## 3 Experimental Setup and Experimental Results  
The experimental setup is shown in Figure 3. The SS 16-4D LDPC-coded generator is introduced before in Figure 2, the output sequences are sent to a Keysight arbitrary waveform generator (AWGen) with sampling rate of 120 GSa/s. Then, the analog signals are sent to an I/Q modulator. The laser source we used is a 10 kHz-linewidth, continuous-wave, tunable source with center frequency of 193.4 THz. The optical signal after the I/Q modulator is amplified by an erbium-doped fiber amplifier (EDFA) of 6 dB noise figure. The boosted signals are coupled with the amplified spontaneous emission (ASE) noise by a 2 \(\times\) 2 3 dB coupler. We employed a variable optical attenuator (VOA) after the ASE noise source to emulate different optical SNR (OSNR) and then send the signal to the 40 km fiber link for the metro network communication. At the receiver side, we first applied a tunable filter (TF) to obtain the signals with our target frequency, and after amplifying the signal with EDFA the signals are sent to the coherent receiver, with the integrated coherent receiver (ICR) board and a 100 GS/s sampling rate oscilloscope from Tektronix, followed by the correlation/matched filter decision circuitry. After the correlation detector we also use the correlation approach to figure out the beginning of the LDPC-coded block (FEC frame). Then, we sent the resulting projections along the SS basis functions to the LDPC decoder to analyze the post-BER performance. The structure of our correlation detector is shown in Figure 4. Both real and imaginary parts of the received signals are split and each branch is multiplied by each order SS to calculate the correlation, followed by summation operation to implement the integration operation. We also figured out the beginning of the LDPC-coded block before the splitter. Then, we sent the resulting projections along the SS basis functions to the LDPC decoder to analyze the post-FEC BER performance. We can see the only DSP we implemented is just the correlation calculation, which strictly speaking does not belong to the DSP, and we use only 3-5 pilot symbols to perform the group sign recovery. Figure 3: Experimental setup for SS-based 16-4D LDPC coded modulation scheme. OADM: optical add-drop multiplexer emulator. The structure of our DSP progress is shown in Figure 5. The received sequences are collected from the real-time oscilloscope (Tektronics with sampling rate of 100 GSa/s). Given that the sampling rates of AWGen and oscilloscope are different, we need to do the resampling first. After that we use the correlation calculation to do the synchronization, to find out the starting point of the codeword. The algorithm we used for the equalization is the least mean squares (LMS) algorithm. Then, we do the phase recovery, using the 4 th power algorithm. After this DSP progress, we normalized the signal and then calculate the log-likelihood ratios (LLRs) for LDPC decoding. The transmitted and received waveforms in a back-to-back (b2b) configuration are shown in Figure 6 with 250 samples. The red curve represents our generated sequence of I channel, while the blue curve is our received sequence without any noise loaded for I channel. We can see our sequence is recovered efficiently. Figure 4: The configuration of the correlation detector. Figure 5: The configuration of the DSP progress for 16 QAM. The pre-FEC and post-FEC BER performances of SS-4D modulation and the traditional 16 QAM are summarized in Figure 7a,b. We used both 120 GSa/s and 92 GSa/s Keysight AWGens to implement the 32 samples and 24 samples per Slepian sequence, respectively. The baud rate is 120 G/32 = 3.75 GBaud for 32 samples and 92 G/24 = 3.833 GBaud for 24 samples, respectively; while the baud rate of 16 QAM is also 3.75 GBaud. The length of the codeword we used for both SS and 16 QAM is 5648 bits, with 4239 information bits and 1409 parity check bits, thus, the code rate is 0.75.