FCS in Joint Semantic and Channel Coding

• FCS is a method that promptly selects the first decoded output meeting predefined semantic or reconstruction criteria in joint coding systems.
• It is applied in semantic token and point cloud communications to achieve low-latency, error-resilient data recovery under noisy channel conditions.
• FCS ensures graceful degradation by leveraging angular similarity and Chamfer-distance metrics to maintain semantic coherence and structural integrity.

The First Correct Solution (FCS) principle denotes the selection or use of the earliest feasible answer—whether in decoding, demodulation, or reconstruction—that aligns with any prescribed correctness constraint in communication and coding systems. Within the broader context of joint semantic and channel coding, FCS is intricately linked to the mapping and decoding processes where candidate solutions are considered, and the earliest solution meeting a formal acceptability (correctness) condition is chosen and disseminated for downstream tasks or applications.

1. Definition and Role in Joint Token and Channel Coding

FCS typically refers to the algorithmic output that first satisfies the decodability, semantic integrity, or similarity requirements imposed by a system employing Joint Token and Channel Coding (JTCC). In the context of semantic communications, transmitted content consists of semantic units (e.g., discrete tokens from foundation models or point-cloud patches), and correct decoding involves reconstructing source representations which are acceptably close—semantically or perceptually—to the original input. FCS is operationalized as the first decoded or reconstructed token that satisfies a correctness constraint (such as closest match in semantic embedding space or minimal structural distortion), regardless of whether alternative solutions might subsequently be found or be marginally better according to the same criterion (Bao et al., 10 Jun 2026, Ying et al., 19 Nov 2025).

A plausible implication is that FCS provides irreducible latency and error guarantees in scenarios where early or real-time decisions are required, such as real-time semantic communications under strict delay budgets.

2. Context in Semantic Token Codebook Communication (STCC) and JTCC

Within STCC (Bao et al., 10 Jun 2026), FCS manifests during the decoding phase, where each received noisy channel vector $\mathbf y_i$ is mapped by the decoder $g_\phi$ to an embedding $\hat{\mathbf e}_i$. Token recovery is achieved via nearest-neighbor search in a fixed semantic codebook $\mathcal V$, yielding

$$\hat s_i \;=\;\underset{k\in\mathcal V}{\arg\max}\; \frac{\hat{\mathbf e}_i\cdot E(k)}{\|\hat{\mathbf e}_i\|_2\,\|E(k)\|_2}$$

where the FCS is the first $k$ encountered in the search order that achieves the best (i.e., maximum) cosine similarity. This implements the FCS principle by promptly selecting the first nearest valid token mapping.

In JTCC for point tokens (Ying et al., 19 Nov 2025), the decoder reconstructs point clouds or semantic tokens from soft-quantized or masked channel outputs. The FCS interpretation here is that the first reconstructed token set achieving a minimal Chamfer-Distance loss or sufficient semantic match is accepted and output.

3. Mathematical Formalization and Algorithmic Workflow

The FCS selection is embedded in the argmax or nearest-neighbor operations that underpin decoder implementations in both STCC and JSCCM/JTCC designs. The process is formalized as:

For semantic tokens:

$$\hat s = \operatorname{FCS}\left\{ k \in \mathcal V \;\middle|\; k = \underset{k}{\arg\max}\; \cos(\hat{\mathbf e}, E(k)) \right\}$$

For structural units or point clouds:

$$\hat X = \operatorname{FCS}\left\{ X' \;\middle|\; \mathcal{L}_{\text{CD}}(X, X') \leq \delta \right\}$$

where $\delta$ is a threshold for allowable reconstruction error (e.g., in Chamfer distance or semantic similarity).

FCS enables efficient algorithmic pipelines, as demonstrated in the following inference pseudocode (as given for STCC (Bao et al., 10 Jun 2026)): $g_\phi$1

4. Theoretical Guarantees and Error Regions

The correctness of the FCS is underpinned by topological alignment between the semantic embedding manifold and channel constellation geometry. In STCC, the triple-loss objective aligns angular distances in channel space with semantic distances. The shape of decision regions (angular cones) and their overlaps, governed by loss hyperparameters (notably $\lambda_2$ for cosine margin), controls the probability of an FCS falling outside the target cluster.

A theoretical sketch provided in (Bao et al., 10 Jun 2026) states that isotropic noise in $g_\phi$0 induces spherical perturbations, and the overlap of angular cones provides an upper bound for incorrect FCS outputs—making the selection robust to random channel corruption if the angular margin is sufficient.

5. Implications: Semantic Drift, Structural Distortion, and Performance

The FCS paradigm, when combined with topology-aligned coding, transforms the error characteristics of semantic communication systems:

• Semantic Drift: In symbolic modalities (e.g., text), an incorrect FCS tends to be semantically or syntactically related to the intended token, quantified by the strong correlation between confusion frequency and cosine similarity of embeddings (Bao et al., 10 Jun 2026).
• Structural Distortion: In perceptual modalities (e.g., images, point clouds), the FCS yields reconstructions that are structurally coherent—erroneous outputs are proximal in feature or visual space, as measured by Chamfer-distance or UMAP clustering (Bao et al., 10 Jun 2026, Ying et al., 19 Nov 2025).
• Graceful Degradation and Low-SNR Robustness: FCS contributes to the avoidance of catastrophic failure ("cliff effect") exhibited by classic separate coding. For example, STCC achieves approximately 86.9% token accuracy at 0 dB SNR for text, and over 0.9 semantic similarity under Rayleigh fading at 0 dB, substantially outperforming traditional systems (Bao et al., 10 Jun 2026). JTCC for point clouds yields similar robustness, with ≈1 dB PSNR gain over separate coding baselines and 6× compression in modulated symbol count (Ying et al., 19 Nov 2025).

6. Summary Table: FCS in Semantic Token Coding

Modality	FCS Output Mechanism	Performance Role
Foundation Model Tokens	Cosine NN in embedding codebook	Graceful semantic drift; robust recovery
Point Clouds	Min-CD (Chamfer Distance) match	Structural distortion instead of random errors
Images	Angular NN in latent space	Visual coherence under noise

The concept and operationalization of FCS, as instantiated in leading JTCC/STCC frameworks, yields communication systems that efficiently map semantic units to robust channel representations, ensuring that the first output matching semantic criteria is both computationally efficient and robust to physical channel errors (Bao et al., 10 Jun 2026, Ying et al., 19 Nov 2025).