DELTA-Code: Divisibility in Codes & Learning

• DELTA-Code is a framework uniting algebraic coding theory and reinforcement learning, emphasizing divisibility and differential representations.
• It classifies Δ-divisible linear codes via canonical decompositions, combinatorial design, and geometric interpretations to establish structural bounds.
• The approach benchmarks algorithmic reasoning in LLMs by evaluating learnability and transferability, highlighting new empirical methods and RL strategies.

DELTA-Code denotes a family of methodologies in coding theory, finite geometry, and algorithmic reasoning, unified by the notion of divisibility, differential representations (“delta”), and the paper of learnability and transferability in models and codes. The term is used in multiple research contexts, including the classification of $\Delta$-divisible linear codes, investigations of geometric and combinatorial properties of codes over finite fields, and controlled benchmarks in reinforcement learning for algorithmic discovery in LLMs. DELTA-Code is fundamentally characterized by rigorous algebraic definitions, structured decompositions, and empirical methods for probing new paradigms of learning and generalization.

1. Definitions and Fundamental Concepts

DELTA-Code, in the context of coding theory, refers largely to the paper and classification of $q$-ary $\Delta$-divisible linear codes—codes for which every codeword has Hamming weight divisible by a fixed integer $\Delta$ (Kiermaier et al., 2020, Kurz, 2021). In this setting, the characterization of codes revolves around divisibility constraints and the underlying algebraic structures. The key invariant is the exponent $a$ such that $q^a$ divides $\Delta$; such codes are also referred to as $q^a$-divisible codes.

For a linear code $C \leq \mathbb{F}_q^n$ with $\Delta$-divisibility, the weights of nonzero codewords belong to $\Delta \mathbb{Z}$. This property leads to structural constraints, bounds on dimension, and connections to projective geometry and combinatorial designs. A critical principle is the building-block theorem: up to codimension and coordinate manipulations, all such codes decompose into direct sums and repetitions of canonical codes—primarily simplex codes, first-order Reed–Muller codes, and parity-check codes.

In algorithmic reasoning and reinforcement learning with LLMs, DELTA-Code also refers to the "Distributional Evaluation of Learnability and Transferrability in Algorithmic Coding," a benchmark for probing whether RL can equip pretrained LLMs with genuinely novel procedural strategies and whether these skills transfer to new tasks (Sun et al., 25 Sep 2025).

2. Classification of $\Delta$-Divisible Codes

The classification framework hinges on several algebraic and combinatorial steps (Kiermaier et al., 2020):

• Reduction to Basic Building Blocks: Any $q$-ary $\Delta$-divisible code spanned by codewords of weight $\Delta$ is, up to repetitions and zero appendage, a sum of simplex codes, and for $q=2$, Reed–Muller and parity-check codes.
• Canonical Decomposition: Indecomposable full-length $\Delta$-divisible codes are classified by dimension $k$, exponent $a$, and the repetition factor $T$.
• Inductive Characterization: Employing local support and intersection properties (see Lemma 3.1), one removes and re-adds minimal-weight codewords, constraining possible configurations.
• Projective Codes and Alternative Proofs: Liu’s projective classification theorem is elegantly rederived from these principles, showing that every projective binary $4$-divisible code of length $4\Delta$ must decompose into classified families.

Family Type	Description	Core Example
Simplex	Constant weight, generic $q$	$Sim_q(k)$
Reed-Muller	Binary, structure-rich	$RM_2(k)$
Parity Check	Binary, balanced sum	$PC_2(k)$

This structure underpins the import of DELTA-Code for both theory and enumerative classification. The reduction to canonical forms aids in the exploration of divisibility bounds and the existence spectrum of codes for given $(n, k, q, \Delta)$ parameters.

3. Geometric and Combinatorial Interpretations

$\Delta$-divisible codes are intimately related to geometric configurations in finite projective spaces (Kurz, 2021). Key connections include:

• Partial Spreads and Vector Space Partitions: The "hole set" of incomplete spreads forms a $q^x$-divisible multiset, influencing bounds on spread size.
• Projective $q^x$-divisible Sets: Classification results associate the existence and cardinality of $q^x$-divisible sets with parameters of underlying subspaces and unions. Non-existence proofs are achieved via linear programming methods and combinatorial invariants.
• Cylinder Conjecture: For prime fields, it is conjectured that every $q$-divisible set of $q^r+1$ points in $PG(v-1, q)$ with maximal dimension must be cylindrical—a hypothesis open for prime fields and refuted for non-prime cases.

These interpretations reinforce the significance of DELTA-Code in combinatorial geometry, block designs, and incidence theory, linking coding-theoretic divisibility to geometric partition constraints.

4. Methods, Bounds, and Open Problems

A rigorous toolkit supports DELTA-Code research:

• Divisible Code Bounds: Inequalities such as variants of the Griesmer bound, MacWilliams identities, and algebraic criteria restrict feasible code parameters and nonexistence.
• Linear Programming and Classification: Extended LP bounds and combinatorial analysis eliminate parameter sets, especially for small $n$ and $q$.
• Algorithmic Construction and Computer Classification: Empirical methods include machine computations for small cases and algebraic generation techniques.

Active research directions are dominated by:

• Cylinder Conjecture Elucidation
• Bounds for Partial Spreads
• Extendability of MRD Codes
• Full Classification of Small Parameter Cases
• Structural Investigation of Hole Sets and Tail Conditions

5. DELTA-Code in Algorithmic Reasoning and RL

In LLMs and coding problem benchmarks (Sun et al., 25 Sep 2025), DELTA-Code is formulated as a controlled distributional evaluation. It is constructed to answer:

• Learnability: Can RL techniques enable a model to grok solutions to problem families where pretraining yields $pass@K=0$? The benchmark demonstrates a sharp grokking transition: after prolonged exploration with near-zero rewards, RL-trained models abruptly achieve high pass rates.
• Transferrability: Assess transfer to OOD tasks, along exploratory, compositional, transformative, and cross-family axes. Transferrability is robust for recomposed skills, less so for qualitatively transformative tasks (requiring new solution schemas).

Pivotal RL ingredients include staged warm-up with dense rewards, experience replay, curriculum transfer, and verification-in-the-loop mechanisms. These regimes facilitate the emergence of new reasoning strategies and highlight the influence of reward design and curriculum composition on breaking prior-induced barriers.

6. Practical Implications and Future Directions

DELTA-Code, in both algebraic and algorithmic settings, provides a robust conceptual and empirical framework:

• Facilitates Systematic Classification of codes whose weights are subject to divisibility constraints, impacting coding theory and finite geometry.
• Enables Controlled Benchmarks for probing the limits of RL-driven reasoning in algorithmic problem-solving with LLMs, clarifying boundaries of generalization to OOD tasks.
• Informs Non-Existence Results, Structural Bounds, and Construction Methods in code theory, while advancing understanding of curriculum and reward design for learning new skills.
• Sustains Open Questions, notably in geometric combinatorics and transformative generalization, offering a foundation for future research that may extend DELTA-Code principles to novel domains in symbolic learning, formal logic, or even higher-level algebraic structures.

A plausible implication is that the DELTA-Code philosophy—formulating problems so as to isolate and expose structural limits—may serve as a template for forthcoming investigations in both coding theory and data-driven models. The confluence of algebraic rigor and empirical methodology distinguishes DELTA-Code as a reference construct in modern mathematics and machine learning research.