Introducing 4D Geometric Shell Shaping for Mitigating Nonlinear Interference Noise

Calligraphic letters $\mathcal{X}$ represent sets. Blackboard bold letters $\mathbb{X}$ denote matrices in which $\bm{x}_{i}$ are row vectors denoting the $i^{\text{th}}$ row. Conditional probability density functions (PDFs) are denoted by $f_{\bm{Y}|\bm{X}}(\bm{y}|\bm{x})$, where $\bm{Y}$ and $\bm{X}$ denote random (4D) vectors and $\bm{y}$ and $\bm{x}$ denote their realizations. Probability mass functions (PMFs) are denoted by $P_{\bm{X}}(\bm{x})$, expectations are denoted by $E[\cdot]$ and $\overline{(\cdot)}$ denotes binary negation. The squared Euclidean norm of a matrix is denoted by $||\mathbb{X}||^{2}=||\bm{x}_{1}||^{2}+\ldots+||\bm{x}_{M}||^{2}$, where $||\bm{x}_{i}||^{2}=x_{1i}^{2}+\ldots+x_{4i}^{2}$. The indicator function is denoted by $\mathds{1}[\cdot]$, which is 1 when its argument is true and 0 otherwise, and $\mathbb{R}$ and $\mathbb{N}$ denote the set of all real and natural numbers, respectively, with $\mathbb{R}_{>0}$ denoting the set of all positive real numbers.

Throughout this paper, we consider a constellation with $N=4$ real dimensions denoted by the $4\times M$ real-valued matrix $\mathbb{X}=[\bm{x}_{1},\bm{x}_{2},\ldots,\bm{x}_{M}]^{\top}$, where $M=2^{m}$ is the constellation cardinality, $\bm{x}_{i}\triangleq(x_{1i},x_{2i},x_{3i},x_{4i})\in\mathbb{R}^{4}$ is a vector denoting the $i$-th constellation point of $\mathbb{X}$ with $\bm{x}_{i}\neq\bm{x}_{j}$ for $i\neq j$. The 2D coordinates of the x- and y-polarization are represented by $x_{1i},x_{2i}$ and $x_{3i},x_{4i}$, respectively, and the rows of $\mathbb{X}$ are labeled using a fixed binary labeling. In other words, the $i$-th symbol $\bm{x}_{i}$ is associated with a unique binary label $\bm{b}_{i}=(b_{1i},b_{2i},\ldots,b_{mi})$.

A number of metrics exist for evaluating the performance of a fiber optical system. For systems based on SMD, the mutual information (MI) is often used. Using Monte-Carlo simulations, the MI can be approximated as

where $P_{\bm{X}}(\bm{x}_{j})$ is the probability of the symbol $\bm{x}_{j}$, $D$ is the number of transmitted symbols and $q_{\bm{Y}|\bm{X}}$ is the auxiliary channel, which is an approximation of the actual channel law $f_{\bm{Y}|\bm{X}}$ from which the samples $\bm{y}_{1},\bm{y}_{2},\ldots,\bm{y}_{D}$ are taken from. We use mismatched decoding [38] in this paper for which $q_{\bm{Y}|\bm{X}}$ in (1) is considered to be the AWGN channel, i.e.,

where $\sigma^{2}$ is the total noise variance of the 4D AWGN channel. The expression in (1) is a modified version of [39, Eq. (30)] that takes probabilities into account (for PS).

For bit-interleaved coded modulation (BICM) systems with BMD and probabilistic shaping, an AIR is the $\text{R}_{\text{BMD}}$ [40, eq. (1)],[3, eq. (6)]. When bit levels are independent, which is the case for conventional uniform signaling, the BMD rate reduces to the well known generalized MI [41, Thm. 4.11, Coroll. 4.12],[42].

We approximate the BMD rate via Monte-Carlo simulations as [3, Eq. (8)]

where $C_{k}$ is the random variable representing the transmitted bit at bit position $k$, $I(C_{k};\bm{Y})$ is the bit-wise MI between $C_{k}$ and output $\bm{Y}$, $c_{k,i}$ are the transmitted coded bits and $L_{k,i}$ are the log-likelihood ratios (LLRs) defined as

where $\mathcal{J}_{b}^{k}$ is the set of constellation point indices with $b\in\{0,1\}$ at bit position $k$.

To evaluate the performance predictions made by the above-mentioned AIRs, post-forward error correction (FEC) bit error rates (BERs) will also be calculated. The system model is shown in Fig. 1 indicating the performance metrics which are considered in this paper.

The constellation $\mathbb{X}$ is typically designed to maximize a certain performance metric [9]. In this paper, $\text{R}_{\text{BMD}}$ is the chosen metric and the optimal constellation is denoted by $\mathbb{X}^{*}$. The resulting optimization problem is defined as

where $\text{R}_{\text{BMD}}$ is given by 4 and 5, and $\mathcal{X}$ is the set containing all $4\times M$ real-valued matrices satisfying a variance (power) constraint, i.e.,

The optimization problem in 6 has four DOFs per constellation point, one for each dimension, resulting in $4M$ DOFs. We call this the unconstrained optimization and denote it by 4D-GS. The DOFs are directly related to the dimensionality $4$ and the constellation cardinality $M=2^{m}$. From this we see that for increasing constellation sizes, the DOFs grow exponentially and the optimization becomes challenging.

In this paper, instead of solving the unconstrained optimization in (6), we define a set of constraints which reduce the DOFs, while minimizing the potential loss in performance. The optimization under these constraints is defined as

where the optimization space is constrained to $\mathcal{X}_{\text{GSS}}\subset\mathcal{X}$. As we will show below, the imposed constraints make the optimization problem in (8) to only have $28$ DOFs instead of $1024$ DOFs for the chosen system $(m=8)$.

In this paper we propose to impose three constraints on the constellation, which we call (i) “uniform $t$-shell division”, (ii) “X-Y symmetry”, and (iii) “orthant¹¹1An orthant is a generalization in $N$-dimensional Euclidean space of what a quadrant is in the 2D plane. symmetry”. We call the optimization under these constraints 4D-GSS. In what follows, we explain these three constraints and how they lead to $28$ DOFs for $m=8$. As we will show in Sec. IV-B, the loss in performance by introducing these three constraints is minimal.

Each of the three constraints mentioned in Sec. II-C is associated with a part of the binary labeling, which is considered to be fixed. Fig. 2 provides an example of how the bit allocation is predefined under the considered constraints in a 4D constellation with $m=8$ bits and $t=4$ shells. The left side of Fig. 2 shows 16 constellation points belonging to a single orthant with their corresponding binary labels. The binary labels are grouped into sets of bits referred to by (a) through (d). The four bits in (a) define an orthant. In Fig. 2 only the first orthant is shown and thus, all $\bm{x}_{i}$ have the same binary label (all zeros) for (a). The two bits in (b) determine the shell, with each shell having the same amount of points ($2$ in this case). The bits in (c) select between two points on a shell, and (d) selects between two X-Y symmetric points.

The three GSS constraints described above reduce the number of 4D constellation points to be optimized from $2^{8}=256$ to only $2^{m-5}=8$ (filled circles in Fig. 2) with a reduction in DOFs from $1024$ to $28$. In what follows, we formally describe the set $\mathcal{X}_{\text{GSS}}$ together with two symmetry operations that make up these three GSS constraints.

The shell constraint forces $2^{m-5}$ constellation points to be equally divided on the $t$ concentric 4D shells in the all positive $\mathbb{R}^{4}_{>0}$ space (first orthant). This uniformly divided part of this constraint is provided by 9b. The upper limit for $t$ is given by $2^{m-5}$ (see (10)), which is equivalent to having a dedicated shell for each constellation point. In the case of $t=1$, this constraint turns into a constant modulus constraint, effectively creating a generalization of the format proposed in [10]. By forcing each point to be on top of a certain shell, we take away one additional DOF per constellation point, such that $3$ DOFs per constellation point remain. However, $t$ extra DOFs are added due to the number of the shells. This results in $3\cdot 2^{m-5}+t$ DOFs. The advantage of having an integer power of two ($t=2^{p}$) shells is that $p$ out of $m-5$ bits can be used to select the shell. In Fig. 2, $p=2$. This offers the possibility of achieving rate-adaptivity by adding PS on top of GSS using the PAS architecture [1], which is called hybrid shaping. In this case, only the bits which select the shell are shaped. The remaining $m-p-5$ uniform bits select the specific constellation points on a shell.

In the remainder of this section we will assume an identical setup to the one in the example in Fig. 2. As a result $|\mathcal{X}_{\text{GSS}}|=8$ and the $8$ constellation points are labeled by $\bm{l}_{i}=[b_{5i},b_{6i},b_{7i}]\in\{0,1\}^{3}$.

X-Y symmetry mirrors the points in the $\mathcal{X}_{\text{GSS}}$ set over its two polarizations. A single bit added to the labeling end is used to distinguish between the two X-Y symmetric points. The X-Y symmetry also ensures identical average transmit power over the two polarizations. In Fig. 2, this mirroring causes for example $\bm{x}_{7}$ to become $\bm{x}_{15}$ and the extra bit added to the binary labels is $d$ and $\overline{d}$, resp. After applying the X-Y symmetry operation to $\mathcal{X}_{\text{GSS}}$, we must apply the orthant symmetry operation, defined as follows.

The first orthant $\mathbb{R}_{>0}^{4}$ in Example 1 only considers 4D symbols with all their components to be positive. Orthant symmetry is achieved by mirroring the $2^{m-4}$ constellation points (after applying the X-Y symmetry operation) with respect to the origin along the axes of the $4$ dimensions and by changing the bits $b_{1},b_{2},b_{3},b_{4}$ (see 12), where each of these bits is associated with the sign of one real dimension. An advantage of assigning bits in this manner is that it ensures the orthants themselves are Gray-labeled (adjacent orthants differ only in one bit), which provides higher AIRs compared to other labeling strategies when used in BICM systems [43]. In the context of optical communications, the mirroring procedure was first used in [11, Sec. II-B].

By using the three constraints described above, the DOFs are reduced from an unconstrained $1024$ $(4\cdot 2^{8})$ to $28$ $(3\cdot 2^{8-5}+4)$. Fig. 3 compares the DOFs when using 4D-GSS compared to the unconstrained case of 4D-GS for increasing constellation sizes. Table I shows the DOFs and the number of 4D shells for a number of different constellation types at a fixed SE of $m=8$. This table also shows the amount of discrete 4D energy levels. One of the properties of GSS constellations is the ability to directly and efficiently control the energy of symbols in 4D (due to indexing a power of two shells), which is much more difficult in conventional QAM and GS constellations.

For designing constellations with high nonlinear tolerance in mind, a suitable transmission scenario needs to be chosen which is expected to have high NLIN. For this reason an unamplified 400ZR link is chosen [37], for which the transmitted constellation is PM-16QAM, which matches the desired cardinality of the considered 4D-GSS constellation.

The considered system transmits a dual-polarized, single channel waveform of $2^{20}$ symbols with a fixed random seed over a single span of standard single-mode fiber (SSMF). This setup uses the Manakov equation as the fiber model [44, Sec. IV] and is simulated via the split-step Fourier method (SSFM) with 1000 steps per span using uniform step size and fiber parameters $\alpha=0.2$ dB/km, $\beta_{2}=-21.68$ ps²/km and $\gamma=1.20$ (W$\cdot$km^-1). Increasing the number of steps per span above 1000 has been verified to provide identical results for the highest launch powers and distances considered in this paper. The symbol rate is matched to the 400ZR specification at 59.84 Gbaud. At the transmitter, the generated symbol sequences are upsampled to 4 samples per symbol and pulse shaped using a root-raised-cosine filter with a roll-off of 1%. At the receiver, the waveforms are ideally compensated for chromatic dispersion, matched filtered, downsampled to 1 sample per symbol and corrected for phase rotation by utilizing cross-correlation between the input and output symbol sequences. Demapping is done in 4D using 5 to calculate the LLRs.

Since an unamplified link is used, noise sources due to transmitter and receiver impairments are emulated instead of simulating optical erbium-doped fiber amplifier (EDFA) amplification. For this, worst-case design parameters from the 400ZR specification will be used as a guideline.

At the transmitter side, the 400ZR specification requires the in-band optical signal-to-noise ratio (OSNR) to have a minimum value of 34 dB/0.1nm. At the receiver side, the concatenated FEC scheme in the 400ZR standard is specified to operate error-free (post-FEC BER = $10^{-15}$) when a pre-FEC BER of $1.25\cdot 10^{-2}$ or lower is achieved. The receiver sensitivity requires least $-20$ dBm of power to be present at the input of the receiver. These previous two conditions combined with the minimum of 34 dB OSNR at the transmitter guarantee error-free operation.

In simulations, the transmitter is emulated by adding AWGN to the transmitted waveform such that the OSNR value is equal to the in-band OSNR limit. For the receiver it is possible to calculate the necessary AWGN addition using [45, Eq. (18)]

where $P_{e}$ is the targeted BER, $Q(\cdot)$ is the Q-function and $E_{b}/N_{0}$ is the accompanying SNR per bit under a Gray-coded $M$-QAM assumption in an AWGN channel. For 16QAM and a BER of $1.25\cdot 10^{-2}$, 15 provides an $E_{b}/N_{0}$ of 7.53 dB, which translates to a signal-to-noise ratio (SNR) of 13.5 dB. Assuming that an input power of $-20$ dBm is present at the receiver in a back-to-back scenario, the amount of noise power added in the receiver is equal to $-33.5$ dBm. This is added as AWGN after simulating the optical fiber.

400ZR uses a concatenated FEC scheme as defined in [37, Sec. 10] consisting of an outer staircase code (SCC) with hard-decision (HD) decoding of rate $0.937$, and an inner Hamming code with soft-decision (SD) decoding of rate $0.930$, resulting in a total overhead of $14.8\%$. The SCC in 400ZR is defined to be taken from [46, Annex A], which describes a $(255,239)$ SCC with blocks of size $512\times 510$ and a $(1022,990)$ Bose-Chaudhuri-Hocquenghem (BCH) code as the component code. The FEC code is a double-extended $(128,119)$ Hamming code using a parity-check matrix as described in [37, Sec. 10.5].

Instead of implementing the full concatenated FEC as described above, in this paper we only implemented the Hamming code and a SD decoder based on a Chase-I decoder [47]. A post-FEC BER after the Hamming decoder of $4.5\cdot 10^{-3}$ is targeted, which is the required pre-FEC BER for the $(255,239)$ SCC to achieve a BER of $10^{-15}$ at the output. The scrambling and interleaving steps are approximated by using a bit-wise fixed random permutation after FEC encoding and the inverse operation before FEC decoding. The full system diagram is shown in Fig. 4.

In 17, the optimization problem for determining $\mathbb{X}^{*}$ was defined. If the constraints from Sec. II are applied, the optimization only needs to be performed for the 28 resulting DOFs. To enforce the shell constraints, each point out of the 8 points in $\mathcal{X}_{\text{GSS}}$ is now represented in spherical coordinates $(r_{i},\theta_{j},\phi_{j},\omega_{j})$, where $i=1,2,3,4$ and $j=1,2,\ldots,8$. The optimization problem can now be defined as

where the parameters $(\bm{r},\bm{\theta},\bm{\phi},\bm{\omega})$ are constrained such that the corresponding points are in $\mathcal{X}_{\text{GSS}}$.

Since there are no existing constellations which strictly adhere to the chosen constraints, selecting an initialization for the optimization procedure is not straightforward. It was determined on a trial-and-error basis that there were no clear differences in resulting performance after optimization between randomly initialized constellations and constellations which were initialized with a distinct structure. For that reason it was chosen to initialize all parameters at the halfway point between the upper and lower bounds, which are shown in 16.

Optimization over the system in Fig. 4 is performed using a patternsearch optimizer [48], which is a derivative-free multidimensional optimization algorithm. Patternsearch automatically finds the optimal way to spread out the constellation. During optimization, the number of transmitted symbols is lowered to $2^{18}$ using a different random seed compared to the results generation. The number of steps per span for the SSFM is kept at 1000. To enhance stability during the optimization procedure, fixed random seeds are used for the sequence generation and AWGN noise additions.

Conventional PM-16QAM, as used in the 400ZR standard, is considered as a baseline. Next to that, PS is applied on top of PM-16QAM by shaping each real dimension with an ideal amplitude shaper, where amplitudes are randomly drawn from a predefined distribution, which is identical over all dimensions. The distribution is optimized for each pair of launch power and transmission distance. This optimization is performed using the same patternsearch optimizer as Sec. III-D. Since PM-16QAM only has two amplitude values per dimension, the optimization effectively only has a single DOF. The proposed 4D-GSS-4 constellations are also optimized and evaluated for each pair of launch power and distance. Lastly, a 4D sphere packed constellation is also considered, specifically the 256 point Welti constellation (w4-256) [49]. Sphere packed constellations are optimal for the AWGN channel for a given SE and will provide insight into the maximum expected performance in the linear region later in this paper. For $\text{R}_{\text{BMD}}$ evaluation, w4-256 uses the optimized binary labeling from [50].

It is well known that symbol-level interleaving, as described in one of the DSP steps in the 400ZR standard [37, Sec. 11.1], has a negative impact on the performance of PS constellations implemented with a finite blocklength amplitude shaper [51]. This is due to symbol-level interleaving breaking time-domain structures of probabilistically shaped symbol sequences and can be mitigated by employing extra pre- and post-interleaving steps in the DSP chain [52]. In this paper, the coded bits of all considered constellations are independent. PM-16QAM-PS uses an ideal amplitude shaper, which does not implement an actual shaping algorithm, resulting in independent bits. 4D-GSS and the two other baselines all use uniform signaling and are thus also not affected by such dependencies. Since all considered constellations are not affected by symbol-level interleaving, only a bit-wise permutation as discussed in Sec. III-C was implemented.

Fig. 5(a) shows $\text{R}_{\text{BMD}}$ results for a distance sweep between 120 and 180 km. In the inset of Fig. 5(a), 4D-GSS-4 achieves $1.6\%$ gain in $\text{R}_{\text{BMD}}$ and $1.7\%$ gain in distance compared to PM-16QAM around 160km. These gains represent the increase in data rate and distance as a result of the shaping gain. Since in this specific scenario, the 400ZR link is loss-limited, results are shown at optimal launch power until the distance is too large to satisfy the received power requirement of at least $-20$ dBm. The point after which this occurs is indicated by solid markers. Beyond these markers, the constellations are pushed to launch powers above the optimal value to satisfy the received power requirement. In Fig. 5(a), it is shown that the distance for which 4D-GSS-4 can operate at optimal launch power is approximately $5$ km larger than the other considered constellations. This increase in maximum optimal launch power is also reflected in Fig. 5(b) where 4D-GSS-4 has $0.5$ dB higher optimal launch power ($P^{*}$) compared to the baselines and hence, the highest nonlinear tolerance among the considered schemes. The region where NL tolerance is observed is indicated in yellow. In the linear domain, 4D-GSS-4 has similar performance to PM-16QAM and loses in performance compared to PM-16QAM-PS. Even though an optimized binary labeling is used for the w4-256 constellation, since it is not designed to maximize $\text{R}_{\text{BMD}}$, it performs poorly.

To evaluate the performance losses induced by the GSS constraints, optimized 4D constellations which have only orthant symmetry as the constraint (denoted with 4D-OS) are evaluated around the optimal launch power. It is shown in Fig. 5(b) that removing the shell constraints and X-Y constraint from 4D-GSS-4 has a negligible impact on performance. Furthermore, it has been shown in [11, Fig. 6] that lifting the orthant symmetry constraint has a marginal impact on performance. All this indicates that the proposed symmetry constraints as used in 4D-GSS-4 have very little impact on total performance, but do reduce the optimization complexity significantly. Fig. 5(b) also includes MI results for 4D-GSS-4 (denoted by 4D-GSS-4 MI). This indicates the theoretical upper limit for the $\text{R}_{\text{BMD}}$ where it is clear that quite a large gap still exists between the $\text{R}_{\text{BMD}}$ and the MI for 4D-GSS-4.

Possible explanations of the increased nonlinear tolerance of 4D-GSS-4 can be observed from Fig. 6, which shows the peak-to-average power rating (PAPR) (in linear units) and the fourth order standardized moment (i.e. the kurtosis²²2The kurtosis of a distribution depends on the dimensionality [53, Sec. III], where maximum kurtosis is achieved for a Gaussian distribution. Maximum kurtosis values for 2D and 4D constellations are $2$ and $1.5$ respectively. In this paper, only 4D kurtosis values are shown. It is also common practice to compare the kurtosis of an $N$-D distribution to that of an $N$-D univariate normal distribution. This is typically done by defining ‘excess kurtosis’, which is the kurtosis of the distribution minus the kurtosis of the Gaussian distribution, then comparing it to zero.) of the considered constellations. PAPR is a rough indicator for evaluating nonlinear tolerance [11, Sec. II-D] and also influences the amount of distortion resulting from limited equipment dynamic range and linearity, especially in orthogonal frequency division multiplexing transmission [54, Sec. V-A]. Kurtosis is considered to be more directly related to nonlinearities, where channel input distributions with high kurtosis lead to higher NLIN power [3, Sec. III-B]. Fig. 6(a) shows that PM-16QAM has a PAPR of 1.8, while 4D-GSS-4 constellations have a PAPR of 1.25 on average over the considered distances, which is a reduction of $31\%$ compared to PM-16QAM. Similarly, Fig. 6(b) shows a decrease in the kurtosis from 1.16 for PM-16QAM to 1.10 for 4D-GSS-4, which is a reduction of $5\%$.

Fig. 7 shows the post-FEC BER of 4D-GSS-4 compared to PM-16QAM when the inner Hamming code is SD-decoded as described in Sec. III-C, together with HD decoding of the same code. The outer SCC has a FEC limit of $4.5\cdot 10^{-3}$, which is used as the minimum required BER after the Hamming code for error-free operation. A gain of $2\%$ in transmission distance between 4D-GSS-4 and PM-16QAM is achieved at the SCC-FEC limit, which is very close to the observed $1.6\%$ gain in $\text{R}_{\text{BMD}}$. The HD-decoded Hamming code shows similar gains between the two constellation types but cannot satisfy the SCC-FEC limit over similar distances. When only the Hamming codes are considered without the outer SCC, gains increase to in-between $2.5\%$ and $3.3\%$ depending on the specific distance and code, as indicated by the $10^{-5}$ BER line.

It was observed in Fig. 5(b) that the $\text{R}_{\text{BMD}}$ for the optimized 4D-GSS-4 constellations was consistently lower than the MI by about $0.06$ bits/4D-sym. This could indicate an issue with the binary labeling since PM-16QAM(-PS) did not show such a gap. To find the potential upper performance limit of 4D-GSS-4, the optimization procedure was repeated using the MI as the performance metric using 1 for evaluating the MI. This results in the following optimization problem

Fig. 8(a) shows MI results for a distance sweep between 120 and 180 km. The inset of Fig. 8(a) shows gains of $3\%$ in distance and $2.5\%$ in MI for 4D-GSS-4 vs. PM-16QAM and is slightly outperformed by w4-256. Again, Fig. 8(b) shows that 4D-GSS-4 has the largest optimal launch power. Moreover, due to the rapidly-vanishing MI of w4-256, 4D-GSS-4 outperforms w4-256 at very high powers ($P>14$). However, for lower powers ($P<14$), w4-256 achieves larger MI than 4D-GSS-4. The observed gap in performance for 4D-GSS-4 when comparing MI to $\text{R}_{\text{BMD}}$ indicates a possible issue where the proposed GSS framework does not provide a structure suitable for a good binary labeling.

Results in Fig. 9 show similar trends to Fig. 6, with the main difference being that the MI optimized 4D-GSS-4 constellations have even lower average PAPR and kurtosis. The difference in PAPR against PM-16QAM increases from $31\%$ to $33\%$, whilst the difference in kurtosis increases from $5\%$ to $6\%$. Against w4-256, 4D-GSS-4 shows a reduction in PAPR of $10\%$ and a reduction in kurtosis of $2\%$, which could contribute to 4D-GSS-4 gaining performance in terms of MI over w4-256 for launch powers larger than $14$ dBm.

To investigate possible causes of the binary labeling penalty for 4D-GSS-4, we look at the bit-wise MI $I(C_{k};\bm{Y})$. Fig. 10 compares the bit-wise MI of 4D-GSS-4, PM-16QAM and w4-256 at the optimal launch powers. The individual bits are denoted by $b_{i}$ for $i=1,\ldots,m$. For PM-16QAM, the bits are reordered such that the bits which determine the signs are the first four bits (same as 4D-GSS-4). This does not effect the performance since PM-16QAM is a Cartesian product of four independent pulse amplitude modulation (PAM)-4 constellations. As a result, this also implies symmetry across all four dimensions and thus, PM-16QAM also has orthant symmetry.

In the binary labels, the bits $b_{1}$ through $b_{4}$ determine the orthant for both PM-16QAM and 4D-GSS-4. The bits $b_{5}$ through $b_{8}$ determine the amplitude of each of the PAM-4 signals for PM-16QAM. For 4D-GSS-4, the bits $b_{5}$ and $b_{6}$ select the shell, bit $b_{7}$ selects between 2 points on a shell, and bit $b_{8}$ selects between X-Y symmetric points.

In terms of bit-wise MI, $b_{1}$ through $b_{4}$ perform identically for both PM-16QAM and 4D-GSS-4, which is expected due to both constellations employing orthant symmetry. Bits $b_{5}$ trough $b_{8}$ have larger bit-wise MI for 4D-GSS-4 compared to PM-16QAM, where bit $b_{7}$ has the lowest value. This suggests that the proposed structure combined with the chosen constellation cardinality does not allow for a good labeling of $b_{7}$. Lastly, as expected, w4-256 has much worse performance in general compared to the other two constellations since the structure of this constellation is not optimized for allowing a good binary labeling at all.