Reversible Encoding Scheme

• Reversible encoding schemes are bijective transformations that ensure lossless data recovery through efficient, invertible algorithms.
• They leverage diverse algebraic and combinatorial frameworks—including self-reciprocal polynomials and module constructions—to optimize error correction and security.
• Applications span generative steganography, ANS-based data hiding, and quantum error correction, illustrating their versatility in communication and storage systems.

A reversible encoding scheme is a class of transformations or codes in which the encoding map from source data to an encoded representation is bijective: for every valid encoded object, there exists a well-defined and efficiently computable inverse map that recovers the original data with no loss of information. Such schemes underpin many foundational areas of information theory, cryptography, steganography, data storage, and error correction, offering precise control over reversibility, security, rate-distortion tradeoffs, robustness to noise and adversarial perturbations, and the imposition of combinatorial or algebraic constraints on the codewords.

1. Structural Principles of Reversible Encoding

The defining property of a reversible encoding scheme is the existence of a one-to-one mapping (bijection) between an input alphabet (messages, data, secrets, etc.) and some output space (codewords, transformed images, or channel symbols), together with explicit and efficient algorithms for both forward encoding and reverse decoding.

In formal terms, let $M$ denote the message space and $C$ the codeword space. A reversible encoding scheme specifies maps $E: M \to C$, $D: C \to M$ with $D \circ E = \text{Id}_M$ and $E \circ D = \text{Id}_C$ (on the range of $E$). In information-theoretic and cryptographic settings, reversibility may be further constrained to preserve auxiliary structure: e.g., coordinate permutations, algebraic symmetries, statistical distributions, or combinatorial orderings.

Reversibility is essential for lossless compression algorithms, data hiding (steganography, watermarking), tamper-evident or attack-resilient codes, and for ensuring data can be reconstructed in the presence of channel noise, errors, or adversarial manipulation.

2. Algebraic and Combinatorial Foundations

Reversible encoding is realized via diverse algebraic and combinatorial frameworks, each tailored to optimize performance, robustness, and the particular invariance or security requirements of the domain:

• Combinatorial bijections: Secret-to-image transformation for generative steganography employs a two-stage bijection—first mapping messages $m\in\{0,1\}^K$ to Gaussian latent vectors $z\in\mathbb{R}^D$ via explicit combinatorial assignments of Gaussian samples (by partition, sorting, and position selection), and then to images via a bijective generative model such as a trained Glow flow ($G$ and $G^{-1}$ are both invertible) (Zhou et al., 2022).
• Module-theoretic constructions: Linear codes over finite fields or modules tailored to satisfy specific symmetries (e.g., reversal, complement) admit a classification via the module structure. For instance, reversible and reversible-complementary DNA codes over $\mathbb{F}_4$ correspond to certain $R$-submodules $C\leq\mathbb{F}_4^n$ invariant under the reverse permutation, with explicit generator matrices and enumerative formulas for code counting (García-Claro, 23 Jun 2025).
• Polynomial and palindromic generators: In cyclic codes, reversibility corresponds to generator polynomials being self-reciprocal—i.e., $g(x) = x^{\deg g}g(x^{-1})$, ensuring the code is closed under coordinate reversal. BCH codes with self-reciprocal generators yield codes that are reversible and have favorable error-correcting and cryptographic properties (Li et al., 2016). Generalizations to $m$-quasi-reciprocal (coterm) polynomials extend this principle to more flexible code constructions and DNA coding scenarios (Chen et al., 2018).
• Homophonic and numeral systems: Variable-to-fixed length homophonic coding via dual Shannon–Fano–Elias–Gray codes provides fixed-rate blockwise reversible encoding from uniform to arbitrary channel distributions, crucial for capacity-achieving communication over asymmetric channels (Honda et al., 2016). Reversible data hiding based on Asymmetric Numeral Systems (ANS) exploits precisely invertible integer-based state transitions for efficient payload embedding and extraction, avoiding floating-point precision loss pervasive in arithmetic coding (Wang et al., 2023, Townsend et al., 2022).
• Cellular automata and algebraic dynamics: Frobenius-driven Laplacian cellular automata implement a reversible encoding scheme in which high-entropy states produced by chaotic evolution become exactly recoverable via algebraic collapse (Frobenius return), coupled with spatial redundancy via multi-tile revivals for error-tolerance and majority decoding (Nowak-Kępczyk, 21 Nov 2025).

3. Representative Methodologies and Algorithms

The realization of reversible encoding differs substantially by context, but several recurrent algorithmic motifs appear:

Domain	Main Encoding Principle	Decoding Strategy
Generative steganography	Combinatorial grouping & position arrangement + flow	Bijection inversion + combinatorial decoding
DNA/reversal coding	Module/based generator-matrix construction	Syndrome/coset or direct polynomial inversion
ANS-based data hiding	Integer state-driven symbol walking (ANS core)	State-lifting and reverse mapping
Homophonic coding	Interval-partition or Gray-order mapping	Interval arithmetic inversion
Cellular automata	Iterative linear chaotic transformation	Algebraic unraveling via Frobenius identity

For example, in S2IRT (Zhou et al., 2022), message bits are hidden in the specific combinatorial selection of latent vector positions, and extraction is exact when the latent variable mapping and Glow flow are both inverted with perfect numerical precision. In reversible ANS-based embedding (Wang et al., 2023), the encoding and decoding traverse the same sequence of reversible arithmetic state updates, guaranteeing bit-for-bit recovery.

In module-theoretic schemes for DNA codes (García-Claro, 23 Jun 2025), the R-module generator matrix explicitly encodes messages into codewords, with reversibility (and possibly reverse-complement invariance) assured by construction. Cyclic reversible BCH codes use multiplication by a self-reciprocal generator, with decoding by classic BCH syndrome solvers (Li et al., 2016).

For reversible string reconstruction from erroneous compositions (Chen, 16 Mar 2025), the code construction exploits the monotonicity of prefix/suffix weights and Ordered-Weight code embeddings. Decoding reduces to error correction on prefix sums and difference sequences, robustly recovering the original string under $\Theta(n)$ composition errors.

4. Capacity, Robustness, and Performance Metrics

Reversible encoding schemes are evaluated according to information embedding capacity, extraction accuracy, rate-versus-distortion tradeoffs, error correction capability, and computational efficiency.

• Capacity: In S2IRT, theoretical hiding rates reach up to $16$ bpp, with robust extraction demonstrated at $4$ bpp (Zhou et al., 2022). ANS-based embedding tracks the Shannon rate-distortion bound within $0.01$ bpp, and supports high rates (up to $2.5$ bpp at PSNR $\approx 22$ dB on images) (Wang et al., 2023).
• Extraction accuracy: Lossless recovery is characterized by metrics such as $IE_A=1-\frac{ED(m,m')}{\max(|m|,|m'|)}$, where $ED$ is the edit distance; S2IRT achieves $IE_A \approx 1.0$ in practice (Zhou et al., 2022). For error-tolerant DNA and string codes, robust decoding holds for code distances proportional to the number of attacked coordinates (Chen, 16 Mar 2025, García-Claro, 23 Jun 2025).
• Error tolerance: Codes constructed for prefix-suffix composition recovery protect against up to $\Theta(n)$ adversarial errors with polynomial-time decoding (Chen, 16 Mar 2025). In Laplacian CA-based encoding, spatial redundancy via multi-tile revivals enables majority-voting recovery under weak channel noise (Nowak-Kępczyk, 21 Nov 2025).
• Computational efficiency: Many module- and polynomial-based codes admit $O(n\log n)$ or polynomial-time encoding/decoding; ANS-based numeration schemes require only integer arithmetic and bit shifts (Wang et al., 2023, Townsend et al., 2022).

5. Specialized Structures: Symmetry and Invariance

Domain-dependent constraints on codeword symmetry drive construction choices:

• Reversal invariance: Codes with generator polynomials that are self-reciprocal, or codes based on coterm polynomials, ensure every codeword's reversal remains within the code (Li et al., 2016, Chen et al., 2018). In DNA coding, this fulfills biochemical reverse and reverse-complement constraints, crucial for molecular storage and computation (García-Claro, 23 Jun 2025).
• Distribution shaping and homophony: Homophonic encoding transforms uniform input distributions into prescribed output distributions (e.g., matching channel or storage bias), while maintaining reversibility and blockwise synchronization (Honda et al., 2016).
• Gauge and logical invariance in quantum codes: Reversible encoding schemes for topological quantum codes (toric, subsystem, and Haah's code) ensure logical operators under encoding/decoding are inverted correctly, enabling error-protected quantum memory with single-shot locality (Łodyga et al., 2014).

6. Applications and Contemporary Directions

Reversible encoding schemes are foundational tools across multiple fields:

• Steganography and data hiding: S2IRT demonstrates concealment of messages in generated images with strong undetectability and near-perfect extraction; robust variants (SE-S2IRT) further resist common image transformations (Zhou et al., 2022).
• DNA storage and computation: Reversible and reversible-complementary DNA codes, via module or coterm polynomial methods, meet biochemical and coding-theoretic requirements for robust DNA information storage (García-Claro, 23 Jun 2025, Chen et al., 2018).
• Error and attack resistance: Reversible codes with LCD (complementary dual) structure are secure against side-channel attacks and suitable for both classical and quantum coding (notably, CSS quantum codes) (Li et al., 2016).
• Compression and rate-distortion optimization: Verified reversible compression approaches, using formally verified invertible programming languages, jointly specify and prove correctness of the coding scheme, with implications for reliable storage and secure communication (Townsend et al., 2022).
• Cellular automata and procedural synthesis: The algebraic structure of certain cellular automata enables reversible information encoding and spatial error correction, applicable to procedural generation and tamper-evident storage (Nowak-Kępczyk, 21 Nov 2025).

7. Theoretical Limits and Open Problems

Open research questions pertain to optimal tradeoffs between code rate, minimum distance, and redundancy under various reversible and combinatorial constraints; explicit constructive methods for high-complexity reversibility in settings such as multi-parameter Laplacian CA or high-order DNA block codes; and formal connections between reversible encoding and other symmetries or invariants across classical and quantum domains (Nowak-Kępczyk, 21 Nov 2025, García-Claro, 23 Jun 2025, Li et al., 2016). Explicit determination of code parameters (e.g., dimension and distance) for general BCH and DNA codes with reversibility and complementarity remains active (Li et al., 2016, García-Claro, 23 Jun 2025).

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In summary, reversible encoding schemes underpin modern information-theoretic, cryptographic, and storage technologies, connecting deep algebraic and combinatorial design with demanding requirements for performance under attack, noise, and adversarial conditions. The diversity of methodologies—from combinatorial bijections to module and polynomial constructions to algorithmic dynamics—reflects the maturity and continuing evolution of this field.