 1D-like, bulk-like and surface-like. A possible
interpretation of our numerical results is that the surface-to-bulk ratio has
a peak in the vicinity of the transition. Incidentally, we also find numerical
evidence that for a non-interacting SAW, the fraction of 1D-like monomers is
1/4 in the thermodynamic limit. To the best of our knowledge, this was
previously not discussed in the literature. More work is needed to clarify the
status and physical meaning of these numerical results.

Concerning future work, it would be interesting to explore more realistic
models of magnetic polymers, e.g. by considering Potts or Heisenberg type
models and general dipole-dipole couplings in two and three dimensions. Models
with separate coupling constants might generate richer phase diagrams with
separate globule-coil and magnetic transitions.

Possible experimental realizations of magnetic polymers, for which our model
and its suggested generalizations may be applicable, include magnetic
filaments where magnetic nanoparticles are either cross-linked by a polymer to
form linear structures—these can be realized via e.g. biotemplaing [34]— or
self-organize into one-dimensional like structures at liquid-liquid interfaces
[35]. Monte Carlo simulations of models of magnetic polymers may complement
molecular dynamics studies of magnetic filaments [36].

When this work was completed, we became aware of an independent study of the
same model in Ref. [37]. Our estimates of the location of the transition and
critical exponents and those of Ref. [37] are consistent within the combined
errorbars.

##  V Acknowledgments

We acknowledge financial support by RFBR according to the research project No
19-07-01117. K.F. and I.P. acknowledge support within the Project Teams
framework of MIEM HSE. Numerical simulations were performed using the
computational resources of HPC facilities at HSE University [38]. Multiple
illuminating discussions with Lev Shchur and Yury Budkov are gratefully
acknowledged.

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# Extended Irreducible Binary Sextic Goppa codes

Daitao Huang  College of Computer Science and Technology, Nanjing University
of Aeronautics and Astronautics, Nanjing, 211100, P. R. China
<EMAIL_ADDRESS>and  Qin Yue  Department of Mathematics, Nanjing University
of Aeronautics and Astronautics, Nanjing, 211100, P. R. China
<EMAIL_ADDRESS>
###### Abstract.

Let $n(>3)$ be a prime number and $\mathbb{F}_{2^{n}}$ a finite field of
$2^{n}$ elements. Let $L=\mathbb{F}_{2^{n}}\cup\\{\infty\\}$ be the support
set and $g(x)$ an irreducible polynomial of degree $6$ over
$\mathbb{F}_{2^{n}}$. In this paper, we obtain an upper bound on the number of
extended irreducible binary Goppa codes $\Gamma(L,g)$ of degree $6$ and length
$2^{n}+1$.

###### Key words and phrases:

Binary Goppa codes, Extended Goppa codes, equivalent, Act on Group;

###### 2010 Mathematics Subject Classification:

94B05

This work was supported in part by National Natural Science Foundation of
China (No. 61772015).

##  1\. Introduction

Let $\mathbb{F}_{2^{n}}$ be an extension of the finite field $\mathbb{F}_{2}$.
Many codes defined over $\mathbb{F}_{2}$ are subfield subcodes of another code
defined over the extension $\mathbb{F}_{2^{n}}$. For instance, Alternant codes
are subfield subcodes of Generalized Reed-Solomon codes and classical Goppa
codes are a special case of Alternant codes [7].

Goppa codes are particularly appealing for cryptographic applications.
McEliece was the first to exploit the potential of Goppa codes for the
development of a secure cryptosystem [10]. Goppa codes are especially suited
to this purpose because they have few invariants and the number of
inequivalent codes grows exponentially with the length and dimension of the
code. Indeed, since the introduction of code-based cryptography by McEliece in
1978 [5], Goppa codes still remain among the few families of algebraic codes
which resist to any structural attack. This is one of the reasons why every
improvement of our knowledge of these codes is of particular interest. In the
McEliece cryptosystem, one chooses a random Goppa code as a key hence it is
important that we know the number of Goppa codes for any given set of
parameters. This will help in the assessment of how secure the McEliece
cryptosystem is against an enumerative attack. An enumerative attack on the
McEliece cryptosystem finds all Goppa codes for a given set of parameters and
tests their equivalence with the public code [6]. There have been many
attempts to count the number of Goppa codes.

In [1, 2, 3], it has been shown that two Alternant codes
$\mathcal{A}_{k}(v,L)$ and $\mathcal{A}_{k}(v^{\prime},L^{\prime})$ are equal
even if the parameters $(v,L)$ are different from $(v^{\prime},L^{\prime})$.
Based on these result, many extended Goppa codes become equivalent by a parity
check in [13, 15]. Moreno [11] proved that there is only one extended
irreducible binary degree $3$ Goppa code of any length and that there are four
quartic Goppa codes of length $33$. Ryan [15] focused on the quartic case of
length $2^{n}+1$, where $n(>3)$ is prime, and gave a upper bound on the number
of inequivalent irreducible Goppa codes. Some similar topic were investigated
in [16, 17, 18]. In [8], Magamba and Ryan obtained an upper bound on the
number of extended irreducible $q$-ary Goppa codes of degree $r$ and length
$q^{n}+1$, where $p,r(>2)$ are two prime numbers and $q=p^{t}$. In [12],
Musukwa gave an upper bound on the number of inequivalent extended irreducible
binary Goppa codes of degree $2p$ and length $2^{n}+1$, where $n$ and $p$ are
two odd primes such that $p\neq n$ and $p\nmid(2^{n}\pm 1)$.

In this paper, we shall investigate extended irreducible binary Goppa codes of
degree $r$, where $r$ is a product of two distinct prime numbers. In the case:
$r=6$, we obtain upper bound of number of extended irreducible binary sextic
Goppa codes of length $2^{n}+1$, where $n(>3)$ is a prime.

The paper is organized as follows. In Section $2$, we remind some definitions
of Alternant codes, Goppa codes, and extended Goppa codes. Moreover, we recall
sufficient condition of equivalent extended Goppa codes by projective linear
group acting on the projective line $\mathbb{F}_{2^{n}}\bigcup\\{\infty\\}$.
In Section $3$, we investigate projective semi-linear group acting on the
elelment of degree $6$ over $\mathbb{F}_{2^{n}}$ with $n(>3)$ a prime. In
Section $4$, we obtain the upper bound on the number of inequivalent extended
Goppa codes by applying the Cauchy Frobenius Theorem. We conclude the paper in
Section $5$.

##  2\. Preliminaries

Let $\mathbb{F}_{2^{n}}$ be the finite field of order $2^{n}$ and
$\overline{\mathbb{F}}_{2^{n}}=\mathbb{F}_{2^{n}}\cup\\{\infty\\}$ a set of
coordinates for the projective line. We will introduce some basic knowledge in
the following.

###  2.1. Alternant code, Goppa Code and extended Goppa Code

In the subsection, we describe concepts of Alternant codes, Coppa codes, and
extended Goppa codes. In detail, see [3, 15].

######  Definition 2.1.

Let $L=(\alpha_{0},\ldots,\alpha_{m-1})$ be an $m$-tuple of distinct elements
of $\mathbb{F}_{2^{n}}$, $v=(v_{0},\ldots,v_{m-1})$ an $m$-tuple of non-zero
elements of $\mathbb{F}_{2^{n}}$, and $r$ an integer less than $m$. The
Alternant code $\mathcal{A}_{r}(v,L)$ is defined as follows:

$\mathcal{A}_{r}(v,L)=\\{x=(x_{0},\ldots,x_{m-1})\in\mathbb{F}_{2}^{m}:H_{r}(v,L)x^{T}=0\\},$

where the parity-check matrix is

$H_{r}(v,L)=\left(\begin{array}[]{cccc}v_{0}&v_{1}&\ldots&v_{m-1}\\\
v_{0}\alpha_{0}&v_{1}\alpha_{1}&\ldots&v_{m-1}\alpha_{m-1}\\\
\vdots&\vdots&&\vdots\\\
v_{0}\alpha_{0}^{r-1}&v_{1}\alpha_{1}^{r-1}&\ldots&v_{m-1}\alpha_{m-1}^{r-1}\\\
\end{array}\right).$

######  Definition 2.2.

Let $L=(\alpha_{0},\ldots,\alpha_{m-1})$ be an $m$-tuple of distinct elements
of $\mathbb{F}_{2^{n}}$ and $g(x)\in\mathbb{F}_{2^{n}}[x]$ a polynomial of
degree $r(<m)$ such that $g(\alpha_{i})\neq 0$ for $i=0,1,\ldots,m-1$. The
Goppa code $\Gamma(g,L)$ with the Goppa polynomial $g(x)$ and the support $L$
is defined as follows:

$\Gamma(g,L)=\left\\{x=(x_{0},x_{2},\ldots,x_{m-1})\in\mathbb{F}_{2}^{m}:\sum_{i=0}^{m-1}\frac{x_{i}}{x-\alpha_{i}}\equiv
0\pmod{g(x)}\right\\}.$

If $g(x)$ is irreducible over $\mathbb{F}_{2^{n}}$, $\Gamma(g,L)$ is called
irreducible.

######  Remark 2.3.

Let $\Gamma(g,L)$ be the Goppa code in Definition 2.2.

(1) $\Gamma(g,L)=\mathcal{A}_{r}(v_{g,L},L)$, where
$v_{g,L}=(g(\alpha_{0})^{-1},g(\alpha_{1})^{-1},\ldots,g(\alpha_{m-1})^{-1})$.

(2) The expurgated Goppa code $\widetilde{\Gamma}(g,L)$ of $\Gamma(g,L)$ is
defined as follows:

$\widetilde{\Gamma}(g,L)=\left\\{x=(x_{0},\ldots,x_{m-1})\in\Gamma(g,L):\sum\limits_{i=0}^{m-1}x_{i}=0\right\\}.$

In fact, $\widetilde{\Gamma}(g,L)=\mathcal{A}_{r+1}(v_{g,L},L)$, where
$v_{g,L}=(g(\alpha_{0})^{-1},g(\alpha_{1})^{-1},\ldots,g(\alpha_{m-1})^{-1})$.

(3) The extended Goppa code $\overline{\Gamma}(g,L)$ of $\Gamma(g,L)$ is
defined as follows:

$\overline{\Gamma}(g,L)=\left\\{x=(x_{0},\ldots,x_{m-1},x_{m}):(x_{0},\ldots,x_{m-1})\in\Gamma(g,L),\sum\limits_{i=0}^{m}x_{i}=0\right\\}.$

######  Definition 2.4.

Let $L=(\alpha_{0},\ldots,\alpha_{m-1})$ be an $m$-tuple of distinct elements
of $\mathbb{F}_{2^{n}}$,
$\overline{L}=L\bigcup\\{\infty\\}=(\alpha_{0},\ldots,\alpha_{m-1},\infty)$,
$v=(v_{0},\ldots,v_{m})$ an $(m+1)$-tuple of non-zero elements of
$\mathbb{F}_{2^{n}}$, and $r$ an integer less than $m+1$. The Alternant code
$\mathcal{A}_{r}(v,\overline{L})$ is defined as follows:

$\mathcal{A}_{r}(v,\overline{L})=\\{x=(x_{0},\ldots,x_{m})\in\mathbb{F}_{2}^{m+1}:H_{r}(v,\overline{L})x^{T}=0\\},$

where the parity-check matrix is

$H_{r}(v,\overline{L})=\left(\begin{array}[]{cccc}v_{0}&\ldots&v_{m-1}&0\\\
\vdots&&\vdots&\vdots\\\
v_{0}\alpha_{0}^{r-2}&\ldots&v_{m-1}\alpha_{m-1}^{r-2}&0\\\
v_{0}\alpha_{0}^{r-1}&\ldots&v_{m-1}\alpha_{m-1}^{r-1}&v_{m}\\\
\end{array}\right).$

######  Remark 2.5.

Let $\Gamma(g,L)$ be the Goppa code in Definition 2.2,
$g(x)=\sum\limits_{i=0}^{r}g_{i}x^{i}$ a polynomial of degree $r$, and
$\overline{L}=L\cup\\{\infty\\}$. Then the extended Goppa code
$\overline{\Gamma}(g,L)$ of $\Gamma(g,L)$ is just the Altrnant code
$\mathcal{A}_{r+1}(v_{g,\overline{L}},\overline{L})$, where
$v_{g,\overline{L}}=(g(\alpha_{0})^{-1},\ldots,g(\alpha_{m-1})^{-1},g(\infty)^{-1})$,
$g(\infty)=g_{r}$.

###  2.2. Action of groups

We will recall the actions of the projective linear group and the projective
semi-linear group on $\mathbb{F}_{2^{n}}$ and
$\overline{\mathbb{F}}_{2^{n}}=\mathbb{F}_{2^{n}}\cup\\{\infty\\}$,
respectively.

There are some matrix groups as follows:

(1) the affine group

$AGL_{2}(\mathbb{F}_{2^{n}})=\left\\{M=\left(\begin{array}[]{cc}a&b\\\
0&1\end{array}\right):a\in\mathbb{F}_{2^{n}}^{*},b\in\mathbb{F}_{2^{n}}\right\\};$

(2) the general linear group

$GL_{2}(\mathbb{F}_{2^{n}})=\left\\{M=\left(\begin{array}[]{cc}a&b\\\ c&d\\\
\end{array}\right):a,b,c,d\in\mathbb{F}_{2^{n}},ad-bc\neq 0\right\\};$

(3) the projective linear group

$PGL_{2}(\mathbb{F}_{2^{n}})=GL_{2}(\mathbb{F}_{2^{n}})/\\{aE_{2}:a\in\mathbb{F}_{2^{n}}^{*}\\},$

where $E_{2}$ is the $2\times 2$ identity matrix;

(4) the projective semi-linear group

$P\Gamma L_{2}(\mathbb{F}_{2^{n}})=PGL_{2}(\mathbb{F}_{2^{n}})\times G,$

where $G=Gal(\mathbb{F}_{2^{6n}}/\mathbb{F}_{2})=\langle\sigma\rangle$ is the
Galois group, $\sigma(x)=x^{2}$ for $x\in\mathbb{F}_{2^{6n}}$, and its
operation $\cdot$ is as follows:

$(\widetilde{A},\sigma^{i})\cdot(\widetilde{B},\sigma^{j})=(\widetilde{A\cdot\sigma^{i}(B)},\sigma^{i+j}),0\leq
i,j\leq 6n-1.$

Let $M=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\in
GL_{2}(\mathbb{F}_{2^{n}})$. Then the projective linear group
$PGL_{2}(\mathbb{F}_{2^{n}})$ acts on $\overline{\mathbb{F}}_{2^{n}}$ as
follows:

$\displaystyle PGL_{2}(\mathbb{F}_{2^{n}})\times\overline{\mathbb{F}}_{2^{n}}$
$\displaystyle\rightarrow$ $\displaystyle\overline{\mathbb{F}}_{2^{n}}$
$\displaystyle(\widetilde{M},\zeta)$ $\displaystyle\mapsto$
$\displaystyle\widetilde{M}(\zeta)=M(\zeta)=\frac{a\zeta+b}{c\zeta+d},$

where $\frac{1}{0}=\infty$ and $\frac{1}{\infty}=0$; the projective semi-
linear group $P\Gamma L_{2}(\mathbb{F}_{2^{n}})$ acts on
$\overline{\mathbb{F}}_{2^{n}}$ as follows:

$\displaystyle P\Gamma
L_{2}(\mathbb{F}_{2^{n}})\times\overline{\mathbb{F}}_{2^{n}}$
$\displaystyle\rightarrow$ $\displaystyle\overline{\mathbb{F}}_{2^{n}}$
$\displaystyle((\widetilde{M},\sigma^{i}),\zeta)$ $\displaystyle\mapsto$
$\displaystyle(\widetilde{M},\sigma^{i})(\zeta)=M(\sigma^{i}(\zeta))=\frac{a\zeta^{2^{i}}+b}{c\zeta^{2^{i}}+d}.$

In [3], there is a result as follows.

######  Lemma 2.6.

Let $g(x)$ be a polynomial of degree $r$ over $\mathbb{F}_{2^{n}}$ and
$L=(\alpha_{0},\ldots,\alpha_{m-1})$ an ordered tuples of $m$ distinct points
in the projective line set $\overline{\mathbb{F}}_{2^{n}}$. Let
$M=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\in
GL_{2}(\mathbb{F}_{2^{n}})$,
$L^{\prime\prime}=({M}(\alpha_{0}),\ldots,{M}(\alpha_{m-1}))$,
$g^{\prime}(x)=(-cx+a)^{r}g({M}^{-1}(x))$, $M^{-1}(x)=\frac{dx-b}{-cx+a}$, and
$g(-\frac{d}{c})\neq 0$ if $c\neq 0$. Then the Alternant code
$\mathcal{A}_{r+1}(v_{g,L},L)$ is equal to the Alternant code
$\mathcal{A}_{r+1}(v_{g^{\prime},L^{\prime\prime}},L^{\prime\prime})$.

By Lemma 2.6, we have the following corollary.

######  Corollary 2.7.

Let $g(x)=\sum_{k=0}^{r}g_{k}x^{k}$ be a polynomial of degree $r$ over
$\mathbb{F}_{2^{n}}$ and $L=(\alpha_{0},\ldots,\alpha_{m-1})$ an ordered
tuples of $m$ distinct points in the projective line set
$\overline{\mathbb{F}}_{2^{n}}$. Let $M=\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in GL_{2}(\mathbb{F}_{2^{n}})$, $\tau\in
G=Gal(\mathbb{F}_{2^{6n}}/\mathbb{F}_{2})$,
$L^{\prime}=(\tau(\alpha_{0}),\ldots,\tau(\alpha_{m-1}))$,
$L^{\prime\prime}=({M}(\tau(\alpha_{0})),\ldots,{M}(\tau(\alpha_{m-1})))$,
$\tau g(x)=\sum_{k=0}^{r}\tau(g_{k})x^{k}$, $(\tau
g)^{\prime}(x)=(-cx+a)^{r}\cdot\tau g(M^{-1}(x))$, and $(\tau
g)(-\frac{d}{c})\neq 0$ if $c\neq 0$. Then the Alternant code
$\mathcal{A}_{r+1}(v_{g,L},L)$ is equal to the Alternant code
$\mathcal{A}_{r+1}(v_{(\tau g)^{\prime},L^{\prime\prime}},L^{\prime\prime})$.

###### Proof.

Since the Alernant code $\mathcal{A}_{r+1}(v_{g,L},L)$ is a subfield subcode,
the Alternant code is equal to the Alternant code $\mathcal{A}_{r+1}(v_{\tau
g,L^{\prime}},L^{\prime})$. By Lemma 2.6, the Alternant code
$\mathcal{A}_{r+1}(v_{\tau g,L^{\prime}},L^{\prime})$ is equal to the
Alternant code $\mathcal{A}_{r+1}(v_{(\tau
g)^{\prime},L^{\prime\prime}},L^{\prime\prime})$. ∎

In Corollary 2.7, if $\tau$ is equal to the identity transformation, then
Lemma 2.6 is direct from Corollary 2.7. If $g(x)$ is irreducible over
$\mathbb{F}_{2^{n}}$ and has a root $\alpha$ in an extension over
$\mathbb{F}_{2^{n}}$, then $(\tau g)^{\prime}(x)=(-cx+a)^{r}\cdot\tau
g(M^{-1}(x))$ is the irreducible polynomial of degree $r$ over
$\mathbb{F}_{2^{n}}$ and has a root $\beta$ in an extension with

$\beta=(\widetilde{M},\tau)\alpha=M(\tau(\alpha))=\frac{a\tau(\alpha)+b}{c\tau(\alpha)+d}.$

###  2.3.  Equivalent extended Goppa codes

Let $g(x)$ be irreducible of degree $r$ over $\mathbb{F}_{2^{n}}$ and
$L=(\alpha_{0},\ldots,\alpha_{2^{n}-1})$ an ordered subset of $2^{n}$ distinct
points in ${\mathbb{F}}_{2^{n}}$. Suppose that $g(x)$ has a root $\alpha$ in
an extension over $\mathbb{F}_{2^{n}}$. Then

$\Gamma(g,L)=\\{x=(x_{0},x_{1},\ldots,x_{2^{n}-1})\in\mathbb{F}_{2}^{2^{n}}:H(\alpha)x^{T}=0\\},$

where the parity check matrix is

$H(\alpha)=(\frac{1}{\alpha-\alpha_{0}}~{}\frac{1}{\alpha-\alpha_{1}}\ldots\frac{1}{\alpha-\alpha_{2^{n}-1}}).$

See more detail in [4].

For convenience, we will denoted by

$C(\alpha)=\Gamma(g,L)\ \ \ \overline{C(\alpha)}=\overline{\Gamma}(g,L),$

where $\overline{\Gamma}(g,L)$ is the extended Goppa code of $\Gamma(g,L)$.

The following Lemma is similar to that in [15]. For completeness, we give the
proof.

######  Lemma 2.8.

Let $L=(\alpha_{0},\ldots,\alpha_{2^{n}-1})$ and
$\overline{L}=(\alpha_{0},\ldots,\alpha_{2^{n}-1},\infty)$ be ordered tuples
of $2^{n}$ and $2^{n}+1$ distinct points in the projective line
$\overline{\mathbb{F}}_{2^{n}}$, respectively. Let $\alpha$ be a root of an
irreducible polynomial $g(x)=\sum_{i=0}^{r}g_{i}x^{i}$ of degree $r$ over
$\mathbb{F}_{2^{n}}$. Let $M=\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in GL_{2}(\mathbb{F}_{2^{n}})$, $\tau\in
G=Gal(\mathbb{F}_{2^{6n}}/\mathbb{F}_{2})$, $(\widetilde{M},\tau)\in P\Gamma
L_{2}(\mathbb{F}_{2^{n}})$,
$\overline{L}^{\prime\prime}=(M(\tau(\alpha_{0})),\ldots,M(\tau(\alpha_{2^{n}-1})),M(\infty))$,
$\tau g(x)=\sum_{i=0}^{r}\tau(g_{i})x^{i}$, and $(\tau
g)^{\prime}(x)=(-cx+a)^{r}\cdot\tau g(M^{-1}(x))$ a root
$\beta=(\widetilde{M},\tau)\alpha=\frac{a\tau(\alpha)+b}{c\tau(\alpha)+d}$.
Then the Alternant code $\mathcal{A}_{r+1}(v_{g,\overline{L}},\overline{L})$
is equal to the Alternant code $\mathcal{A}_{r+1}(v_{(\tau
g)^{\prime},\overline{L}^{\prime\prime}},\overline{L}^{\prime\prime})$ and the
extended Goppa code $\overline{C(\alpha)}$ is permutation equivalent to the
extended Goppa code $\overline{C(\beta)}$, denoted by
$\overline{C(\alpha)}\cong\overline{C(\beta)}$.

###### Proof.

Let
$\overline{L}=(\alpha_{0},\ldots,\alpha_{2^{n}-1},\infty)=L\cup\\{\infty\\}$
be an ordered tuple of $2^{n}+1$ distinct points in the projective line
$\overline{\mathbb{F}}_{2^{n}}=\mathbb{F}_{2^{n}}\cup\\{\infty\\}$ and $g(x)$
irreducible of degree $r$ over $\mathbb{F}_{2^{n}}$. Then by Remark 2.5,

$\mathcal{A}_{r+1}(v_{g,\overline{L}},\overline{L})=\overline{\Gamma}(g,L),$

where $\mathcal{A}_{r+1}(v_{g,\overline{L}},\overline{L})$ is the Alternant
code and $\overline{\Gamma}(g,L)$ is the extended code of $\Gamma(g,L)$. Let
$M=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\in
GL_{2}(\mathbb{F}_{2^{n}})$, $\tau\in
G=Gal(\mathbb{F}_{2^{6n}}/\mathbb{F}_{2})$, $(\widetilde{M},\tau)\in P\Gamma
L_{2}(\mathbb{F}_{2^{n}})$, $\tau g(x)=\sum_{k=0}^{r}\tau(g_{k})x^{k}$, $(\tau
g)^{\prime}(x)=(-cx+a)^{r}\cdot\tau g(M^{-1}(x))$ a root
$\beta=(\widetilde{M},\tau)\alpha=\frac{a\tau(\alpha)+b}{c\tau(\alpha)+d}$,
and

$\overline{L}^{\prime\prime}=(\widetilde{M},\tau)(\overline{L})=(M(\tau(\alpha_{0}),M(\tau(\alpha_{1})),\ldots,M(\tau(\alpha_{2^{n}-1}),M(\infty)).$

Then by Corollary 2.7,

$\mathcal{A}_{r+1}(v_{g,\overline{L}},\overline{L})=\mathcal{A}_{r+1}(v_{(\tau
g)^{\prime},\overline{L}^{\prime\prime}},\overline{L}^{\prime\prime}).$ (2.1)

Let $g(x)$ have a root $\alpha$ in an extension over $\mathbb{F}_{2^{n}}$.
Then $(\tau g)^{\prime}(x)$ is irreducible over $\mathbb{F}_{2^{n}}$ and has a
root
$\beta=(\widetilde{M},\tau)\alpha=\frac{a\tau(\alpha)+b}{c\tau(\alpha)+d}$ in
an extension over $\mathbb{F}_{2^{n}}$. Note that $\overline{L}$ and
$\overline{L}^{\prime\prime}$ arrange in distinct orders of the projective set
$\overline{\mathbb{F}}_{2^{n}}$. Hence by Remark 2.5 and (2.1),

$\overline{\Gamma}(g,L)\cong\overline{\Gamma}((\tau g)^{\prime},L),\ i.e.,\
\overline{C(\alpha)}\cong\overline{C(\beta)},$

i.e., the extended Goppa code $\overline{C(\alpha)}$ is permutation equivalent
to the extended Goppa code $\overline{C(\beta)}$. ∎

In [15], Ryan used Lemma 2.8 to give an upper bound on the number of extended
irreducible binary quartic Goppa codes of length $2^{n}+1$, where $n(>3)$ is a
prime number. In this paper, we shall obtain an upper bound on the number of
extended irreducible binary sixtic quatic Goppa codes of length $2^{n}+1$,
where $n(>3)$ is a prime number. By $3|(2^{n}+1)$ for any $n(>3)$ a prime, we
prove that there are orbits of length $3$ in Propositions 4.3, which is
important and different from [15].

##  3\.  prime $n>3$

In this section, we investigate irreducible binary sextic Goppa codes over
$\mathbb{F}_{2^{n}}$. We always assume that $n(>3)$ is a prime number.

######  Definition 3.1.

The set $\mathbb{S}=\mathbb{S}(n,6)$ is the set of all elements in
$\mathbb{F}_{2^{6n}}$ of degree $6$ over $\mathbb{F}_{2^{n}}$.

In fact,
$\mathbb{S}=\mathbb{F}_{2^{6n}}\setminus\\{\mathbb{F}_{2^{2n}}\cup\mathbb{F}_{2^{3n}}\\}$,
then

$|\mathbb{S}|=2^{6n}-2^{2n}-2^{3n}+2^{n}.$

######  Lemma 3.2.

The projective semi-linear group $P\Gamma L_{2}(\mathbb{F}_{2^{n}})$ acts on
on the set $\mathbb{S}$:

$\displaystyle\pi:P\Gamma L_{2}(\mathbb{F}_{2^{n}})\times\mathbb{S}$
$\displaystyle\rightarrow$ $\displaystyle\mathbb{S}$
$\displaystyle((\widetilde{M},\sigma^{i}),\alpha)$ $\displaystyle\mapsto$
$\displaystyle(\widetilde{M},\sigma^{i})(\alpha)=\frac{a\alpha^{2^{i}}+b}{c\alpha^{2^{i}}+d}=\beta,$

where $M=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\in
GL_{2}(\mathbb{F}_{2^{n}})$,
$G=Gal(\mathbb{F}_{2^{6n}}/\mathbb{F}_{2})=\langle\sigma\rangle$.

Moreover, by Lemma 2.8 the extended Goppa code $\overline{C(\alpha)}$ is
permutation equivalent to the extended Goppa code $\overline{C(\beta)}$, i.e.,
$\overline{C(\alpha)}\cong\overline{C(\beta)}$.

By Lemma 3.2, there is an equivalent relation $\sim$ in $\mathbb{S}$: for
$\alpha,\beta\in\mathbb{S}$,

$\alpha\sim\beta\Longleftrightarrow\exists(\widetilde{M},\sigma^{i})\in
P\Gamma L_{2}(\mathbb{F}_{2^{n}}),\ (\widetilde{M},\sigma^{i})\alpha=\beta;$

moreover, by Lemma 2.8

$\alpha\sim\beta\Longrightarrow\overline{C(\alpha)}\cong\overline{C(\beta)}.$
(3.1)

There is an interesting question: does the converse proposition of (3.1) hold?

By Lemma 3.2, the orbit of $\alpha\in\mathbb{S}$ is

$\displaystyle\Omega_{\alpha}=\\{\beta\in\mathbb{S}:\alpha\sim\beta\\}$ (3.2)
$\displaystyle=$
$\displaystyle\left\\{(\widetilde{M},\sigma^{i})\alpha=\frac{a\alpha^{2^{i}}+b}{c\alpha^{2^{i}}+d}:\forall
M=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\in
GL_{2}(\mathbb{F}_{2^{n}}),0\leq i\leq 6n-1\right\\}$ (3.5) $\displaystyle=$
$\displaystyle\left\\{\sigma^{i}(\widetilde{M}(\alpha))=(\frac{a\alpha+b}{c\alpha+d})^{2^{i}}:\forall
M=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\in
GL_{2}(\mathbb{F}_{2^{n}}),0\leq i\leq 6n-1\right\\}.$ (3.8)

Then the number of different orbits in $\mathbb{S}$ under the action of the
group $P\Gamma L_{2}(\mathbb{F}_{2^{n}})$ is the upper bound of inequivalent
extended irreducible binary sixtic Goppa codes.

To compute the number of orbits in $\mathbb{S}$ under the action of the group
$P\Gamma L_{2}(\mathbb{F}_{q})$, by (3.2) we shall divide the action into two
actions.

(1) We consider the projective linear group $PGL_{2}(\mathbb{F}_{2^{n}})$
acting on $\mathbb{S}$. Then

$\mathbb{S}=\bigcup_{\alpha\in I}^{\cdot}O_{\alpha},\
\Omega=\\{O_{\alpha}:\alpha\in I\\},$ (3.9)

where $\Omega$ is the set of all distinct orbits in $\mathbb{S}$ under the
action of the projective linear group $PGL_{2}(\mathbb{F}_{2^{n}})$. In fact,
$\Omega$ is a partition of $\mathbb{S}$.

(2) We consider the Galois group
$G=Gal(\mathbb{F}_{2^{6n}}/\mathbb{F}_{2})=\langle\sigma\rangle$ acting on
$\Omega$. Then there is a partition of $\Omega$:

$\Omega=\bigcup_{\alpha\in J}^{\cdot}\Omega_{\alpha},$ (3.10)
$\Omega_{\alpha}=G(O_{\alpha})=\\{\sigma^{i}(O_{\alpha}):0\leq i\leq
6n-1\\}\subset\Omega,$

where $\\{\Omega_{\alpha}:\alpha\in J\\}$ is the set of all distinct orbits in
$\mathbb{S}$ under the action of the projective semi-linear group $P\Gamma
L_{2}(\mathbb{F}_{2^{n}})$. In fact, $|J|$ is just the number of different
orbits in $\mathbb{S}$ under the action of the group $P\Gamma
L_{2}(\mathbb{F}_{2^{n}})$.

In order to count the size of the set $J$, we shall use Cauchy Frobenius
Theorem about orbits in [9].

######  Lemma 3.3.

Let $G$ be a finite group acting on a set $X$. For any $g\in G$, let $F(g)$
denote the set of elements of $X$ fixed by $g$. Then the number of distinct
orbits in $X$ under the action of the group $G$ is
$\frac{1}{|G|}\sum\limits_{g\in G}|F(g)|$.

We firstly state the main result in the following theorem.

######  Theorem 3.4.

Let $n>3$ be a prime number. The number of extended irreducible binary sextic
Goppa codes of length $2^{n}+1$ over $\mathbb{F}_{2^{n}}$ is at most
$\frac{2^{3n}+2^{2n}+3\cdot 2^{n}+12n-18}{6n}$.

To prove Theorem 3.4, we shall show some propositions.

###  3.1.  $PGL_{2}(\mathbb{F}_{2^{n}})\times\mathbb{S}\rightarrow\mathbb{S}$

Consider the projective linear group $PGL_{2}(\mathbb{F}_{2^{n}})$ acting on
the set $\mathbb{S}$ as follows:

$\displaystyle PGL(\mathbb{F}_{2^{n}})\times\mathbb{S}$
$\displaystyle\rightarrow$ $\displaystyle\mathbb{S}$
$\displaystyle(\widetilde{M},\alpha)$ $\displaystyle\mapsto$
$\displaystyle\widetilde{M}(\alpha)=M(\alpha)=\frac{a\alpha+b}{c\alpha+d},$

where $M=\left(\begin{array}[]{cc}a&b\\\ c&d\\\ \end{array}\right)\in
GL(\mathbb{F}_{2^{n}})$. Clearly, it is a faithful action. Then

$\mathbb{S}=\bigcup_{\alpha\in
I}^{\cdot}O_{\alpha},\Omega=\\{O_{\alpha}:\alpha\in I\\},$

where $\Omega$ is the set of all distinct orbits in $\mathbb{S}$ under the
action of the projective linear group $PGL_{2}(\mathbb{F}_{2^{n}})$. In fact,
$\Omega$ is a partition of $\mathbb{S}$. We shall calculate the size of the
set $I$.

It is clear that

$|PGL_{2}(\mathbb{F}_{2^{n}})|=\frac{|GL_{2}(\mathbb{F}_{2^{n}})|}{|\\{aE_{2}:a\in\mathbb{F}_{2^{n}}^{*}\\}|}=2^{3n}-2^{n}.$

For $\alpha\in I$, let $H_{\alpha}$ denote the stabilizer of $\alpha$ under
the action of $PGL_{2}(\mathbb{F}_{2^{n}})$, then

$H_{\alpha}=\left\\{\widetilde{M}(\alpha)=\alpha:\widetilde{M}\in
PGL_{2}(\mathbb{F}_{2^{n}})\right\\}=\left\\{\widetilde{E_{2}}\right\\}$

and

$|O_{\alpha}|=\frac{|PGL_{2}(\mathbb{F}_{2^{n}})|}{|H_{\alpha}|}=2^{3n}-2^{n}.$

By $|\mathbb{S}|=2^{6n}-2^{2n}-2^{3n}+2^{n}$,

$|I|=\frac{|\mathbb{S}|}{|O_{\alpha}|}=2^{3n}+2^{n}-1.$

Moreover, we investigate the structure of each orbit $O_{\alpha}$. Note that
the projective linear group $PGL_{2}(\mathbb{F}_{2^{n}})$ acts transitively on
the orbit $O_{\alpha}$.

Since there is an injective homomorphism of two groups:

$\displaystyle AGL_{2}(\mathbb{F}_{2^{n}})$ $\displaystyle\rightarrow$
$\displaystyle PGL_{2}(\mathbb{F}_{2^{n}})$ $\displaystyle
M=\left(\begin{array}[]{cc}e&f\\\ 0&1\end{array}\right)$
$\displaystyle\mapsto$ $\displaystyle\widetilde{M},$

the affine group $AGL_{2}(\mathbb{F}_{2^{n}})$ is viewed as the subgroup of
$PGL_{2}(\mathbb{F}_{2^{n}})$.

Hence there is an action of $AGL_{2}(\mathbb{F}_{2^{n}})$ on $O_{\alpha}$:

$\displaystyle AGL_{2}(\mathbb{F}_{2^{n}})\times O_{\alpha}$
$\displaystyle\rightarrow$ $\displaystyle O_{\alpha}$ $\displaystyle(M,\beta)$
$\displaystyle\mapsto$ $\displaystyle e\beta+f,$

where $M=\left(\begin{array}[]{cc}e&f\\\ 0&1\end{array}\right)\in
AGL_{2}(\mathbb{F}_{2^{n}})$, $\beta=\frac{a\alpha+b}{c\alpha+d}\in
O_{\alpha},a,b,c,d\in\mathbb{F}_{2^{n}},ad-bc\neq 0$. We shall investigate all
distinct orbits in $O_{\alpha}$ under the action of the affine group
$AGL_{2}(\mathbb{F}_{2^{n}})$.

For $\beta=\frac{a\alpha+b}{c\alpha+d}\in O_{\alpha}$, the orbit of $\beta$ in
$O_{\alpha}$ under the action of the affine group
$AGL_{2}(\mathbb{F}_{2^{n}})$ is

$A({\beta})=\left\\{M(\beta)=e\beta+f:M=\left(\begin{array}[]{cc}e&f\\\
0&1\end{array}\right)\in AGL_{2}(\mathbb{F}_{2^{n}})\right\\}.$

The following lemma is from [14, 15].

######  Lemma 3.5.

$O_{\alpha}=(\bigcup\limits_{\gamma\in\mathbb{F}_{2^{n}}}^{\cdot}A(\frac{1}{\alpha+\gamma}))\bigcup\limits^{\cdot}A(\alpha),$

which are disjoint unions. Moreover, there is a partition of $O_{\alpha}:$

$\Delta_{\alpha}=\left\\{A(\alpha),A(\frac{1}{\alpha+\gamma}):\gamma\in\mathbb{F}_{2^{n}}\right\\}.$
(3.12)

###  3.2. $G\times\Omega\rightarrow\Omega$

Let $G=Gal(\mathbb{F}_{2^{6n}}/\mathbb{F}_{2})=\langle\sigma\rangle$ be the
Galois group of $\mathbb{F}_{2^{6n}}$ over $\mathbb{F}_{2}$ and $\Omega$
defined as (3.3). Now consider the group $G$ acting on the set $\Omega$:

$\displaystyle\varphi:G\times\Omega$ $\displaystyle\rightarrow$
$\displaystyle\Omega$ $\displaystyle(\sigma^{i},O_{\alpha})$
$\displaystyle\mapsto$
$\displaystyle\sigma^{i}(O_{\alpha})=O_{\sigma^{i}(\alpha)}.$

Now we shall count the number of distinct orbits in $\Omega$ under the action
of $G$. By Lemma 3.3, we need to calculate that for $\sigma^{i}\in G$, $0\leq
i\leq 6n-1$,

$|F(\sigma^{i})|=\left|\left\\{O_{\alpha}\in\Omega:\sigma^{i}(O_{\alpha})=O_{\alpha}\right\\}\right|.$
(3.13)

######  Remark 3.6.

(1) In fact, for $0\leq i\leq 6n-1$, let $d=\gcd(6n,i)$, then
$\langle\sigma^{i}\rangle=\langle\sigma^{d}\rangle$. Thus,

$\\{O_{\alpha}\in\Omega:\sigma^{i}(O_{\alpha})=O_{\alpha}\\}=\\{O_{\alpha}\in\Omega:\sigma^{d}(O_{\alpha})=O_{\alpha}\\}$

and

$|F(\sigma^{i})|=|F(\sigma^{d})|.$

(2) Let $o(\sigma^{i})$ denote the order of $\sigma^{i}$ in $G$, then
$o(\sigma^{i})\mid|G|$. By $|G|=6n$ and $n$ a prime,

$o(\sigma^{i})\in\\{1,2,3,6,n,2n,3n,6n\\}.$

Suppose that $o(\sigma^{i})\in\\{6n,3n\\}$ and
$\sigma^{i}(O_{\alpha})=O_{\alpha}$. Then
$H=\langle\sigma^{i}\rangle=\langle\sigma^{k}\rangle$, $k\in\\{1,2\\}$, and
$\sigma^{k}(O_{\alpha})=O_{\alpha}$. Hence
$\frac{a\alpha+b}{c\alpha+d}=\alpha^{2^{k}}$, which is impossible by $\alpha$
of degree $6$ over $\mathbb{F}_{2^{n}}$. Therefore

$|F(\sigma^{i})|=0\mbox{ if }o(\sigma^{i})\in\\{6n,3n\\}.$

In the next section, we shall discuss other cases.

##  4\. The action of $G$ on $\Omega$

######  Proposition 4.1.

Let $G=Gal(\mathbb{F}_{2^{6n}}/\mathbb{F}_{2})=\langle\sigma\rangle$.

(1) If $o(\sigma^{i})=1$, then $|F(\sigma^{i})|=|\Omega|=2^{3n}+2^{n}-1$.

(2) If $o(\sigma^{i})=2$, then $|F(\sigma^{i})|=2^{2n}-1$.

###### Proof.

(1) If $o(\sigma^{i})=1$, then for all $O_{\alpha}\in\Omega$,
$\sigma^{i}(O_{\alpha})=O_{\alpha}$ and
$|F(\sigma^{i})|=|\Omega|=2^{3n}+2^{n}-1$.

(2) By $o(\sigma^{i})=2$,
$H=\langle\sigma^{i}\rangle=\langle\sigma^{3n}\rangle$. Suppose that
$\sigma^{3n}(O_{\alpha})=O_{\alpha}$ for $O_{\alpha}\in\Omega$. Then the group
$H$ acts on the partition $\Delta_{\alpha}$ of $O_{\alpha}$ in (3.12) as
follows:

$\displaystyle H\times\Delta_{\alpha}$ $\displaystyle\longrightarrow$
$\displaystyle\Delta_{\alpha}$ $\displaystyle(\sigma^{3ni},A(\alpha))$
$\displaystyle\longmapsto$ $\displaystyle\sigma^{3ni}(A(\alpha)),i=0,1.$

Denote by $O_{A(\alpha)}$ the orbit of $A(\alpha)$ under the action of the
subgroup $H=\langle\sigma^{3n}\rangle$. Then there is a class equation:

$2^{n}+1=|\Delta_{\alpha}|=\sum|O_{A(\alpha)}|.$

By $|H|=2$, there is $A(\alpha)\in\Delta_{\alpha}$ such that
$|O_{A(\alpha)}|=1$, i.e., $\sigma^{3n}(A(\alpha))=A(\alpha)$. Hence

$\sigma^{3n}(\alpha)=\alpha^{2^{3n}}=c_{1}\alpha+c_{2},$

with $c_{1}\in\mathbb{F}_{2^{n}}^{*},c_{2}\in\mathbb{F}_{2^{n}}.$ Then
$\alpha=\sigma^{6n}(\alpha)=c_{1}\sigma^{3n}(\alpha)+c_{2}=c_{1}^{2}\alpha+c_{1}c_{2}+c_{2}$.
Thus $c_{1}=1$ and

$\alpha^{2^{3n}}=\alpha+c_{2}$

with $c_{2}\neq 0$ (if $c_{2}=0$, then $\alpha\in\mathbb{F}_{2^{3n}}$ which is
contradictory.). Consequently,

$(c_{2}^{-1}\alpha)^{2^{3n}}+c_{2}^{-1}\alpha+1=0.$

If $c_{2}^{-1}\alpha$ is viewed as $\alpha$ then $\alpha$ satisfies the
equation:

$x^{2^{3n}}+x+1=0.$

Hence there is a factorization:

$x^{2^{3n}}+x+1=\prod\limits_{c\in\mathbb{F}_{2^{3n}}}(x+\alpha+c).$

It is clear that $\alpha^{2^{6n}}=\alpha$ and $\alpha^{2^{3n}}=\alpha+1$, so
all roots $\alpha+c\in\mathbb{F}_{2^{6n}}\setminus\mathbb{F}_{2^{3n}}$,
$c\in\mathbb{F}_{2^{3n}}$.

Moreover, if $\alpha$ is a root of the polynomial $x^{2^{3n}}+x+1$ and
$\alpha\in\mathbb{F}_{2^{2n}}$, then $\alpha$ is a root of the polynomial
$x^{2^{n}}+x+1$ and there is a factorization:

$x^{2^{n}}+x+1=\prod_{c\in\mathbb{F}_{2^{n}}}(x+\alpha+c)$

and all roots $\alpha+c\in\mathbb{F}_{2^{2n}}$, $c\in\mathbb{F}_{2^{n}}$.

Hence there are $2^{3n}-2^{n}$ roots of $x^{2^{3n}}+x+1$ in $\mathbb{S}$.

Conversely, if $\alpha\in\mathbb{S}$ is a root of $x^{2^{3n}}+x+1$, then
$\sigma^{3n}(A(\alpha))=A(\alpha)$.

Therefore $\alpha\in\mathbb{S}$ is a root of $x^{2^{3n}}+x+1=0$ if and only if
$\sigma^{3n}(A(\alpha))=A(\alpha)$; moreover, $A(\alpha)$ has $2^{n}$ roots
$\alpha+c,c\in\mathbb{F}_{2^{n}}$, of the polynomial $x^{2^{3n}}+x+1$.

Suppose that $\sigma^{3n}(A(\alpha))=A(\alpha)$, $\alpha^{2^{3n}}+\alpha+1=0$,
and $\sigma^{3n}(A(\frac{1}{\alpha+\gamma}))=A(\frac{1}{\alpha+\gamma})$. Then

$(\frac{1}{\alpha+\gamma})^{2^{3n}}=\frac{1}{\alpha+1+\gamma}=\frac{b_{1}}{\alpha+\gamma}+b_{2},0\neq
b_{1},b_{2}\in\mathbb{F}_{2^{n}},$

which is contradictory with $\alpha$ of degree $6$ over $\mathbb{F}_{2^{n}}$.

In conclusion, $\sigma^{3n}(O_{\alpha})=O_{\alpha}$ if and only if
$\sigma^{3n}(A(\alpha))=A(\alpha)$ if and only if there are $2^{n}$ roots of
$x^{2^{3n}}+x+1$ in $O_{\alpha}$. Then

$|F(\sigma^{3n})|=\frac{2^{3n}-2^{n}}{2^{n}}=2^{2n}-1.$

∎

######  Lemma 4.2.

$\left|\left\\{\beta\in\mathbb{S}:\beta^{2^{2n}}+\frac{1}{\beta}+1=0\right\\}\right|=2^{2n}-2^{n}-2.$

###### Proof.

Now we investigate roots of the following equation

$x^{2^{2n}}+\frac{1}{x}+1=0.$ (4.1)

Suppose that $\beta$ is a root in (4.1) and

$B=\left(\begin{array}[]{ll}1&1\\\ 1&0\end{array}\right).$ (4.2)

Then $\beta^{2^{2n}}=B(\beta)$. By $o(B)=3$,
$\beta^{2^{6n}}=B^{3}(\beta)=\beta$ and $\beta\in\mathbb{F}_{2^{6n}}$. In the
following, we shall find all roots $\beta$ in (4.1) such that
$\beta\in\mathbb{F}_{2^{2n}}\cup\mathbb{F}_{2^{3n}}$.

It is clear that $\beta\in\mathbb{F}_{2^{2n}}$ if and only if
$\beta=\frac{1}{\beta}+1$, i.e., $o(\beta)=3$ and $\beta\in\mathbb{F}_{4}$.
Then

$\left|\left\\{\beta\in\mathbb{F}_{2^{2n}}:\beta^{2^{2n}}+\frac{1}{\beta}+1=0\right\\}\right|=\left|\left\\{\beta\in\mathbb{F}_{2^{2n}}:\beta+\frac{1}{\beta}+1=0\right\\}\right|=2.$

If $\beta\in\mathbb{F}_{2^{3n}}$, then $\beta^{2^{n}}+\frac{1}{\beta+1}=0$.
Conversely, if $\beta^{2^{n}}=\left(\begin{array}[]{ll}0&1\\\
1&1\end{array}\right)(\beta)$, then
$\beta^{2^{3n}}=\left(\begin{array}[]{ll}0&1\\\
1&1\end{array}\right)^{3}(\beta)=\beta$ and $\beta\in\mathbb{F}_{2^{3n}}$.
Hence

$\left|\left\\{\beta\in\mathbb{F}_{2^{3n}}:\beta^{2^{2n}}+\frac{1}{\beta}+1=0\right\\}\right|=\left|\left\\{\beta\in\mathbb{F}_{2^{3n}}:\beta^{2^{n}}({\beta}+1)+1=0\right\\}\right|=2^{n}+1.$

Therefore

$\left|\left\\{\beta\in\mathbb{S}:\beta^{2^{2n}}+\frac{1}{\beta}+1=0\right\\}\right|=2^{2n}+1-2-(2^{n}+1)=2^{2n}-2^{n}-2.$

∎

######  Proposition 4.3.

Let $G=Gal(\mathbb{F}_{2^{6n}}/\mathbb{F}_{2})=\langle\sigma\rangle$. If
$o(\sigma^{i})=3$, then $|F(\sigma^{i})|=2^{n}-2$.

###### Proof.

By $o(\sigma^{i})=3$, $H=\langle\sigma^{i}\rangle=\langle\sigma^{2n}\rangle$.
Suppose that $\sigma^{2n}(O_{\alpha})=O_{\alpha}$ for $O_{\alpha}\in\Omega$.
Then the group $H$ acts on the partition $\Delta_{\alpha}$ of $O_{\alpha}$ in
(3.12) as follows:

$\displaystyle H\times\Delta_{\alpha}$ $\displaystyle\longrightarrow$
$\displaystyle\Delta_{\alpha}$ $\displaystyle(\sigma^{2ni},A(\alpha))$
$\displaystyle\longmapsto$ $\displaystyle\sigma^{2ni}(A(\alpha)),i=0,1,2.$

Denote by $O_{A(\alpha)}$ the orbit of $A(\alpha)$ under the action the
subgroup $H=\langle\sigma^{2n}\rangle$. For $A(\alpha)\in\Delta_{\alpha}$,
$|O_{A(\alpha)}|=1$ or $3$ by $|H|=3$.

Suppose that there is $A(\alpha)\in\Delta_{\alpha}$ such that
$|O_{A(\alpha)}|=1$. Then $\sigma^{2n}(A(\alpha))=A(\alpha)$. By the proof of
Proposition 4.1, $\alpha$ is a root of $x^{2^{2n}}+x+1=0$ and
$\alpha^{2^{4n}}=\alpha$, which is a contradiction with $\alpha$ of degree $6$
over $\mathbb{F}_{2^{n}}$. Hence $|O_{A(\alpha)}|=3$ for all
$A(\alpha)\in\Delta_{\alpha}$.

Without loss of generality, let $\sigma^{2n}(A(\alpha))=A(\frac{1}{\alpha})$.
Then

$\alpha^{2^{2n}}=\frac{a}{\alpha}+b=B_{0}(\alpha),\alpha^{2^{6n}}=B_{0}^{3}\alpha=\alpha,$

where $B_{0}=\left(\begin{array}[]{cc}b&a\\\ 1&0\end{array}\right)\in
GL_{2}(\mathbb{F}_{2^{n}})$. Hence
$B_{0}^{3}=\left(\begin{array}[]{cc}b^{3}&ab^{2}+a^{2}\\\
b^{2}+a&ab\end{array}\right)$ is a scalar matrix and
$a=b^{2}\in\mathbb{F}_{2^{n}}^{*}$. If $\frac{\alpha}{b}$ is viewed as
$\alpha$, then $\alpha\in\mathbb{S}$ satisfies the equation (4.1). Hence
$\alpha$, $\alpha^{2^{2n}}=B(\alpha)=\frac{1}{\alpha}+1$,
$\alpha^{2^{4n}}=B^{2}(\alpha)=\frac{1}{\alpha+1}$ are roots of (4.1), where
$B$ is defined as (4.2). So
$\sigma^{2n}(A(\alpha))=A(\frac{1}{\alpha}+1)=A(\frac{1}{\alpha})$,
$\sigma^{2n}(A(\frac{1}{\alpha}))=A(\frac{1}{\alpha+1})$, and
$\sigma^{2n}(A(\frac{1}{\alpha+1}))=A(\alpha)$.

Suppose that $\alpha\in A(\alpha)$ is a solution of (4.1) and
$P(\alpha)=c\alpha+d\in A(\alpha)$ is also a solution of (4.1), where
$P=\left(\begin{array}[]{ll}c&d\\\ 0&1\end{array}\right)$. Then by
$\alpha^{2^{2n}}=B(\alpha)$,
$BP(\alpha)=(P(\alpha))^{2^{2n}}=P(\alpha)^{2^{2n}}=PB(\alpha)$ and

$B^{-1}P^{-1}BP=\frac{1}{a}\left(\begin{array}[]{cc}c^{2}&cd\\\
c+c^{2}+cd&d+1+d^{2}+cd\end{array}\right)$

is a scalar matrix, i.e., $c=1,d=0$. Hence there is a unique solution
$\alpha\in A(\alpha)$ of (4.1), so there are $2^{n}+1$ roots of
$x^{2^{2n}}+x+1$ in $O_{\alpha}$.

Conversely, if $\alpha$ is a solution of (4.1), then
$\sigma^{2n}(O_{\alpha})=O_{\alpha}$.

Consequently, by Lemma 4.2

$|F(\sigma^{2n})|=\frac{2^{2n}-2^{n}-2}{2^{n}+1}=2^{n}-2.$

∎

######  Proposition 4.4.

Let $G=Gal(\mathbb{F}_{2^{6n}}/\mathbb{F}_{2})=\langle\sigma\rangle$. If
$o(\sigma^{i})=6$, then $|F(\sigma^{i})|=0$.

###### Proof.

By $o(\sigma^{i})=6$, $H=\langle\sigma^{i}\rangle=\langle\sigma^{n}\rangle$.
Suppose that $\sigma^{n}(O_{\alpha})=O_{\alpha}$ for $O_{\alpha}\in\Omega$.
Then the group $H$ acts on the partition $\Delta_{\alpha}$ of $O_{\alpha}$ in
(3.12) as follows:

$\displaystyle H\times\Delta_{\alpha}$ $\displaystyle\longrightarrow$
$\displaystyle\Delta_{\alpha}$ $\displaystyle(\sigma^{ni},A(\alpha))$
$\displaystyle\longmapsto$
$\displaystyle\sigma^{ni}(A(\alpha)),i=0,1,\ldots,5.$

Denote by $O_{A(\alpha)}$ the orbit of $A(\alpha)$ under the action the
subgroup $H=\langle\sigma^{n}\rangle$. Then there is a class equation:

$2^{n}+1=|\Delta_{\alpha}|=\sum|O_{A(\alpha)}|.$ (4.3)

For $A(\alpha)\in\Delta_{\alpha}$, $|O_{A(\alpha)}|\in\\{1,2,3,6\\}$ by
$|H|=6$.

Suppose that there is $A(\alpha)\in O_{\alpha}$ such that $|O_{A(\alpha)}|=1$.
Then $\sigma^{n}(A(\alpha))=A(\alpha)$ and $\alpha^{2^{n}}=a\alpha+b$. Hence
$\alpha=\alpha^{2^{6n}}=a^{6}\alpha$, $a=1$, and $\alpha^{2^{2n}}=\alpha$,
which is contradictory.

Suppose that there is $A(\alpha)\in O_{\alpha}$ such that $|O_{A(\alpha)}|=3$.
Without loss of generality, let $\sigma^{n}(A(\alpha))=A(\frac{1}{\alpha})$.
Then

$\alpha^{2^{n}}=\frac{a}{\alpha}+b=B_{0}(\alpha),\alpha^{2^{6n}}=B_{0}^{6}\alpha=\alpha,$

where $B_{0}=\left(\begin{array}[]{cc}b&a\\\ 1&0\end{array}\right)\in
GL_{2}(\mathbb{F}_{2^{n}})$. Hence

$B_{0}^{6}=\left(\begin{array}[]{cc}b^{6}+(ab^{2}+a^{2})(b^{2}+a)&(ab^{2}+a^{2})(b^{3}+ab)\\\
(b^{2}+a)(b^{3}+ab)&(b^{2}+a)(ab^{2}+a^{2})+a^{2}b^{2}\end{array}\right)$

is a scalar matrix and $a=b^{2}\in\mathbb{F}_{2^{n}}^{*}$ or $b=0$.

If $b=0$, then $B_{0}^{2}=\left(\begin{array}[]{cc}a&0\\\
0&a\end{array}\right)$ and $\alpha\in\mathbb{F}_{2^{2n}}$, which is a
contradiction.

If $a=b^{2}\in\mathbb{F}_{2^{n}}^{*}$, then
$(\frac{\alpha}{b})^{2^{n}}=\frac{b}{\alpha}+1=B(\frac{\alpha}{b})$ with
$B=\left(\begin{array}[]{cc}1&1\\\ 1&0\\\ \end{array}\right)$. Since
$(\frac{\alpha}{b})^{2^{3n}}=B^{3}(\frac{\alpha}{b})=\frac{\alpha}{b}$,
$\frac{\alpha}{b}\in\mathbb{F}_{2^{3n}}$, which is a contradiction.

Hence $|O_{A(\alpha)}|$ can not be $1$ and $3$.

Suppose that $|O_{A(\alpha)}|=2$ or $6$ for all $A(\alpha)\in O_{\alpha}$.
Then it is a contradiction in (4.3).

Therefore,

$|F(\sigma^{n})|=0.$

∎

######  Lemma 4.5.

If $\alpha\in\mathbb{S}$ and $\alpha^{2^{6}}=\alpha+b$,
$b\in\mathbb{F}_{2^{n}}$. Then there is an element $c\in\mathbb{F}_{2^{n}}$
such that $(\alpha+c)^{2^{6}}=(\alpha+c)$.

###### Proof.

Let $T^{6n}_{6}:\mathbb{F}_{2^{6n}}\rightarrow\mathbb{F}_{2^{6}}$ and
$T^{n}_{1}:\mathbb{F}_{2^{n}}\rightarrow\mathbb{F}_{2}$ be the trace
functions, then $T^{6n}_{6}(b)=0$ by $b=\alpha^{2^{6}}-\alpha$. Let
$\mathbb{Z}/(n)$ be the residue class ring modulo $n$, then by $\gcd(n,6)=1$
$6\cdot\mathbb{Z}/(n)=\mathbb{Z}/(n)$ and $T^{n}_{1}(b)=T^{6n}_{6}(b)=0$ for
$b\in\mathbb{F}_{2^{n}}$. Let $S=\\{c^{2^{6}}+c:c\in\mathbb{F}_{2^{n}}\\}$,
then $|S|=2^{n-1}$ and $ker(T^{n}_{1})=S$. Hence there is an element
$c\in\mathbb{F}_{2^{n}}$ such that $b=c^{2^{6}}+c$, so
$(\alpha+c)^{2^{6}}=(\alpha+c)$. ∎

######  Proposition 4.6.

Let $G=Gal(\mathbb{F}_{2^{6n}}/\mathbb{F}_{2})=\langle\sigma\rangle$. If
$o(\sigma^{i})=n$, then $|F(\sigma^{i})|=9$.

###### Proof.

By $o(\sigma^{i})=n$, $H=\langle\sigma^{i}\rangle=\langle\sigma^{6}\rangle$.
Suppose that $\sigma^{6}(O_{\alpha})=O_{\alpha}$ for $O_{\alpha}\in\Omega$.
Then the group $H$ acts on the partition $\Delta_{\alpha}$ of $O_{\alpha}$ in
(3.12) as follows:

$\displaystyle H\times\Delta_{\alpha}$ $\displaystyle\longrightarrow$
$\displaystyle\Delta_{\alpha}$ $\displaystyle(\sigma^{6i},A(\alpha))$
$\displaystyle\longmapsto$
$\displaystyle\sigma^{6i}(A(\alpha)),i=0,1,\ldots,n-1.$

Denote by $O_{A(\alpha)}$ the orbit of $A(\alpha)$ under the action of the
subgroup $\langle\sigma^{n}\rangle$. Then there is a class equation:

$2^{n}+1=|\Delta_{\alpha}|=\sum|O_{A(\alpha)}|.$

For $A(\alpha)\in\Delta_{\alpha}$, $|O_{A(\alpha)}|=1$ or $n$ by $|H|=n$,
where $n$ is a prime number. By $n\nmid(2^{n}+1)$, there exists $A(\alpha)$
such that $|O_{A(\alpha)}|=1$.

If $\sigma^{6}(A(\alpha))=A(\alpha)$ for $A(\alpha)\in\Delta_{\alpha}$, then
$\sigma^{6}(\alpha)=a\alpha+b$, $0\neq a,b\in\mathbb{F}_{2^{n}}$. Hence
$\alpha=\sigma^{6n}(\alpha)=\sigma^{6(n-1)}(a\alpha+b)=a^{n}\alpha+k$,
$k\in\mathbb{F}_{2^{n}}$, so $k=0$, $a^{n}=1$. Thus $a=1$ by
$\gcd(n,2^{n}-1)=1$ and

$\alpha^{2^{6}}=\alpha+b.$

By Lemma 4.5, there exists an element $c\in\mathbb{F}_{2^{n}}$ such that
$(\alpha+c)^{2^{6}}=\alpha+c$. If $\alpha+c$ is viewed as $\alpha$, then there
are exactly two roots: $\alpha,\alpha+1$, of $x^{2^{6}}=x$ in $A(\alpha)$.

Moreover, if
$\sigma^{6}(A(\frac{1}{\alpha+\gamma}))=A(\frac{1}{\alpha+\gamma})$ and
$\alpha^{2^{6}}=\alpha$. Then by above,

$(\frac{a}{\alpha+\gamma}+b)^{2^{6}}=\frac{a}{\alpha+\gamma}+b,0\neq
a,b\in\mathbb{F}_{2^{n}}.$

Hence $a^{2^{6}}=a,b^{2^{6}}=b,\gamma^{2^{6}}=\gamma$, and $a=1$, $b=0$ or
$1$, $\gamma=0$ or $1$ by $n(>3)$ a prime. So there are exactly six roots:
$\alpha,\alpha+1,\frac{1}{\alpha},\frac{1}{\alpha}+1,\frac{1}{\alpha+1},\frac{1}{\alpha+1}+1$,
in $O_{\alpha}$ of $x^{2^{6}}=x$.

Conversely, if $\alpha^{2^{6}}=\alpha$, then
$\sigma^{6}(O_{\alpha})=O_{\alpha}$. Thus, $\left|\mathbb{S}\bigcap
O_{\alpha}\right|=6$.

Therefore, $\sigma^{6}(O_{\alpha})=O_{\alpha}$ if and only if there are
exactly six roots in $O_{\alpha}$ of $x^{2^{6}}=x$.

On the other hand,
$\mathbb{F}_{2^{6}}\bigcap\mathbb{S}=\mathbb{F}_{2^{6}}\setminus(\mathbb{F}_{2^{2}}\cup\mathbb{F}_{2^{3}})$
and $\left|\mathbb{F}_{2^{6}}\bigcap\mathbb{S}\right|=54$. Then

$|F(\sigma^{6})|=\frac{\left|\mathbb{F}_{2^{6}}\bigcap\mathbb{S}\right|}{\left|\mathbb{S}\bigcap
O_{\alpha}\right|}=9.$

∎

######  Proposition 4.7.

Let $G=Gal(\mathbb{F}_{2^{6n}}/\mathbb{F}_{2})=\langle\sigma\rangle$. If
$o(\sigma^{i})=2n$, then $|F(\sigma^{i})|=3$.

###### Proof.

By $o(\sigma^{i})=2n$, $H=\langle\sigma^{i}\rangle=\langle\sigma^{3}\rangle$.
Suppose that $\sigma^{3}(O_{\alpha})=O_{\alpha}$. Then the group $H$ acts on
the partition $\Delta_{\alpha}$ of $O_{\alpha}$ in (3.12) as follows:

$\displaystyle H\times\Delta_{\alpha}$ $\displaystyle\longrightarrow$
$\displaystyle\Delta_{\alpha}$ $\displaystyle(\sigma^{3i},A(\alpha))$
$\displaystyle\longmapsto$
$\displaystyle\sigma^{3i}(A(\alpha)),i=0,1,\ldots,2n-1.$

Denote by $O_{A(\alpha)}$ the orbit of $A(\alpha)$ under the action of
$\langle\sigma^{3}\rangle$. Then there is a class equation:

$2^{n}+1=|\Delta_{\alpha}|=\sum|O_{A(\alpha)}|.$

For $A(\alpha)\in\Delta_{\alpha}$, $|O_{A(\alpha)}|\in\\{1,2,n,2n\\}$ by
$|H|=2n$. By $n\nmid(2^{n}+1)$, there exists $A(\alpha)\in\Delta_{\alpha}$
such that $|O_{A(\alpha)}|=1$ or $2$.

If there is $A(\alpha)\in\Delta_{\alpha}$ such that $|O_{A(\alpha)}|=2$, then
$\sigma^{3}(A(\alpha))=A(\frac{1}{\alpha+\gamma})$ and
$\sigma^{6}(A(\alpha))=A(\alpha)$. Without loss of generality, let
$\alpha^{2^{6}}=\alpha$ and $\alpha^{2^{3}}=\frac{a}{\alpha+\gamma}+b,0\neq
a,b\in\mathbb{F}_{2^{n}}$. By

$\alpha=\alpha^{2^{6}}=\frac{a^{2^{3}}}{\frac{a}{\alpha+\gamma}+b+\gamma^{2^{3}}}+b^{2^{3}},$

$b=\gamma^{2^{3}}$, $a=1$, and
$(\alpha+\gamma)^{2^{3}}=\frac{1}{\alpha+\gamma}$. If $\alpha+\gamma$ is
viewed as $\alpha$, then $\sigma^{3}(A(\alpha))=A(\frac{1}{\alpha})$ and
$\alpha^{2^{3}}=\frac{1}{\alpha}$. Hence
$(\frac{1}{\alpha+1})^{2^{3}}=\frac{1}{\alpha+1}+1$ and
$|O_{A(\frac{1}{\alpha+1})}|=1$.

Without loss of generality, let $A(\alpha)\in\Delta_{\alpha}$ such that
$|O_{A(\alpha)}|=1$. Then $\sigma^{3}(A(\alpha))=A(\alpha)$ and
$\sigma^{6}(A(\alpha))=A(\alpha)$, so $\alpha^{2^{6}}=\alpha$ by Proposition
4.6 and $\alpha^{2^{3}}=a\alpha+b$, $0\neq a,b\in\mathbb{F}_{2^{n}}$. Then
$\alpha=\alpha^{2^{6}}=a^{2^{3}}(a\alpha+b)+b^{2^{3}}$, so $a=1$ and $b=1$ by
$\gcd(2^{6}-1,2^{n}-1)=1$. Hence $\alpha^{2^{3}}+\alpha+1=0$.

Suppose that there is also $A(\frac{1}{\alpha+\gamma})$ such that
$|O_{A(\frac{1}{\alpha+\gamma})}|=1$. Then

$\sigma^{3}(\frac{1}{\alpha+\gamma})=\frac{1}{\alpha+1+\sigma^{3}(\gamma)}=\frac{a}{\alpha+\gamma}+b,0\neq
a,b\in\mathbb{F}_{2^{n}}.$

Hence $b=0$, $a=1$, and $\gamma^{8}+\gamma+1=0$, which is contradictory with
$\gamma\in\mathbb{F}_{2^{n}}$.

Hence there is a unique $A(\alpha)\in O_{\alpha}$ such that $|O_{\alpha}|=1$
and there are exactly two roots: $\alpha,\alpha+1$, of $x^{2^{3}}+x+1=0$ in
$O_{\alpha}$.

Conversely, if $\alpha\in\mathbb{S}$ is a root of $x^{2^{3}}+x+1=0$, then
$\sigma^{3}(O_{\alpha})=O_{\alpha}$. Thus, $|\mathbb{S}\bigcap O_{\alpha}|=2$.

Moreover, it is clear that

$\left|\left\\{\alpha\in\mathbb{S}:\alpha^{2^{3}}+\alpha+1=0\right\\}\right|=\left|\left\\{\alpha\in\mathbb{F}_{2^{6}}\setminus\\{\mathbb{F}_{2^{3}}\bigcup\mathbb{F}_{2^{2}}\\}:\alpha^{2^{3}}+\alpha+1=0\right\\}\right|=6.$

Therefore,

$|F(\sigma^{3})|=\frac{6}{2}=3.$

∎

By above propositions, we finally give the proof of Theorem 3.4 as follows.

Proof of Theorem 3.4 For $i$ an integer, denote by $\phi(i)$ the Euler
function. Then, by Remark 3.6(1)

$\displaystyle\sum\limits_{\sigma^{i}\in
G,i=0,\ldots,6n-1}|F(\sigma^{i})|=\sum\limits_{d|6n}|F(\sigma^{d})|\phi(\frac{6n}{d})$
$\displaystyle=$
$\displaystyle|F(\sigma^{0})|\phi(1)+|F(\sigma^{3n})|\phi(2)+|F(\sigma^{2n})|\phi(3)+|F(\sigma^{6})|\phi(n)+|F(\sigma^{3})|\phi(2n)$
$\displaystyle=$ $\displaystyle 2^{3n}+2^{2n}+3\cdot 2^{n}+12n-18.$

By Lemmas 3.3 and 2.8, there are at most $\frac{2^{3n}+2^{2n}+3\cdot
2^{n}+12n-18}{6n}$ extended irreducible binary sextic Goppa codes of length
$2^{n}+1$ over $\mathbb{F}_{2^{n}}$.

##  5\. Conclusion

In this paper, we have obtained an upper bound on the number of extended
irreducible Goppa codes of degree $6$ and length $2^{n}+1$ with $n(>3)$ a
prime number. Our results show that many extended Goppa codes become
equivalent and this supports the idea of mounting an enumeration attack on the
McEliece cryptosystem using extended Goppa codes.

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