Optimal Single-Class Coding

• Optimal single-class coding is defined as a framework where all messages are treated uniformly, using universal prefix, joint source–channel, and index coding without explicit partitioning.
• Recent constructions, such as the Δδ and ν codes, achieve near-optimal expansion factors, narrowing the bounds in universal integer compression.
• Analytical methods including Gallager’s expurgated bounds and graph-based linear coding reveal fundamental limits and guide further improvements in system performance.

Optimal single-class coding denotes the analytic and design framework for coding systems—particularly universal prefix coding, joint source–channel coding, and network index coding—where all messages or source sequences are treated as a single class, without explicit partitioning. In such codes, parameters governing codebook construction, or codeword assignment, are held independent of individual message probabilities or side information. The optimality of single-class constructions is highly context-dependent, varying across source, channel, and network coding settings. Notably, recent advances sharpen understanding of expansion factors in integer coding and expurgated exponents for discrete memoryless source–channel models.

1. Formal Definitions and System Models

Single-class coding arises in a range of problems:

• Universal Coding of Integers (UCI): For alphabet $\mathcal{A} = \mathbb{N}$, a prefix code $\mathcal{C}:\mathcal{A} \to \{0,1\}^*$ is universal if there exists a finite constant $K_{\mathcal{C}}$ such that for every decreasing distribution $P$ with $0<H(P)<\infty$,

$$\frac{A_P(L_{\mathcal{C}})}{\max\{1, H(P)\}} \leq K_{\mathcal{C}}$$

where $A_P(L_{\mathcal{C}})$ is the average codeword length and $H(P)$ is the entropy (Yan et al., 31 Jul 2025).

• Joint Source–Channel Coding (JSCC): For a discrete memoryless source (DMS) $P_V$ and discrete memoryless channel (DMC) $W$, messages $v \in \mathcal{V}^k$ are mapped to codewords $x(v)\in\mathcal{X}^n$ via an encoder, with decoder seeking to minimize $\Pr[V^k \neq \widehat{V}^k]$. In single-class coding, all codewords $x(v)$ are drawn i.i.d. from a fixed $Q \in \mathcal{P}(\mathcal{X})$ (Moeini et al., 21 Jan 2026).
• Index Coding: For a single-class index coding problem (one message/request per receiver), the code does not differentiate among message types; optimal codes may be linear or nonlinear, depending on graph structure (Ong, 2013).

2. Optimal Single-Class Coding in Universal Integer Compression

The optimal expansion factor $C^*$ for universal coding of integers is defined as: $$C^* = \inf_{\mathcal{C} \in \text{UCI}} \sup_P \frac{A_P(L_{\mathcal{C}})}{\max\{1, H(P)\}}$$ where the infimum is over all prefix codes. No prefix code achieves $K^*_{\mathcal{C}}<2$. A construction shows, for $P_m$ with $P_m(1)=1-1/m$ and $P_m(a)=1/(m2^m)$ for $a=2,\ldots,2^m+1$, that $H(P_m)\to 1$ while $A_{P_m}(L_{\mathcal{C}})>2$ for any UCI $\mathcal{C}$ as $m\to\infty$ (Yan et al., 31 Jul 2025).

Two recent codes sharply approach this lower bound:

• $\Delta\delta$ code: Expansion factor $K^*_{\Delta\delta}\leq 2.0821$.
• $\nu$ code: Expansion factor $K^*_{\nu} \leq 2.0386$.

This narrows the known optimal UCI interval to $2 \leq C^* \leq 2.0386$. Neither code achieves expansion below $2.02$ for worst-case $P$; further improvement would require fundamentally new constructions or code families (Yan et al., 31 Jul 2025).

3. Expurgated Exponents for Single-Class Joint Source–Channel Codes

In the context of joint source-channel coding with i.i.d. ensemble, the optimal single-class expurgated error exponent for a rate $t=k/n$ is: $$E_{J,\mathrm{ex}}^{\mathrm{sc}}(t) = \sup_{\rho\geq 1} \Bigg\{ E_x(\rho) - t\,E_s(\rho, P_V) \Bigg\}$$ where

$$E_x(\rho) = \max_{Q\in\mathcal{P}(\mathcal{X})} \left[ -\rho\log\sum_{x,\bar{x}} Q(x)Q(\bar{x}) e^{-d_B(x,\bar{x})/\rho} \right]$$

$$E_s(\rho,P_V) = \log\left( \sum_{v\in\mathcal{V}} P_V(v)^{1/(1+\rho)} \right)^{1+\rho}$$

$$d_B(x,\bar{x}) = -\log \sum_{y} \sqrt{ W(y|x) W(y|\bar{x}) }$$

Gallager-style analysis, leveraging two-level expurgation and parameter optimization, yields the above (Moeini et al., 21 Jan 2026).

This exponent provides a fundamental lower bound for achievable reliability in JSCC with i.i.d. codebooks, independent of partitioning or source-dependent codeword assignments.

4. Optimality Regimes, Comparison with Multi-Class and Partitioned Approaches

Integer Coding

The $\nu$ and $\Delta\delta$ codes currently constitute the best-known single-class universal integer codes. Improvement to $C^*$ below $2$ is impossible for any UCI; reaching the lower bound remains open only up to $2 \leq C^* \leq 2.0386$ (Yan et al., 31 Jul 2025).

JSCC Expurgated Exponents

Two-class partitioning, in which source sequences are grouped into classes with distinct i.i.d. codeword distributions, allows at least as high an exponent: $$E_{J,\mathrm{ex}}^{\mathrm{mc}}(t) \geq E_{J,\mathrm{ex}}^{\mathrm{sc}}(t)$$ If the maximizing distributions coincide, the single-class exponent is attained. Numerically, optimizing over all code distributions leads to equality of exponents for all tested scenarios, though strict improvement from partitioning can occur if class code distributions are not both optimized. For binary input alphabets ($|\mathcal{X}|=2$), partitioning never improves the exponent, since the maximizer is always uniform, rendering the single-class $E_x(Q,\rho)$ function already concave in $\rho$. Whether $$E_{J,\mathrm{ex}}^{\mathrm{mc}}(t)=E_{J,\mathrm{ex}}^{\mathrm{sc}}(t)$$ for all DMS–DMC pairs is unresolved (Moeini et al., 21 Jan 2026).

Index Coding

For single-class index-coding instances described by directed graphs, the minimum codelength $\ell^*$ is at least the size of a maximum acyclic induced subgraph (MAIS). For any graph $G$ with this number reached by deleting at most two vertices, the MAIS lower bound is attained by a linear code: $$\ell^*(G) = \mathrm{MAIS}(G) = |V(G)|-\min\{|V_r|\}$$ where $V_r$ is a minimum vertex set whose removal makes $G$ acyclic. The existence of a matching linear coding solution for these cases demonstrates optimality of single-class, linear constructions in a broad class of networks (Ong, 2013).

5. Coding Constructions and Analytical Techniques

Integer Coding

The $\Delta\delta$ code modifies four codewords of Elias’s $\delta$ code, maintaining prefix-freeness while achieving superior expansion. The $\nu$ code perturbs $L_\delta(a)$ at finitely many $a$ and blocks of indices defined by binary-logarithms. Analytical upper bounds employ piecewise linear inequalities for codeword length and probabilistic concentration over decreasing $P$, culminating in explicit universal bounds (Yan et al., 31 Jul 2025).

JSCC Exponents

The exponent derivation combines Gallager’s expurgated bound for channels with expurgation over source types, leveraging Hölder’s inequality and the Bhattacharyya distance. Specialization to single-class codes (constant $Q$, constant $\rho$) yields the succinct maximization in $E_{J,ex}^{sc}(t)$ (Moeini et al., 21 Jan 2026).

Index Coding

Linear codes are constructed via generator matrices linked to the graph’s acyclic substructures. For $k\leq 2$ vertex removals, row entries are sums of the message and its internal neighbors in the critical subgraph, ensuring recoverability through algebraic cancellation (Ong, 2013).

6. Comparison Table and State of the Art

Context	Optimal Single-Class Metric	Best Known Value/Expression	Notable Result
Universal Integer Coding	Expansion $C^*$	$2 \leq C^* \leq 2.0386$	$\nu$ code is nearly optimal
JSCC Expurgated Exponent	$E_{J,ex}^{sc}(t)$	Supremum formula (see above)	Two-class ≥ single-class; possibly =
Index Coding (MAIS cases)	Codelength $\ell^* = \mathrm{MAIS}$	Linear codes attain lower bound for $|V_r|\leq 2$	Linear coding is optimal

Improvements in single-class constructions are tightly bounded by information-theoretic impossibility results. In all tabulated scenarios, single-class coding achieves, or comes near, the fundamental performance limits, with further gains requiring either problem-specific multi-class techniques or fundamentally new code classes.

7. Open Problems and Future Directions

In universal coding, the existence of a single prefix code achieving $K^* = 2$ remains unresolved; all known codes saturate just above this limit. In JSCC, while two-class partitioning gives no observed numerical advantage under joint optimization, the general equality of exponents is open for broader DMS–DMC combinations and larger alphabets (Moeini et al., 21 Jan 2026). In graph-based network coding, the mathematics of MAIS-based linear-optimality suggests further structure-dependent single-class code optimality criteria (Ong, 2013).

A plausible implication is that, in diverse settings, optimal single-class coding is fundamentally limited by specific combinatorial or convexity constraints, with only marginal gains possible from further fine-tuning unless new insights expose currently hidden structural leverage.