Short-Span PD4L Scheme: Polarization in Coding & TDI

• Short-Span PD4L Scheme is an advanced architecture that employs polarization-driven techniques to optimize channel reliability in polar coding and minimize delay in time-delay interferometry.
• In polar coding, the scheme prunes the polarization tree by eliminating the least reliable indices from the generator matrix, achieving performance gains up to 0.25 dB in IoT and eMBB scenarios.
• In gravitational-wave detection, PD4L TDI designs ensure minimal null frequencies and reduced aliasing, leading to improved phase coherence and reliable parameter estimation in high-frequency analyses.

The Short-Span PD4L Scheme denotes a class of advanced architectures employing polarization-driven (PD) methodologies to achieve robust performance in systems requiring minimized latency, reduced span, and superior handling of impairments (both in classical and quantum communications, as well as time-delay interferometry for gravitational-wave observatories). The "PD4L" notation has arisen in two distinct, highly technical domains: (1) as a shortening technique for polar codes using polarization orderings, and (2) as a minimal time-span, second-generation time-delay interferometry (TDI) configuration with reduced data span and null frequencies. These schemes share a defining feature: performance optimization is accomplished through careful exploitation of polarization (either channel polarization in coding or interferometric link polarization)—always within a structural span of four units, hence the "4L."

1. Channel Polarization and PD-Based Shortening Methodologies

In polar coding, polarization transforms a set of communication channels into a mix of highly reliable and highly unreliable subchannels. The PD shortening scheme, particularly as developed for the Short-Span PD4L application, operates by directly linking the generator matrix’s row indices to the polarization indices of the synthesized channels. Given a block length $n=2^\ell$, the Bhattacharyya parameters $Z(W_n^{(i)})$ for each synthetic channel are recursively estimated using Gaussian Approximation, with:

$$Z(W_n^{(2i-1)}) = \phi^{-1}(1 - [1 - \phi(Z(W_{n/2}^{(i)}))]^2)$$

$Z(W_n^{(2i)}) = 2 \cdot Z(W_{n/2}^{(i)})$

where $\phi(x)$ is a carefully defined piecewise function. After obtaining the polarization vector $b = [Z(W_0), \ldots, Z(W_{n-1})]^T$, indices are sorted so that channels are ordered by reliability, and the least reliable $(n-n')$ indices are selected as the shortening set $p$.

Rows and columns corresponding to $p$ are then eliminated from $G_n$ to yield the shortened generator matrix $G_{n'}$, ensuring that the ordering among remaining channels is preserved. This mechanism prunes the "polarization tree," focusing code resources on the channels most likely to support robust transmission, and is ideal for low-latency, short-block-length systems ("Short-Span PD4L Scheme") (Oliveira et al., 2018).

2. Minimal Time-Span, Null Frequency, and Delay Engineering in PD4L TDI

In time-delay interferometry for space-based gravitational wave (GW) detection, the PD4L scheme is instantiated as a set of second-generation TDI observables with a maximal delay of $3L$ (where $L$ denotes the light travel time along a spacecraft arm), and a total observable span of $4L$. PD4L combines first-generation Beacon and Monitor TDI sequences, exemplified by (for PD4L-1):

$$\overrightarrow{1\,2\,3\,2}\;\overleftarrow{2\,1\,2}\; \overrightarrow{2\,3\,2\,1}\;\overleftarrow{1\,3\,2\,3}\; \overrightarrow{3\,1\,3}\;\overleftarrow{3\,2\,3\,1}$$

The minimal null frequencies—present only at integer multiples $f_\text{null}=m/L$—contrast sharply with the higher-order nulls of Michelson-type TDIs, reducing the deleterious effect of frequency notches and aliasing at high Fourier frequencies. The reduced time span (from $8L \to 4L$) yields practical benefits: minimized data segment margin losses, reduced high-frequency aliasing, and significantly less elongated signal tails (Wang, 6 Feb 2025, Wang, 24 Jul 2025).

3. Spectrum Distance and Performance Metrics

The efficacy of PD-based shortening in polar codes is quantitatively assessed using Spectrum Distance (SD), a metric that measures the weighted Hamming distance spectrum across polarization tree branches. For $l$ polarization levels and $Q$ shortened bits,

$$d = \sum_{k=0}^{l} P_1(l, k, Q) \cdot k, \quad \lambda = \sum_{r=0}^{l} P_0(l, r, Q) \cdot r,$$

where $P_1(l, k, Q)$ and $P_0(l, r, Q)$ quantify the probability distributions for ones and zeros, respectively, and are computable via binomial coefficients. PD-based short-span codes exhibit uniformly superior SD values relative to Column Weight (CW) and Reversal Quasi-Uniform Puncturing (RQUP), with simulation results demonstrating up to 0.25 dB (IoT scenario) and 0.20 dB (eMBB scenario) performance gain in BER/FER, and empirical SD examples as follows:

Scenario	SD_PD	SD_RQUP	SD_CW
IoT (n=512)	4.53	4.46	4.43
eMBB (n=2048)	5.46	5.44	5.43

A higher SD implies a more favorable distance spectrum and enhanced error-rate performance (Oliveira et al., 2018).

4. High-Frequency GW Analysis and Parameter Inference

In GW data analysis, the PD4L TDI channels produce minimal and uniformly spaced nulls, leading to smoother noise power spectral densities and signal responses—particularly beneficial for high-frequency analysis (where $fL \gtrsim 0.1$). During parameter inference for chirping GW signals from massive binary black hole coalescences, the PD4L configuration produces quasi-orthogonal science channels (A, E, T) with Gaussian posterior distributions and low systematic bias. In contrast, prior relay/Michelson TDI schemes with more frequent nulls and longer delays exhibit significant high-frequency aliasing and waveform modulation artifacts, which can degrade parameter recovery (Wang, 6 Feb 2025, Wang, 24 Jul 2025).

Additional advantages include:

• Reduced loss of data at segment boundaries (due to short span)
• Strong suppression of time-domain signal tails
• Improved phase coherence in rapidly evolving GW signals

5. Operational Limitations and Implementation Challenges

Although the PD4L approach yields operational benefits, certain limitations are present:

• Noise stability: The null channel ($T_{PD4L}$) in the PD4L TDI is highly stable, but the science channels demonstrate greater sensitivity in their power spectral derivatives to arm length variations, especially at $u<0.1$.
• Interpolation precision: The aggressive cancellation of signals and noise at low frequencies increases the demand for high-precision (ideally, higher-order) interpolation to reconstruct delayed samples accurately.
• Limited data duration for robust noise inference: Empirical results support reliable noise parameter estimation in PD4L channels for data sets up to four months. For longer durations, the lower stability may become a limiting factor, especially compared to TDI schemes employing the $C^{12}_3$ null stream (Wang, 6 Feb 2025).

6. Comparative Redundancy, Correlation, and Future Applicability

Comparative studies demonstrate that, while TDI schemes (Michelson, relay, and PD4L) exhibit near-identical sensitivity and channel redundancy at low frequencies (owing to strong correlation among their quasi-orthogonal stations), the distinctions at high frequencies are pronounced. Configurations with longer time spans (6L, 8L) display more nulls, greater aliasing, and fluctuating noise spectra, in contrast to the PD4L’s compactness and stability. The quasi-orthogonal channels (A, E) of these TDIs are mathematically related by rotation matrices with minor rescaling, revealing fundamental information redundancy (Wang, 24 Jul 2025).

For GW observatory pipeline design and waveform modeling, the short-span PD4L configuration is a strong candidate due to its minimal nulls, reduced aliasing, and stabilized noise response—favorable characteristics for both frequency-domain analysis and (because all TDI channels become equivalent in time-domain analysis) flexible interoperability.

7. Summary and Domain-Spanning Implications

The Short-Span PD4L Scheme, as developed across multiple research tracks, constitutes an advanced design pattern exploiting polarization orderings or rapid delay compositions ("span 4L") to achieve enhanced coding, communication, or interferometric outcomes. In polar coding, PD-based shortening yields rate-compatible codes with superior SD—well-suited to rigorous, low-latency applications such as 5G scenarios. In space-based GW detection, the PD4L TDI channels achieve superior high-frequency performance, resilience to aliasing, and robust parameter inference metrics, subject to the caveat of interpolation demands and finite stability at extended durations.

A plausible implication is that short-span, polarization-driven schemes such as PD4L are likely to play a central role in both next-generation wireless/optical communication systems and GW observatory architectures where spectrum, noise, and latency constraints critically interact.