CBraMod Encoder: Methods & Applications

• CBraMod encoder is a unified encoding paradigm that implements distinct methods for high-information representation in wireless, neurophysiology, and flash memory domains.
• It leverages sparse-encoded codebook index modulation, criss-cross transformer patching, and minimal-push-up rank modulation to achieve enhanced spectral efficiency and optimized hardware operations.
• The approach delivers practical gains with improved URLLC performance, robust EEG/ECG signal processing, and up to 54% rate increase in memory write efficiency compared to traditional methods.

The term CBraMod encoder refers to structurally distinct methodologies unified by their focus on learning or encoding high-information representations for efficient transmission, robust physiological signal processing, or efficient non-volatile memory writes. Across the literature, "CBraMod" appears in three technical domains: (1) sparse-encoded codebook index modulation for wireless communications (Arslan et al., 2020), (2) criss-cross transformer backbones for EEG/ECG foundation models (Wang et al., 10 Dec 2024, Ghallab et al., 17 Dec 2025), and (3) compressed encoding for rank modulation in flash memory (Gad et al., 2011). Each instantiation implements a discrete, principled encoder that leverages sparsity, structure, or minimality for domain-efficient information transfer.

1. Domain-Specific Definitions and Principles

Sparse-Encoded Codebook Index Modulation (SE-CBIM)

• Principle: Encodes input bits by selecting a codebook index and a sparse activation pattern in a virtual digital domain (VDD), followed by spreading the sparse vector using a selected codebook and OFDM modulation. Maximizes spectral efficiency and provides ultra-reliable low-latency communication (URLLC) without explicit channel coding (Arslan et al., 2020).

Criss-Cross Brain Model (CBraMod) for EEG/ECG

• Principle: Implements a patch-based masked autoencoding strategy for physiological signals (EEG/ECG) using a criss-cross transformer backbone. Separately models spatial (channel/lead-wise) and temporal dependencies, with an asymmetric conditional positional encoding for adaptation to variable layouts or sampling rates (Wang et al., 10 Dec 2024, Ghallab et al., 17 Dec 2025).

Compressed Encoding for Rank Modulation (CBraMod)

• Principle: Implements a "minimal-push-up" strategy for flash memory writing: transition between cell permutations using the smallest possible increase in charge level for each rewrite, outperforming classic push-to-top approaches in terms of code cardinality for bounded cost (Gad et al., 2011).

2. Architectural Components and Encoding Workflows

Sparse-Encoded Codebook Index Modulation

• Input splitting: Bit vector $b \in \{0,1\}^m$ is split into codebook index ($b^{(1)}$) and activation pattern ($b^{(2)}$) sub-blocks; $$m = \log_2 G + 1 + \lfloor \log_2 \binom{M}{K} \rfloor$$.
• Activation pattern encoding: Index $d$ selects a lexicographically ordered sparse activation pattern and symbol-set (unit-norm), expressed as a K-sparse vector $s \in \mathbb{C}^{M \times 1}$.
• Spreading: Chosen codebook $C_g$ spreads $s$, producing $x_F = C_g s$.
• OFDM modulation: Apply IFFT and add cyclic prefix for channel robustness.
• Complexity: Dominated by $O(N \log N)$ (IFFT), $O(NK)$ (spreading), and constant per-pattern mapping costs.

Criss-Cross Transformer Encoders for EEG/ECG

• Patchification: Raw signal $S \in \mathbb{R}^{C \times T}$ split into $n = \lfloor T/t \rfloor$ non-overlapping patches per channel/lead, yielding $\{x_{i,j}\}$.
• Patch embedding: Time-domain branch (1D conv-GN-GELU) and frequency-domain branch (FFT+linear) sum to yield patch token $e_{i,j}$ of dimension $d$.
• Positional encoding: Asymmetric conditional positional encoding via 2D depthwise convolution across $(C \times n)$ grid for contextual adaptation.
• Criss-cross transformer: $M$ stacked blocks, each with $K$ attention heads (split $K/2$ spatial, $K/2$ temporal). Attention alternates across channel stripes (spatial) and patch stripes (temporal).
• Self-supervised training: Masked autoencoding (EEG) or dual-masking (ECG) with MSE loss only on masked elements or leads (Wang et al., 10 Dec 2024, Ghallab et al., 17 Dec 2025).
• Embedding fusion (for multi-modal): Independently pretrained CBraMod encoders for EEG and ECG; channel/patchwise pooling, $\ell_2$ normalization, and concatenation for downstream MLP classification.

Minimal-Push-Up Rank Modulation Encoder

• Initialization: Assign virtual levels $\ell_i = n+1-\sigma^{-1}(i)$ for initial permutation $\sigma$.
• Sequential update: For each target permutation $\pi$, sequentially raise cell $c=\pi(k)$ to just above $\ell_{\pi(k+1)}$ processing ranks $k=n-1 \to 1$.
• Output: The level vector $\ell'$ enforces the desired permutation at minimal level cost. The worst-case cost $C=\ell_{\pi(1)}-n$.
• Partition construction: Codes constructed to maximize possible messages under cost-constrained transitions, typically via dominating-set partitions of $S_n$.

3. Mathematical Formulations and Pseudocode

Encoder	Key Variable(s)	Core Mapping/Pseudocode
SE-CBIM	$b$, $C_g$, $I_d$	Bit split $\to$ AP encode $\to$ spread via $C_g$
Criss-Cross EEG/ECG	$\{x_{i,j}\}$, $e_{i,j}$	Patchify $\to$ time+freq embed $\to$ transformer
Minimal-Push-Up	$\sigma$, $\pi$, $\ell$	For $k=n-1$ to 1: $$\ell_{\pi(k)} \leftarrow \max(\ell_{\pi(k)}, \ell_{\pi(k+1)}+1)$$

SE-CBIM AP Encoding Pseudocode:

def AP_Encode(M, K, b1, b2, d): C0 = nCr(M, K) if d < C0: ell = 1 j = d else: ell = 2 j = d - C0 I = lex_pattern(M, K, j) s = zeros(M) for t in range(K): s[I[t]] = b1[t] if ell == 1 else b2[t] return s

Minimal-Push-Up Pseudocode:

for i in 1..n: l[sigma(i)] = n+1-i for k in range(n-1, 0, -1): c = pi(k) next = pi(k+1) l[c] = max(l[c], l[next] + 1) cost = l[pi(1)] - n

4. Key Performance Metrics and Analytical Comparisons

SE-CBIM (Sparse-Encoded Codebook Index Modulation)

• Spectral efficiency:

$$\eta = \frac{m}{N+L} = \frac{\log_2 G + \lfloor\log_2 \binom{M}{K}\rfloor + 1}{N+L}$$

• Latency & encoding complexity: Asymptotic complexity is $O(N\log N)$ (IFFT-dominated for fixed $K \ll N$); AP-mapping via precomputed lookup is constant per nonzero (Arslan et al., 2020).
• URLLC-favorable: Ultra-reliability and low-latency via sparsity-exploitative CS solvers for detection; no explicit channel coding required.

Criss-Cross CBraMod (EEG/ECG Foundation Models)

• Generalization: ACPE ensures compatibility across diverse channel layouts and sampling rates.
• Ablation findings: Dual-masking (ECG) offers statistically significant improvement in reconstruction loss over patch-only masking (final MSE 0.11 vs 0.17) (Ghallab et al., 17 Dec 2025).
• Downstream performance: Multi-modal (ECG+EEG) CBraMod fusion is competitive with or superior to prior state-of-the-art on emotional recognition benchmarks (e.g., DREAMER dataset), with best AUC for arousal and dominance, and strong F1 across all categories (Ghallab et al., 17 Dec 2025).
• Efficiency: Cross-attention fusion did not significantly outperform concat+MLP fusion but doubled inference time.

Minimal-Push-Up Encoding for Rank Modulation

• Rate enhancement: Max code size for $r=1$ cost is $M = \left(\frac{3}{4}\right)2^{n-1}$; rate $R = 1 - \frac{1}{n}\log_2(\frac{8}{3})$ bits/cell (Gad et al., 2011).
• Comparison to push-to-top: Larger number of low-cost transitions; up to 54% rate increase for $n=5$ over classic push-to-top.
• Computational cost: Both classic and minimal-push-up encodings admit $O(n)$–$O(n\log n)$ implementations.

5. Applications and Adaptation Across Modalities

• SE-CBIM: Targeted at high-efficiency, low-latency wireless transmission environments, especially 5G URLLC (Arslan et al., 2020).
• Criss-Cross CBraMod: Designed for generalizable, foundational physiological signal representations, specifically EEG for BCI, and ECG for multi-modal emotion recognition and health monitoring. Effective as a common backbone for cross-modal fusion with minimal demand for modality-specific adaptation (Wang et al., 10 Dec 2024, Ghallab et al., 17 Dec 2025).
• Minimal-Push-Up Encoder: Optimized for robust, high-endurance rank-modulation storage in multi-level flash memory (Gad et al., 2011).

6. Implementation and Design Considerations

SE-CBIM/URLLC

• Codebook design: Codebooks are generated with i.i.d. Bernoulli (±1) entries offline. Symbol-sets are unit-energy, orthogonal or well-separated for robust detection.
• Transmitter/receiver symmetry: Activation patterns ordered and shared; practical for channel resources aligned on both sides.

Criss-Cross EEG/ECG Models

• Transformer configuration: For ECG, typical settings include 12 transformer layers, 8 heads per layer, 256 token dimension (Ghallab et al., 17 Dec 2025).
• Positional encoding adaptation: 2D depthwise convolution with asymmetric kernels allows the model to seamlessly transition between different clinical layouts (e.g., lead placements, sampling rates).

Minimal-Push-Up Codes

• Dominating-set construction: Specific coset-based methods guarantee optimal covering for small $n$; for general $n$, sphere-packing style code combination is used.

7. Impact and Significance

The CBraMod encoder, in its respective domain incarnations, systematically advances the state of the art by aligning computational or statistical efficiency with hardware, channel, or cross-modal constraints:

• Communications: Delivers maximum spectral efficiency for sparse VDD coding, enabling channel coding redundancy reduction for URLLC scenarios (Arslan et al., 2020).
• Neurophysiology: Supplies a reusable, generalizable foundation model (for both EEG and ECG) that simplifies multi-modal fusion by encoding spatial and temporal structure with minimal design overhead, yielding empirically validated improvements in emotion recognition and healthcare tasks (Wang et al., 10 Dec 2024, Ghallab et al., 17 Dec 2025).
• Non-volatile memory: Provides a rigorously optimal write strategy for rank-modulation, bridging combinatorial code construction and hardware-centric endurance optimization (Gad et al., 2011).

These methodological principles enable practical gains—higher throughput, better generalization, and longer device lifetimes—across starkly different engineering problems, unified by the defining structural tenets of the CBraMod encoder paradigm.