Superposition Block Markov Coding

• Superposition Block Markov Coding is a layered transmission framework that organizes information across blocks using state-dependent coding to optimize multiuser communications.
• It employs hierarchical codebook generation combining superposition and Marton coding to achieve higher rates than traditional Gelfand-Pinsker approaches.
• The technique underpins BMST codes and other modern constructions, offering near-capacity performance in applications like fading, broadcast, and cooperative channels.

Superposition Block Markov Coding is a layered transmission and decoding framework that generalizes superposition coding and block Markov encoding, with foundational applications in compound/broadcast channels, multiuser information theory, and modern error-control coding. It operates by constructing codebooks and transmissions where information is organized across multiple blocks and superimposed layers, thereby enabling improved management of interference, cooperative communications, and robust encoding against channel states known (possibly noncausally) at the transmitter.

1. Foundational Principles and Channel Models

The central premise is to exploit encoder-side state knowledge (often noncausally available) and the layered structure of superposition coding:

• In the compound channel or two-receiver broadcast channel model, a DM (discrete memoryless) state sequence $S$ is available noncausally to the transmitter, which must send a common message $M$ to both receivers, each facing potentially different state-conditioned channels (Nair et al., 2010).
• A naive extension of the single-user Gelfand-Pinsker result yields $$R_{GP} = \max_{p(u|s), x(u,s)} \min \{ I(U;Y_1) - I(U;S), I(U;Y_2) - I(U;S) \}$$, but in general this is suboptimal.

Superposition Block Markov Coding (SBMC) constructs layered codebooks (common and receiver-specific, with auxiliaries $W$, $U$, $V$) using a combination of superposition coding (for shared/common parts) and Marton coding (for splitting into private parts). This approach surpasses the single-auxiliary Gelfand-Pinsker strategy in achievable rate, flexibility, and often achieves capacity for special classes.

2. Layered Codebook Generation and Encoding

The codebook is constructed in a hierarchical multistage fashion:

• Common codebook: Index codewords by a random variable $W$ for every message $m$, generating $w^n(m, l_0)$ via $p_W(w)$.
• Satellite codebooks: For each $(m, l_0)$, generate receiver-specific codewords $u^n(m, l_0, l_1)$ (via $p_{U|W}(u|w)$) and $v^n(m, l_0, l_2)$ (via $p_{V|W}(v|w)$), supporting Marton coding.
• Layering via superposition: $W$ acts as a base layer common to both receivers; $U$ and $V$ allow for individualized tailoring to each receiver channel.

Encoding uses the noncausal state knowledge for joint typicality:

1. Search for $l_0$ such that $(w^n(m, l_0), s^n)$ is jointly typical.
2. With $w^n(m, l_0)$ fixed, find $(l_1, l_2)$ such that $$(w^n(m, l_0), s^n, u^n(m, l_0, l_1), v^n(m, l_0, l_2))$$ is jointly typical.
3. Transmit $x = f(w, u, v, s)$, compensating for the state, in the spirit of dirty-paper coding.

3. Decoding, Indirect Decoding, and Error Analysis

Decoding in SBMC is performed via indirect decoding using typicality checks:

• Decoder 1: Searches for unique $(m, l_0, l_1)$ such that $(w^n(m, l_0), u^n(m, l_0, l_1), y_1^n)$ is jointly typical; recovers $m$ through its dedicated ($U$) layer.
• Decoder 2: Analogously, $(w^n(m, l_0), v^n(m, l_0, l_2), y_2^n)$; recovers $m$ via the $V$ layer.

Indirect decoding exploits the independence of the codebook layers: each receiver requires only the common plus one private codeword. Careful error event analysis yields rate constraints:

• Covering constraints: $T_0 > I(W;S)$, $T_1 > I(U;S|W)$, $T_2 > I(V;S|W)$, $T_1+T_2 > I(U;S|W) + I(V;S|W) + I(U;V|W,S)$.
• Packing constraints: $R+T_0+T_1 < I(W,U;Y_1)$, $R+T_0+T_2 < I(W,V;Y_2)$.

Elimination of auxiliaries via Fourier-Motzkin yields:

$$C \geq \max_{p(w,u,v|s), x=f(w,u,v,s)} \min \left\{ I(W,U;Y_1)-I(W,U;S), I(W,V;Y_2)-I(W,V;S), \frac{1}{2}\big[ I(W,U;Y_1)-I(W,U;S) + I(W,V;Y_2)-I(W,V;S) - I(U;V|W,S) \big] \right\}$$

4. Performance Advantages, Achievable Rates, and Optimality

Contrasted to the conventional single-auxiliary Gelfand-Pinsker coding, SBMC with superposition/Marton coding achieves strictly higher rates:

• By splitting auxiliaries to target each receiver, the encoder can “pre-compensate” state-dependent channel effects separately.
• The penalty term $I(U;V|W,S)$ and layered mutual information expressions allow for rates that strictly exceed $R_{GP}$.
• Example: In deterministic channels, $R_{GP} \approx 0.41$ is exceeded by the new scheme achieving up to $0.5$; in compound Gaussian channels, clean separation yields dirty-paper coding capacity for both receivers.

Optimality is established for deterministic channels (when $I(Y_1;Y_2|S)=0$, $C=\max_{p(x|s)}\min\{H(Y_1|S), H(Y_2|S)\}$ by choosing $W=\emptyset$, $U=Y_1$, $V=Y_2$) and for certain compound Gaussian channels, where appropriate auxiliary choices recover compound Costa rates.

5. Block Markov Extensions, Decoding Strategies, and Modern BMST Codes

The SBMC paradigm underpins the development of Block Markov Superposition Transmission (BMST) codes (Ma et al., 2013, Liang et al., 2014, Huang et al., 2015, Huang et al., 2015, Liang et al., 2015, Ma et al., 2016, Zhao et al., 2016, Cai et al., 2017):

• BMST construction: "Superimposing" basic (short) codes via interleaving and addition mod-$q$ over multiple blocks (see $$c^{(t)} = v^{(t)} + \sum_{i=1}^m \text{interleaver}_i(v^{(t-i)})$$) creates a spatially coupled superposition code.
• Encoding complexity remains comparable to that of the basic code; decoding uses iterative sliding-window algorithms with tunable delay, ensuring near-optimal performance at low BER and predictable extra coding gain (e.g., $10\log_{10}(m+1)$ dB).
• Universality: BMST applies to linear, nonlinear, and group codes, supporting rate-compatibility and flexible adaptation for block/fading channels.

EXIT chart analyses adapted for BMST (incorporating MI-BER coupling) establish decoding thresholds and allow design parameter optimization for waterfall/error floor performance (Huang et al., 2015, Huang et al., 2015).

6. Generalized Frameworks and Groupcasting

Recent theoretical development extends superposition coding with rate-splitting and binning for concurrent groupcasting over broadcast channels (Romero et al., 2020):

• Messages indexed by arbitrary receiver groups are layered according to a “superposition order” on an enlarged set; each message is split into sub-messages (rate-splitting), enabling joint decoding of desired and partial interfering components.
• Achievability leverages recursively structured codebooks and recursive mutual covering lemmas, and binning provides codebook diversity for matching arbitrary input distributions, enhancing flexibility in multiuser scenarios.
• The polyhedral characterizations and packing constraints generalize the SBMC paradigm to arbitrary numbers of receivers and message sets.

7. Connections to Coding Theory, Practical Implementations, and Applications

SBMC directly informs the construction and analysis of modern codes:

• BMST–BCH, HT–coset, and RUN codes use block Markov superposition to attain near-capacity at low BER and manageable complexity, with proven net coding gains in challenging environments (e.g., optical transport, fading channels) (Liang et al., 2015, Cai et al., 2017, Liang et al., 2014, Huang et al., 2015).
• Systematic BMST–R produces rate-compatible codes with waterfall performance and error floors analytically tied to encoding memory and code rate (Ma et al., 2016).
• Partially BMST (PBMST) selectively superimposes parity-check portions, optimizing joint source–channel coding for Gaussian sources with nested lattice codes (Zhao et al., 2016).
• BMST and SBMC are widely used for low-latency, ultra-reliable communications, paralleling advances in spatially coupled LDPC and other graph-based codes.

In summary, Superposition Block Markov Coding synthesizes layered encoding with state-dependent transmission, indirect decoding, and joint typicality analysis. Its legacy includes not only sharp capacity theorems in multiuser information theory but also a suite of practical high-performance codes with widespread applicability in cooperative, block-fading, and groupcast/broadcast environments.