Seeded Modular Code Design

• Seeded modular code design is a framework that constructs larger code structures by instantiating them with smaller, pre-specified modules or algebraic templates.
• It integrates methodologies from coding theory, secure computation, and neural code generation to optimize error-correction, secrecy gains, and distributed computation.
• The approach enables efficient, scalable, and maintainable system designs through modular architectures applied in lattice constructions, program synthesis, and design-to-code transformations.

Seeded modular code design refers to a class of methods in which code artifacts, codewords, or program modules are constructed in a compositional and hierarchical fashion, where each larger unit is “seeded” or instantiated using smaller, previously specified modules or algebraic templates. The seeding process encodes essential structural, algebraic, or functional properties, enabling modularity, reusability, and systematic control over code properties such as error-correction, security, or design abstraction. This paradigm intersects advanced topics in coding theory, program synthesis, secure computation, and neural code generation, providing a flexible and rigorous framework for constructing robust and scalable systems.

1. Algebraic Origins and Modular Lattice Construction

The modular approach to code construction is exemplified in the algebraic structure of lattices developed via number-theoretic generalizations of Construction A (Hou et al., 2016). In this framework, a linear code over a finite field serves as the “seed” for the construction of a lattice over a number field. Formally, given a Galois number field $K$ (totally real or CM), with ring of integers $\mathcal{O}_K$, and a prime ideal $\mathfrak{p} \subset \mathcal{O}_K$, the reduction map

$$\rho_{\mathfrak{p}}: \mathcal{O}_K^N \to (\mathcal{O}_K/\mathfrak{p})^N \cong \mathbb{F}_p^{fN}$$

is used to lift an $\mathbb{F}_p$-linear code $C \subset \mathbb{F}_p^{fN}$ to a lattice $L = \rho_{\mathfrak{p}}^{-1}(C)$ in $K^N$. The lattice is seeded both by the code $C$ and by an algebraic element $a\in K$ (often from the codifferent), which defines a positive-definite bilinear form:

$$b_a(x, y) = \sum_{i=1}^N \operatorname{Tr}_{K/\mathbb{Q}}\left(a x_i \overline{y}_i\right)$$

Explicit generator and Gram matrices are derived, whose determinants and inner product structures encode code minima and symmetries (Equations (4)-(9)), permitting precise control over invariants like minimal norm, theta series, and secrecy gain. The minimal norm $\mu(L)$ and secrecy gain $\chi_w(L)$ quantify the code’s robustness for communication and secrecy applications. Seeding with suitably chosen self-dual or self-orthogonal codes enables construction of (uni)modular lattices with extremal invariants, as illustrated by explicit constructions of modular lattices in dimensions 8, 12, etc.

2. Modular Code Design for Security and Channel Separation

In the context of information-theoretic security, seeded modular code design appears in modular BRI (biregular irreducible function) schemes (Wiese et al., 2018). Here, the code architecture is split into two composable modules:

• A security component seeded by a biregular irreducible function $f$.
• An error-correcting code tailored for the main communication channel.

The biregular irreducible function is implemented via edge-disjoint decompositions of complete bipartite graphs into Ramanujan biregular graphs, and the key security metric, the semantic security leakage, is provably upper-bounded as a function of (1) the channel properties (through an $E$-smooth conditional Rényi-2-divergence term) and (2) the expansion property of $f$, measured by the second-largest eigenvalue $\varphi_2$ of the associated stochastic matrix (Theorem 15). This modular structure achieves optimal trade-offs between rate and leakage decay, with the existence of optimal “nearly Ramanujan” sequences. The separation between error-correction and security generation is shown to achieve secrecy capacity for discrete and Gaussian wiretap channels, even when the seed is generated locally and reused.

3. Efficient and Secure Distributed Computation via Seeded Modularity

The principle of seeding manifests in distributed coded computation through Modular Polynomial (MP) codes and Generalized Gap Additive Secure Polynomial (GGASP) codes (Karpuk et al., 2023). MP codes exploit the fact that, for matrix multiplication, only a seed set of polynomial coefficients (“useful” coefficients corresponding to desired submatrix products) need to be decoded. This is achieved by the mod-$M$ transform:

$$\hat{h}(x) = \frac{1}{M} \sum_{m=0}^{M-1} \zeta^m h(\zeta^m x)$$

where $\zeta$ is a primitive $M$-th root of unity. The resulting design seeds interpolation to a strict subset of coefficients, reducing the recovery threshold and the decoding complexity—$O(ab \cdot \max(N, P\log P))$ for output submatrix size $ab$—relative to grid or entangled polynomial codes. GGASP codes implement seeded “gaps” in degree support to further lower the recovery threshold. Experiments demonstrate that, for various ($K, M, L, T$) regimes, the rate and robustness of MP and GGASP codes often outperform other polynomial codes. Notably, MP codes exhibit graceful degradation under certain straggler patterns, as only a subset of worker responses may suffice for recovery.

4. Program Synthesis and Hierarchical Specification Seeding

Seeded modularity extends to program synthesis in the form of modular system synthesis (Park et al., 2023). The process decomposes program construction into layered, compositional sub-problems, where each module is synthesized from two sorts of specifications:

• An implementation-specific specification (relying on the module and lower-level modules).
• An implementation-agnostic “seeded” specification abstracted from the lower layer.

Each newly synthesized module's specification acts as a “seed” or primitive for higher-level synthesis (“one module’s semantics is another module’s primitives”). The methodology relies on information hiding; module interfaces and abstract algebraic specifications (expressed, e.g., as term-rewriting systems or via regular-tree grammars) shield higher layers from lower-level implementation details, preserving uniform problem structure and bounding the search space. Automated tools such as MoSSKit (JLibSketch for implementation, Spyro for specification mining) support this approach.

5. Seeded Modular Code Generation with Neural Systems

Recent neural systems extend modular seeding principles to LLM-driven code generation.

• Chain-of-Revision Modularization (CodeChain) (Le et al., 2023): The process begins with chain-of-thought (CoT) prompting to prompt modular decomposition. Multiple code samples are generated, sub-modules are clustered via embedding (e.g., StarCoder, CodeT5+), and centroids of clusters serve as reusable module seeds. These are injected into successive prompts, driving an iterative cycle of self-revision. Formalism:

$$C_k = \arg\min_{S \in \text{cluster } k} \| \text{embedding}(S) - \mu_k \|$$

Empirical pass@1 improvements of up to 76% on CodeContests are achieved, and ablation confirms that representative module seeding and iteration are critical to performance gains.

• Hierarchical Modular Prompting (MoT) (Pan et al., 16 Mar 2025): MoT decomposes code generation tasks into a three-level Multi-Level Reasoning (MLR) graph, where high-/intermediate-/detailed-level nodes seed modular reasoning subproblems. Each graph node encapsulates task purpose, rationale, and execution strategy, and the modularized plan tightly aligns code generation with reasoning structure. MoT outperforms CoT and SCoT, delivering Pass@1 scores up to 95.1% on HumanEval+ and significant gains (e.g., ≥32.85% on MBPP over baselines).

These seeded approaches facilitate error isolation, local refinement, and compositional reuse—a plausible implication is improved maintainability, scalability, and testability of machine-generated code.

6. Modular Code Extraction in Design-to-Code Transformation

In design-to-code conversion, modularity is enforced through structured preprocessing and component extraction (Muhammad et al., 22 Jul 2025). LOCOFY Large Design Models (LDMs) implement:

• Design Optimiser: XGBoost-based module trained on expert-annotated ground truth, restructures sub-optimal designs into semantically organized hierarchies suitable for modularization.
• Tagging and Feature Detection: Fine-tuned YOLO-type architectures (Jasmine) identify UI elements and their relationships with high consistency, grouping them for subsequent modular extraction.
• Auto Components: Repeated UI structures across screens are automatically grouped, abstracted, and instantiated as reusable components with dynamically inferred properties.

The inference pipeline deterministically processes input designs stepwise to generate modular, production-grade code. The Preview Match Score (PMS) metric quantitatively validates code quality:

$\text{PMS} = \frac{M}{N} \times 100\%$

where $M$ is the number of well-matched nodes (in terms of position and size within 3% tolerance), further demonstrating structural fidelity and modularity.

7. Theoretical Foundations, Performance, and Implications

Seeded modular code design frameworks rigorously separate concerns, enabling optimized control of invariants (e.g., minimal norm, secrecy gain, rate, recovery threshold), reducing computational complexity, and providing clear abstraction boundaries. Capacity-achieving codes for secure channels, optimal distributed matrix codes, and scalable program synthesis are realized within this paradigm. Empirical evidence across domains—coding theory, computer security, distributed computing, program synthesis, and neural code generation—demonstrates that seeded modular approaches consistently yield superior or competing performance on key metrics (e.g., pass@1, PMS, recovery threshold) compared to non-modular or monolithic baselines.

Limitations include practical computational barriers for certain algebraic constructions (e.g., efficient instantiation of Ramanujan biregular irreducible functions), potential exponential growth in search space if abstraction boundaries are not precisely drawn, and bottlenecks in automatic specification synthesis for modular synthesis frameworks. Future research is directed toward more efficient algorithmic implementations, integration with formal verification, compositional synthesis in reactive/data-driven contexts, and expandability to novel domains while maintaining the seeded modular structure.

Seeded modular code design thus represents a mathematically and computationally principled methodology for scalable, robust, and cognitively aligned code generation and program synthesis, with strong theoretical guarantees and demonstrated empirical validity across multiple research and application domains.