Core-Spreading Technique

• Core-spreading technique is a family of algorithmic strategies that optimize the distribution and transformation of key information elements for robust system performance.
• It employs methods like bit-flip descent for GNSS/CDMA code design and DPSS-based spreading in OFDM systems to effectively reduce interference and improve signal integrity.
• In distributed settings, core-spreading enables resilient consensus using localized updates and tunable parameters, balancing convergence speed with disturbance tolerance.

Core-spreading techniques encompass a family of algorithmic and coding design strategies that optimize the distribution or transformation of core information elements—such as code symbols, graph states, or subcarrier resources—to achieve system-level objectives such as robust communication, interference mitigation, or resilient distributed computation. Applications span large-scale sequence optimization in GNSS and PNT systems, multi-user interference suppression in OFDM-based integrated sensing and communications, and self-stabilizing distributed algorithms in aggregate computing. Core-spreading unifies these domains through the objective of shaping or propagating signals or information so as to achieve collective performance guarantees under constraints inherent to the application domain.

1. Optimization of Spreading Codes in GNSS and CDMA Systems

The core-spreading technique for large-scale GNSS code optimization addresses the selection of binary code families $\{ s_i \}_{i=1}^n$ of length $T$ with the aim of minimizing both aperiodic autocorrelation sidelobes and pairwise cross-correlation magnitudes. The objective is quantified by the $\ell_1$-type cost functional

$$C(s_1,\dots,s_n) =\sum_{i=1}^n\sum_{\tau\neq0}\bigl|R_{s_i s_i}(\tau)\bigr| \;+\;\sum_{1\le i<j\le n}\sum_{\tau=0}^{T-1}\bigl|R_{s_i s_j}(\tau)\bigr|,$$

where $R_{s_i s_j}(\tau)$ is the (a)periodic cross-correlation between codes $i$ and $j$ at lag $\tau$.

The bit-flip descent method incrementally optimizes this cost by selectively flipping code bits: at each iteration, a small, randomly-sampled subset of candidate bits $M \ll nT$ is considered, and the bit whose flip yields the largest negative change in $C$—computed efficiently via incremental formulas that localize changes to correlations involving only the modified code—is selected and updated if the cost is reduced. This greedy, sample-driven descent permits optimization over massive codebooks (e.g., $n=100$, $T=4092$, total bits $> 400$k) at practical computational cost, leveraging parallel computation and adaptive sample sizing (Yang et al., 2024).

The framework generalizes to sequence design problems for CDMA, MIMO radar, and future GNSS signals, and admits extensions such as constraint enforcement by forbidding illegal flips and escape mechanisms from local minima via restarts or simulated annealing.

2. Spreading in OFDM-Based Multi-User ISAC and Interference Mitigation

Core-spreading in the context of multi-user OFDM-based ISAC systems specifically targets the suppression of inter-band (IB) cross-correlation interference. While OFDMA with disjoint subcarrier sets guarantees mutual channel orthogonality under periodic (CP-protected) cross-correlation, the need for aperiodic correlation in ranging operations induces nonzero spectral leakage between non-overlapping bands, thus elevating the integrated sidelobe level (ISL).

The core-spreading layer, realized as an orthogonal transformation (typically truncated discrete prolate spheroidal sequences, DPSS) applied to each user’s subcarrier QAM symbols, sequentially concentrates the energy within the assigned band while minimizing sidelobe leakage. The associated spreading matrices $\mathbf{P}^{(i)}$ for user $i$ are constructed by taking the leading $\lfloor \eta L \rfloor$ DPSS eigenvectors (with $0 < \eta \le 1$), where $L$ is the subcarrier block size and $\eta$ is the spectral utilization factor. The impact on IB energy $E_{IB}$ is analytically upper-bounded by inter-band Frobenius norms, showing explicit reduction relative to unspread OFDM (Said et al., 4 May 2025).

The principal trade-off is between ISL suppression (favoring small $\eta$) and data throughput (favoring $\eta \approx 1$); simulation results confirm a regime near $\eta = 0.9$ yields up to $4$–$6$ dB ISL reduction with limited capacity loss, enabling robust localization in spectrum-constrained multi-user environments.

3. The Core-Spreading Technique in Distributed Information Propagation

In aggregate computing, the core-spreading technique refers to a distributed algorithmic block for resilient information dissemination over a network. Each node $i$ maintains a state $x_i(t)$, iteratively updated via local Bellman-Ford-style minimizations and bounded-increment “raise” steps. The approach mandates three tunable parameters:

• $\delta > 0$ (minimum raise): forces progress away from underestimates,
• $M \geq 0$ (modulation threshold): above which standard Bellman-Ford recursion resumes,
• $D \geq 0$ (dead zone): grants disturbance tolerance for noisy-perturbed topology and link values.

For a node $i$ at time $t$, the update is:

1. Compute candidate $$\tilde x_i = \min(\min_{k\in N(i)} f(x_k(t), e_{ik}), s_i)$$.
2. If $x_i(t) \geq M$ or $|x_i(t) - \tilde x_i| \leq D$, set $x_i(t+1) \leftarrow \tilde x_i$; else $x_i(t+1) \leftarrow g(x_i(t))$, where $g(x) \geq x + \delta$.

This algorithm is globally uniformly asymptotically stable in the absence of perturbations, and achieves bounded steady-state error under persistent disturbances, with error scales dictated by the product of the maximal link noise and the network diameter. The explicit trade-off is between convergence speed (large $\delta$, small $D$) and disturbance tolerance (large $D$) (Mo et al., 2021).

4. Computational Strategies and Complexity

The bit-flip descent for code design achieves an $O(M n T)$ per-iteration cost—a substantial improvement over the $O(n^2T^2)$ of exhaustive recomputation. Key to this efficiency is the incremental update rule, affecting only $O(nT)$ correlation entries per bit flip. Block-flip generalizations further scale this for very large system parameters.

In OFDM-based spreading, the construction of DPSS basis vectors and assembly of the spreading matrices typically requires eigendecomposition of Dirichlet kernel submatrices of dimension comparable to the band size, after which the per-symbol computational overhead is linear in the number of selected basis vectors.

Distributed core-spreading relies on strictly local computation and communication, with the performance bounds governed by the graph’s effective diameter and the “progressivity” of the underlying update function.

5. Practical Performance and Trade-Offs

Empirical results confirm rapid convergence and tangible performance improvements:

• GNSS code optimization yields $40\%$ cross-correlation peak reduction for GPS L1 C/A analogues ($n=63, T=1023$) with convergence in minutes on modern multicore hardware, and approximately $37\%$ $\ell_1$ cost reduction for Galileo E1 analogues ($n=100, T=4092$) (Yang et al., 2024).
• In multi-user ISAC, adding DPSS spreading preserves low sidelobes under strong interference even as standard OFDM exhibits $5$ dB ISL increases per $5$ dB interference increment; substantial EISL reduction is observed at modest spectral back-off $\eta$ (Said et al., 4 May 2025).
• Distributed spreading stabilizes to a unique fixed point exactly in unperturbed settings, and remains within an explicit bound under bounded perturbations, with convergence and tolerance directly tunable via the three design parameters (Mo et al., 2021).

The core architectural compromise is consistently between convergence/accuracy and resource utilization: in code design, between computational cost and global optimality; in OFDM spreading, between interference suppression and data capacity; in distributed propagation, between speed and robustness of consensus.

6. Generalizations and Applicability

The core-spreading paradigm extends to any system where a set of elements (sequence entries, subcarriers, node states) must be jointly optimized, transformed, or propagated to satisfy additive cost criteria tied to pairwise or groupwise interactions. Applications beyond GNSS and ISAC include CDMA codebook design, MIMO waveform shaping, and robust aggregation in decentralized learning or sensing.

Constraint handling is naturally integrated by prohibiting flips or transformations that would violate application-specific restrictions (e.g., constant-weight codes, zeros at specified lags). Stochastic variants (random restarts, simulated annealing) can provide partial escape from suboptimal local minima in non-convex settings.

7. Summary Table: Core-Spreading Applications

Application Domain	Core-Spreading Role	Key Performance Target
GNSS/CDMA	Sequence design via bit-flip descent	Minimized autocorrelation/cross-corr
OFDM ISAC (multi-user)	DPSS-based user spreading	Suppressed IB interference/ISL
Distributed Aggregation	Stable, bounded consensus dynamics	Fast, resilient information spread

The breadth of the core-spreading concept highlights its centrality to the design of modern robust, scalable, and interference-tolerant information and signal systems.