One-Shot Random Coding Bound

• One-shot random coding bound is a nonasymptotic framework that evaluates finite-blocklength communication using randomized code ensembles and information density.
• The strong functional representation uses Poisson processes to convert stochastic channels into deterministic mappings, enabling exact channel simulation and efficient variable-length coding.
• This framework extends to complex network settings and quantum channels, unifying techniques like hypothesis testing and structured coding under a common achievability strategy.

The one-shot random coding bound is a nonasymptotic achievability statement for communication, compression, or simulation when a source or channel is used once, or when a finite-blocklength instance is analyzed directly rather than through asymptotic typicality. In the literature, the term covers several closely related constructions: stochastic encoders and decoders weighted by information density, Poisson functional representations and Poisson matching arguments, hypothesis-testing bounds, and Rényi-entropy or decoupling formulations. A particularly influential formulation is the one developed from the strong functional representation lemma, which yields explicit universal constants for exact channel simulation, variable-length lossy source coding, multiple description coding, Gray–Wyner coding, and a reduction of the Gelfand–Pinsker problem to point-to-point coding (Li et al., 2017).

1. Nonasymptotic meaning and canonical form

One-shot information theory studies source coding and channel coding when the signal blocklength is $1$; in that regime, each source and channel can be used only once, and the sources and channels are arbitrary and not required to be memoryless or ergodic (Liu, 24 Nov 2025). A one-shot random coding bound is then an achievability inequality obtained by averaging performance over a randomized code ensemble and then, typically, derandomizing or extracting a deterministic code.

A common single-user form is an information-density bound. For a channel $P_{Y|X}$, an input law $P_X$, and message set size $M$, one Poisson-matching-based formulation gives

$$P_e \le \mathbb{E}\!\left[\min\{M\cdot 2^{-\iota_{X;Y}(X;Y)},1\}\right],$$

where $$\iota_{X;Y}(x;y)=\log \frac{dP_{Y|X}(y|x)}{dP_Y(y)}$$ (Liu, 24 Nov 2025). An earlier stochastic-coder technique yields the closely related point-to-point lower bound on correct decoding

$$\mathbb{E}P[C]\ge \mathbb{E}_{q_{XY}}\!\left[\frac{1}{1+(M-1)2^{-(X;Y)}}\right],$$

with stochastic likelihood or stochastic mutual-information decoding and Jensen’s inequality as the key analytic step (Yassaee et al., 2013).

These formulations already display the characteristic structure of one-shot random coding: a finite-blocklength trade-off expressed directly through information quantities rather than asymptotic rates. In later work, the same pattern reappears in multiterminal settings, but with more elaborate denominators, auxiliary variables, or logarithmic penalties.

2. Strong functional representation as a coding mechanism

The strong functional representation lemma (SFRL) states that for random variables $X$ and $Y$ on a Polish space with Borel probability measures and $I(X;Y)<\infty$, there exists a random variable $P_{Y|X}$0 such that $P_{Y|X}$1, $P_{Y|X}$2, and

$P_{Y|X}$3

Equivalently, since $P_{Y|X}$4 and $P_{Y|X}$5,

$P_{Y|X}$6

If $P_{Y|X}$7 and $P_{Y|X}$8 are discrete with finite alphabets, $P_{Y|X}$9 can be chosen with cardinality $P_X$0 (Li et al., 2017).

The proof uses a marked Poisson point process representation. One draws a rate-$P_X$1 Poisson point process $P_X$2 with i.i.d. marks $P_X$3, defines $P_X$4, sets

$P_X$5

and outputs $P_X$6. Bounding $P_X$7 and then using a maximum-entropy argument yields the additive constant $P_X$8 in the SFRL bound (Li et al., 2017).

This construction matters because it turns a probabilistic channel $P_X$9 into a deterministic map fed by independent randomness. In the discrete case, the same idea admits a simpler representation with independent exponentials $M$0 and

$M$1

after appropriate scaling by $M$2 (Li et al., 2017). The paper interprets the Poisson representation as a form of “soft random coding”: instead of nearest-codeword selection in a conventional random codebook, the encoder selects an index by a likelihood-weighted competition among random candidates.

3. Exact channel simulation and variable-length lossy coding

In one-shot exact channel simulation, Alice observes $M$3, Alice and Bob share unlimited common randomness $M$4, Alice sends a prefix-free message $M$5, and Bob must generate $M$6 so that the joint law $M$7 matches a target channel $M$8. Choosing $M$9 from the SFRL and Huffman-coding $$P_e \le \mathbb{E}\!\left[\min\{M\cdot 2^{-\iota_{X;Y}(X;Y)},1\}\right],$$0 gives

$$P_e \le \mathbb{E}\!\left[\min\{M\cdot 2^{-\iota_{X;Y}(X;Y)},1\}\right],$$1

A converse lower bound is immediate from the Markov chain $$P_e \le \mathbb{E}\!\left[\min\{M\cdot 2^{-\iota_{X;Y}(X;Y)},1\}\right],$$2 given $$P_e \le \mathbb{E}\!\left[\min\{M\cdot 2^{-\iota_{X;Y}(X;Y)},1\}\right],$$3: $$P_e \le \mathbb{E}\!\left[\min\{M\cdot 2^{-\iota_{X;Y}(X;Y)},1\}\right],$$4 For arbitrary inputs $$P_e \le \mathbb{E}\!\left[\min\{M\cdot 2^{-\iota_{X;Y}(X;Y)},1\}\right],$$5 and channel capacity $$P_e \le \mathbb{E}\!\left[\min\{M\cdot 2^{-\iota_{X;Y}(X;Y)},1\}\right],$$6, the same approach yields

$$P_e \le \mathbb{E}\!\left[\min\{M\cdot 2^{-\iota_{X;Y}(X;Y)},1\}\right],$$7

and in the discrete case the common randomness can be chosen with $$P_e \le \mathbb{E}\!\left[\min\{M\cdot 2^{-\iota_{X;Y}(X;Y)},1\}\right],$$8 in the distribution-dependent setting or $$P_e \le \mathbb{E}\!\left[\min\{M\cdot 2^{-\iota_{X;Y}(X;Y)},1\}\right],$$9 in the capacity-based worst-case-input setting (Li et al., 2017).

The same lemma gives a one-shot variable-length lossy source-coding theorem. For source $$\iota_{X;Y}(x;y)=\log \frac{dP_{Y|X}(y|x)}{dP_Y(y)}$$0, distortion $$\iota_{X;Y}(x;y)=\log \frac{dP_{Y|X}(y|x)}{dP_Y(y)}$$1, and rate-distortion function

$$\iota_{X;Y}(x;y)=\log \frac{dP_{Y|X}(y|x)}{dP_Y(y)}$$2

the achievable expected description length satisfies

$$\iota_{X;Y}(x;y)=\log \frac{dP_{Y|X}(y|x)}{dP_Y(y)}$$3

The proof first selects $$\iota_{X;Y}(x;y)=\log \frac{dP_{Y|X}(y|x)}{dP_Y(y)}$$4 with $$\iota_{X;Y}(x;y)=\log \frac{dP_{Y|X}(y|x)}{dP_Y(y)}$$5, applies SFRL to obtain $$\iota_{X;Y}(x;y)=\log \frac{dP_{Y|X}(y|x)}{dP_Y(y)}$$6 with $$\iota_{X;Y}(x;y)=\log \frac{dP_{Y|X}(y|x)}{dP_Y(y)}$$7, and then uses a two-point Carathéodory mixture so that the entropy and distortion constraints hold simultaneously before applying Huffman coding (Li et al., 2017).

The Poisson representation yields an explicit operational picture. The “soft codebook”

$$\iota_{X;Y}(x;y)=\log \frac{dP_{Y|X}(y|x)}{dP_Y(y)}$$8

is shared conceptually, the encoder chooses

$$\iota_{X;Y}(x;y)=\log \frac{dP_{Y|X}(y|x)}{dP_Y(y)}$$9

and $$\mathbb{E}P[C]\ge \mathbb{E}_{q_{XY}}\!\left[\frac{1}{1+(M-1)2^{-(X;Y)}}\right],$$0 can be encoded with a Zipf-based prefix code. In blocklength $$\mathbb{E}P[C]\ge \mathbb{E}_{q_{XY}}\!\left[\frac{1}{1+(M-1)2^{-(X;Y)}}\right],$$1, the excess logarithmic term scales like $$\mathbb{E}P[C]\ge \mathbb{E}_{q_{XY}}\!\left[\frac{1}{1+(M-1)2^{-(X;Y)}}\right],$$2, so the per-symbol overhead vanishes (Li et al., 2017).

4. Multiterminal and state-dependent regimes

The same representation method extends to several network source-coding problems. For one-shot multiple description coding, if there exists $$\mathbb{E}P[C]\ge \mathbb{E}_{q_{XY}}\!\left[\frac{1}{1+(M-1)2^{-(X;Y)}}\right],$$3 such that

$$\mathbb{E}P[C]\ge \mathbb{E}_{q_{XY}}\!\left[\frac{1}{1+(M-1)2^{-(X;Y)}}\right],$$4

and

$$\mathbb{E}P[C]\ge \mathbb{E}_{q_{XY}}\!\left[\frac{1}{1+(M-1)2^{-(X;Y)}}\right],$$5

with

$$\mathbb{E}P[C]\ge \mathbb{E}_{q_{XY}}\!\left[\frac{1}{1+(M-1)2^{-(X;Y)}}\right],$$6

then the tuple $$\mathbb{E}P[C]\ge \mathbb{E}_{q_{XY}}\!\left[\frac{1}{1+(M-1)2^{-(X;Y)}}\right],$$7 is achievable under the stated distortion constraints (Li et al., 2017).

For the Gray–Wyner system, if $$\mathbb{E}P[C]\ge \mathbb{E}_{q_{XY}}\!\left[\frac{1}{1+(M-1)2^{-(X;Y)}}\right],$$8, $$\mathbb{E}P[C]\ge \mathbb{E}_{q_{XY}}\!\left[\frac{1}{1+(M-1)2^{-(X;Y)}}\right],$$9, and $X$0 satisfy the distortion requirements, then one-shot achievability holds under

$X$1

$X$2

$X$3

In both multiple description and Gray–Wyner settings, the added logarithmic terms scale like $X$4 at blocklength $X$5, so they disappear asymptotically relative to $X$6 (Li et al., 2017).

The Gelfand–Pinsker problem admits a particularly transparent reduction. For fixed $X$7 and $X$8, SFRL applied to $X$9 yields $Y$0 with

$Y$1

Therefore

$Y$2

At blocklength $Y$3, one obtains

$Y$4

and standard channel coding over $Y$5 recovers the classical Gelfand–Pinsker rate $Y$6 as $Y$7 (Li et al., 2017). The interpretation given in the paper is that SFRL “orthogonalizes” $Y$8 from $Y$9 by introducing $I(X;Y)<\infty$0 independent of $I(X;Y)<\infty$1.

5. Poisson matching, general networks, and codebook diversity

A later unifying step is the Poisson matching lemma (PML), which replaces both packing and covering lemmas. For measures $I(X;Y)<\infty$2 and Poisson functional representatives $I(X;Y)<\infty$3,

$I(X;Y)<\infty$4

This single inequality yields one-shot achievability results for DMCs, channels with state known at the encoder, lossy source coding with decoder side information, joint source-channel coding, broadcast channels, Berger–Tung coding, multiple access channels, channel resolvability, and wiretap channels (Li et al., 2018).

For general acyclic noisy networks, the exponential process refinement lemma extends the same philosophy. The deterministic one-shot error theorem takes the generic form

$I(X;Y)<\infty$5

with explicit multiplicative terms $I(X;Y)<\infty$6 built from products of information-density differences and logarithmic factors $I(X;Y)<\infty$7 (Liu et al., 2024). Point-to-point channel coding is recovered as the specialization

$I(X;Y)<\infty$8

A distinct but related development appears in one-shot broadcast joint source-channel coding with multiple decoders. With disjoint codebooks at $I(X;Y)<\infty$9 decoders, the error probability satisfies

$P_{Y|X}$00

With a shared codebook, the baseline bound becomes

$P_{Y|X}$01

and a hybrid grouped scheme interpolates between them: $P_{Y|X}$02 The paper interprets the multiplicative $P_{Y|X}$03 or $P_{Y|X}$04 as a codebook-diversity gain distinct from channel diversity, and the second-order expansion exhibits a $P_{Y|X}$05 or $P_{Y|X}$06 improvement in the backoff term (Rowan et al., 15 Jan 2026).

6. Tightness, structured-code variants, and quantum generalizations

One recurrent issue is whether the logarithmic overhead in one-shot random coding is an artifact. In the SFRL framework the relevant quantity is the excess functional information

$P_{Y|X}$07

For discrete $P_{Y|X}$08, there is a general lower bound on $P_{Y|X}$09; for $P_{Y|X}$10, equality holds and the Poisson functional representation attains the infimum; and there exist examples with $P_{Y|X}$11 arbitrarily large for which

$P_{Y|X}$12

Thus the $P_{Y|X}$13 term is essentially tight, within approximately $P_{Y|X}$14 bits of the SFRL upper bound (Li et al., 2017). A common misconception is therefore that one-shot overhead should collapse to a universal constant independent of mutual information; the examples in the paper rule that out.

Another misconception is that one-shot random coding is synonymous with unstructured i.i.d. coding. For Abelian group codes, one-shot achievability is expressed through group-constrained hypothesis-testing mutual information, with classical and classical-quantum bounds of the form

$P_{Y|X}$15

in the classical case, and an analogous quantum bound with an additional prefactor in the cq case (Pang et al., 2024). The paper uses a random homomorphism ensemble and a group-averaged auxiliary distribution, showing that algebraic structure alters both achievability and converse terms through subgroup/coset penalties.

Quantum one-shot random coding bounds extend the same nonasymptotic philosophy. For a classical-quantum channel $P_{Y|X}$16, one recent theorem proves

$P_{Y|X}$17

for $P_{Y|X}$18, where $P_{Y|X}$19, $P_{Y|X}$20 is the Petz–Rényi mutual information, and the decoder is an integral $P_{Y|X}$21-PGM equivalent to a randomized Holevo–Helstrom measurement (Cheng et al., 8 Jul 2025). The same work recovers the optimal error exponent of classical-quantum channels for rates above the critical rate and extends to fully quantum channels, constant-composition codes, and compression with quantum side information.

Taken together, these developments suggest a broad classification of one-shot random coding bounds. One branch uses explicit random-code ensembles and stochastic or threshold decoders; a second uses representation lemmas and Poisson processes; a third uses hypothesis testing or Rényi quantities, especially in structured or quantum settings. Across these branches, the central object remains the same: a finite-blocklength achievability estimate that can be evaluated directly in the one-shot regime and that reproduces first-order and, in many cases, second-order asymptotics when specialized to memoryless models.