State-Dependent Memoryless Channel (SDMC)

• SDMC is a discrete memoryless channel whose behavior depends on a state process, affecting coding schemes and capacity formulations.
• It underpins models like the Gel’fand–Pinsker channel, multiuser MACs, relay and interference channels, and finite-state POST channels.
• Finite-blocklength analysis, dispersion measures, and auxiliary variable-based coding enable robust pre-coding, interference cancellation, and joint state communication.

A state-dependent memoryless channel (SDMC) is a discrete memoryless channel whose behavior depends on a random state process. With input alphabet $\mathcal{X}$, output alphabet $\mathcal{Y}$, and state alphabet $\mathcal{S}$, a single use is governed by a conditional distribution $W(y\mid x,s)$, and for i.i.d. states $P_{S^n}(s^n)=\prod_{i=1}^n P_S(s_i)$ the $n$-use law factorizes as $$P_{Y^n\mid X^n,S^n}(y^n\mid x^n,s^n)=\prod_{i=1}^n W(y_i\mid x_i,s_i)$$. This framework includes the Gel'fand–Pinsker channel with noncausal state information at the encoder, multiuser state-dependent MAC, relay, and interference channels, as well as finite-state constructions such as POST channels in which the previous output acts as the current state (Tan, 2012, Zhang et al., 9 Mar 2026).

1. Formal model and information patterns

The canonical SDMC specifies $\mathcal{X}$, $\mathcal{Y}$, $\mathcal{S}$, a state law $\mathcal{Y}$0, and a state-dependent transition law $\mathcal{Y}$1. For coding schemes that correlate the input with the state, the single-letter joint law is commonly written as

$\mathcal{Y}$2

or, with an auxiliary random variable $\mathcal{Y}$3,

$\mathcal{Y}$4

In the noncausal encoder-side information setting, the encoder observes the entire state sequence $\mathcal{Y}$5 before transmission and chooses $\mathcal{Y}$6, whereas the decoder observes $\mathcal{Y}$7 but not $\mathcal{Y}$8. In causal and strictly causal settings, the encoder observes $\mathcal{Y}$9 or $\mathcal{S}$0 at time $\mathcal{S}$1, respectively. These distinctions recur throughout SDMC theory because they determine whether the state can be pre-coded against, described to the decoder, or merely exploited through adaptation (Tan, 2012, Choudhuri et al., 2012).

The same template extends directly to multiuser channels. In a state-dependent MAC or wiretap MAC, the law becomes $\mathcal{S}$2, with i.i.d. states and encoder mappings that may depend on causal or noncausal channel state information. In relay models, the physical channel is $\mathcal{S}$3, with distinct state-information patterns at the source, relay, and destination. In helper models, an additional terminal observes the state causally or noncausally and provides rate-limited assistance to the encoder or to both terminals. These constructions preserve the memoryless conditional factorization while changing which terminals know which parts of the state process (Chen et al., 2023, Li et al., 2011, Lapidoth et al., 2024).

A finite-state specialization appears in POST channels, where the state is not i.i.d. but is generated by the channel itself via $\mathcal{S}$4. The per-time law is then $\mathcal{S}$5, and the state update is $\mathcal{S}$6. This embeds a finite-state channel into the SDMC template by treating the current channel law as memoryless given the current state, even though the state evolves as a Markov process (Zhang et al., 9 Mar 2026).

2. Capacity formulas and canonical coding principles

For the Gel'fand–Pinsker channel, where the state is i.i.d., independent of the message, known noncausally to the encoder, and unknown to the decoder, the classical capacity is

$\mathcal{S}$7

Here $\mathcal{S}$8 is an auxiliary random variable, and $\mathcal{S}$9 is the actual channel input. The term $W(y\mid x,s)$0 measures decodability from the channel output, while $W(y\mid x,s)$1 is the rate cost of correlating the code with the state. The operational interpretation is binning: the encoder selects a codeword that is jointly compatible with the realized state, so that the decoder recovers the message without explicitly learning the state (Tan, 2012).

This structure underlies the standard intuition of pre-coding against the state. In additive Gaussian channels with additive interference known to the transmitter, it reduces to Costa’s dirty-paper coding phenomenon: the encoder can pre-cancel the interference so that the first-order capacity matches that of the corresponding state-free channel. The same idea reappears in more elaborate SDMCs whenever a suitable auxiliary variable can absorb the state dependence into codebook structure rather than leaving it as residual uncertainty at the decoder (Tan, 2012).

Other state-information patterns lead to different single-letter formulas. When the state is known only at the decoder, the capacity is

$W(y\mid x,s)$2

When the state is known at both encoder and decoder, the capacity is

$W(y\mid x,s)$3

For causal encoder-side information, Shannon strategy enters: the encoder chooses the channel input as a function of an auxiliary $W(y\mid x,s)$4 and the current state, and the capacity-distortion theory recovers the causal-message capacity as

$W(y\mid x,s)$5

By contrast, strictly causal state information does not increase point-to-point message capacity for memoryless channels, although it can still be valuable for state communication or in multiuser settings (Lapidoth et al., 2023, Choudhuri et al., 2012, Li et al., 2011).

The main coding primitives of SDMCs are therefore stable across models: auxiliary random variables, random binning, superposition, and state-dependent mappings $W(y\mid x,s)$6 or $W(y\mid x,s)$7. What changes across regimes is whether the state is neutralized, described, exploited causally, or converted into common randomness.

3. Finite blocklength, dispersion, and second-order behavior

Beyond asymptotic capacity, SDMC analysis also includes finite-blocklength performance. For the Gel'fand–Pinsker channel, the $W(y\mid x,s)$8-capacity is

$W(y\mid x,s)$9

with average error probability at most $P_{S^n}(s^n)=\prod_{i=1}^n P_S(s_i)$0. A central object is the net information density

$P_{S^n}(s^n)=\prod_{i=1}^n P_S(s_i)$1

where

$P_{S^n}(s^n)=\prod_{i=1}^n P_S(s_i)$2

Under a capacity-achieving choice of $P_{S^n}(s^n)=\prod_{i=1}^n P_S(s_i)$3 and $P_{S^n}(s^n)=\prod_{i=1}^n P_S(s_i)$4, the corresponding dispersion is

$P_{S^n}(s^n)=\prod_{i=1}^n P_S(s_i)$5

The finite-blocklength analysis in "On Dispersions of Discrete Memoryless Channels with Noncausal State Information at the Encoder" provides a lower bound for the $P_{S^n}(s^n)=\prod_{i=1}^n P_S(s_i)$6-capacity of the Gel'fand–Pinsker channel and hence an upper bound on the dispersion, yielding a normal-approximation perspective analogous to ordinary DMC theory but with the state-dependent information density replacing the usual information density (Tan, 2012).

The coding proof combines Gel'fand–Pinsker binning with non-asymptotic decoding analysis. Codewords are chosen to be compatible with the realized state sequence, decoding is based on typicality or information-density thresholding, and Berry–Esseen type arguments are applied to sums of $P_{S^n}(s^n)=\prod_{i=1}^n P_S(s_i)$7. This places SDMC finite-blocklength analysis squarely in the broader dispersion program, while making clear that the second-order term depends not only on channel randomness but also on the state-correlation structure induced by the auxiliary variable (Tan, 2012).

The same paper also extends finite-blocklength analysis to a degraded broadcast version of the Gel'fand–Pinsker model. There the encoder uses two auxiliaries, $P_{S^n}(s^n)=\prod_{i=1}^n P_S(s_i)$8 and $P_{S^n}(s^n)=\prod_{i=1}^n P_S(s_i)$9, with superposition coding and Gel'fand–Pinsker coding across layers. The resulting inner bound on the $n$0-capacity region involves vector information densities and corresponding dispersion terms. This makes explicit that in multiuser SDMCs the second-order behavior is intrinsically multidimensional, with coupled fluctuations across users rather than a single scalar dispersion parameter (Tan, 2012).

4. Multiuser, secrecy, relay, and helper-assisted extensions

A large part of SDMC research concerns multiuser generalizations. In the state-dependent two-user interference channel with noncausal state known at both transmitters and unknown at both receivers, achievable rate regions have been obtained by combining Han–Kobayashi rate splitting with Gel'fand–Pinsker coding. Two coding schemes were proposed for the discrete memoryless case: simultaneous encoding for the sub-messages in one scheme and superposition encoding in the other, both with rate splitting and Gel'fand–Pinsker coding. In the Gaussian case, an active interference cancellation mechanism was introduced as a generalized dirty-paper coding technique to partially eliminate the state effect at the receivers (Zhang et al., 2010, Zhang et al., 2011).

Relay models show a different use of state information. In the state-dependent relay channel with strictly causal channel state information at the relay and no state information at the source and destination, the source and relay are connected by finite-capacity orthogonal links that support message cooperation and state cooperation. Two achievable schemes exploit both resources, and a noisy-network-coding-inspired transmission scheme performs better than a strategy based on block Markov coding and backward decoding. Capacity results are identified for some special cases, including deterministic and Gaussian models, illustrating when state cooperation alone, message cooperation alone, or their combination is sufficient (Li et al., 2011, Li et al., 2011).

Secrecy introduces another layer of SDMC structure. For the discrete memoryless state-dependent multiple access channel with an external eavesdropper and causal CSI at the encoders, strong secrecy is imposed through $n$1. The resulting inner bound on the secrecy capacity region is the convex hull of three regions corresponding to two block Markov coding schemes and a wiretap-only construction. One scheme combines backward decoding, Wyner–Ziv coding, secret key agreement, and wiretap coding; the other extends point-to-point causal-CSI wiretap coding to the MAC. In some degraded-message-set cases, both schemes are capacity-achieving, showing that the state can function simultaneously as interference, coordination resource, and secret-key source (Chen et al., 2023).

Helper models interpolate between direct state knowledge and no state knowledge. With a rate-limited cribbing helper that observes the state causally and past inputs strictly causally, the capacity is

$n$2

over distributions of the form $n$3, with the Markov chain $n$4, and it is sufficient to consider deterministic mappings $n$5 and $n$6. With a message-cognizant helper that observes the state noncausally and sends a rate-limited description to both encoder and decoder, the capacity is

$n$7

These formulas formalize two different ways of exploiting auxiliary terminals: by learning the message indirectly through cribbing, or by directly mixing message and state information in the helper description (Lapidoth et al., 2024, Lapidoth et al., 2023).

5. Feedback, asynchronism, POST channels, and adversarial states

Feedback interacts with SDMCs in several distinct ways. In asynchronous state-dependent channels, the encoder may observe a delayed version of the state sequence while the true delay is unknown within a finite set. For the asynchronous Gel'fand–Pinsker channel, achievable rates were derived for both noncausal and causal asynchronous side information. With feedback, the capacity of the asynchronous Gel'fand–Pinsker channel equals the ordinary Gel'fand–Pinsker capacity, because the delay can be learned through a training phase and the problem reduces to the synchronous case (Yemini et al., 2014).

Variable-length feedback coding yields a complementary line of results. For state-dependent discrete memoryless channels with noiseless feedback, necessary and sufficient conditions for positivity of the zero-error capacity were obtained across multiple state-information regimes. Whenever the zero-error capacity is positive, it equals the conventional vanishing-error capacity. At the same time, the vanishing-error capacity is not increased by the use of feedback and variable-length coding. Thus, in SDMCs, feedback and variable length can fundamentally change zero-error feasibility without changing the first-order Shannon limit (Kovačević et al., 2017).

In finite-state SDMCs, feedback can also fail to provide any gain beyond the strictly memoryless case. For approximately memoryless surjective POST channels, where $n$8 and every state-dependent kernel $n$9 is uniformly close to a reference DMC $$P_{Y^n\mid X^n,S^n}(y^n\mid x^n,s^n)=\prod_{i=1}^n W(y_i\mid x_i,s_i)$$0, there exists $$P_{Y^n\mid X^n,S^n}(y^n\mid x^n,s^n)=\prod_{i=1}^n W(y_i\mid x_i,s_i)$$1 such that

$$P_{Y^n\mid X^n,S^n}(y^n\mid x^n,s^n)=\prod_{i=1}^n W(y_i\mid x_i,s_i)$$2

for all $$P_{Y^n\mid X^n,S^n}(y^n\mid x^n,s^n)=\prod_{i=1}^n W(y_i\mid x_i,s_i)$$3-centered $$P_{Y^n\mid X^n,S^n}(y^n\mid x^n,s^n)=\prod_{i=1}^n W(y_i\mid x_i,s_i)$$4-approximately memoryless surjective POST channels. This extends Shannon’s classical theorem that feedback does not increase the capacity of a DMC to an open neighborhood of memoryless channels inside the POST class (Zhang et al., 9 Mar 2026).

Adversarial variants replace the stochastic state law by worst-case or jammer-controlled behavior. In the state-myopic arbitrarily varying channel, the encoder observes a noisy noncausal version $$P_{Y^n\mid X^n,S^n}(y^n\mid x^n,s^n)=\prod_{i=1}^n W(y_i\mid x_i,s_i)$$5 of the adversarial state sequence through a DMC $$P_{Y^n\mid X^n,S^n}(y^n\mid x^n,s^n)=\prod_{i=1}^n W(y_i\mid x_i,s_i)$$6, and the randomized coding capacity is

$$P_{Y^n\mid X^n,S^n}(y^n\mid x^n,s^n)=\prod_{i=1}^n W(y_i\mid x_i,s_i)$$7

with the capacity under stochastic encoders equal to the randomized coding capacity whenever $$P_{Y^n\mid X^n,S^n}(y^n\mid x^n,s^n)=\prod_{i=1}^n W(y_i\mid x_i,s_i)$$8. In the state-aware adversary setting, the GP-AVC capacity is given by a worst memoryless state-dependent channel induced by memoryless jamming, and for the Gaussian dirty-paper AVC the randomized capacity is

$$P_{Y^n\mid X^n,S^n}(y^n\mid x^n,s^n)=\prod_{i=1}^n W(y_i\mid x_i,s_i)$$9

the same as an AWGN channel with channel noise and jamming noise, even though the adversary knows the state noncausally (Budkuley et al., 2018, Budkuley et al., 2015).

6. State communication, sensing, and rate-distortion viewpoints

Not all SDMC problems treat the state as a nuisance. In causal state communication, the objective is to reconstruct the state sequence itself, possibly while sending an independent message. For strictly causal encoder state information, block Markov encoding that communicates a description of the previous block’s state sequence is optimal for minimum state-estimation distortion. The corresponding strictly causal capacity-distortion function is

$\mathcal{X}$0

over $\mathcal{X}$1 and reconstruction functions $\mathcal{X}$2 satisfying the distortion constraint. For causal state information, Shannon strategy extends the theory, and the capacity-distortion function becomes

$\mathcal{X}$3

over $\mathcal{X}$4, $\mathcal{X}$5, $\mathcal{X}$6, and $\mathcal{X}$7. These results turn SDMC state knowledge into a source-communication resource rather than merely a means of pre-canceling interference (Choudhuri et al., 2012).

Joint sensing and communication produces a related but distinct perspective. In the state-dependent memoryless MAC with generalized feedback studied for joint radar and communication, two transmitters send messages while estimating their respective channel states through feedback. An achievable capacity-distortion tradeoff region was derived and shown, for a binary erasure MAC with binary states, to outperform a resource-sharing scheme. The formulation treats the state as a physical quantity to be inferred while data transmission continues over the same SDMC (Kobayashi et al., 2019).

A more explicit rate-distortion theory for sensing over SDMCs appears in vehicular ISAC models. With channel law $\mathcal{X}$8, where $\mathcal{X}$9 is the communication output and $\mathcal{Y}$0 is a sensing echo, the estimation rate-distortion function for the state estimator is

$\mathcal{Y}$1

A modified Blahut–Arimoto type algorithm was proposed for computing this function, and a capacity-rate-distortion tradeoff region was defined to unify communication and sensing within a single optimization framework. The numerical results show that coding can improve estimation rate for certain channels, which suggests that waveform design for SDMC-based ISAC cannot be reduced to classical communication-only or sensing-only criteria (Feng et al., 18 Sep 2025).

The same tendency is visible in recent multiuser state-estimation problems. For the SD-DMMAC with generalized feedback and causal side information at the encoders, an achievable rate-distortion region for joint state and message communication was obtained using Markov coding and backward-forward two-stage decoding. In its single-user specializations, the construction recovers familiar SDMC capacity-distortion tradeoffs; in the full MAC, it couples common-message transmission, private-message transmission, first-stage state descriptions, and refinement layers for estimation (Li et al., 2024).

Taken together, these results show that SDMCs support at least three distinct operational roles for state: nuisance parameter to be pre-coded against, latent variable to be reconstructed, and structured source of coordination or secrecy. The breadth of the framework is precisely what makes SDMCs central across contemporary information theory: the same conditional law $\mathcal{Y}$2, or its multiuser generalizations, can encode interference mitigation, helper-assisted communication, secrecy, finite-blocklength analysis, feedback phenomena, and integrated sensing-and-communication within a single state-dependent memoryless abstraction.