Formalizing building-up constructions of self-dual codes through isotropic lines in Lean

Abstract: The purpose of this paper is two-fold. First we show that Kim's building-up construction of binary self-dual codes is equivalent to Chinburg-Zhang's Hilbert symbol construction. Second we introduce a $q$-ary version of Chinburg-Zhang's construction in order to construct $q$-ary self-dual codes efficiently. For the latter, we study self-dual codes over split finite fields (\F_q) with (q \equiv 1 \pmod{4}) through three complementary viewpoints: the building-up construction, the binary arithmetic reduction of Chinburg--Zhang, and the hyperbolic geometry of the Euclidean plane. The condition that (-1) be a square is the common algebraic input linking these viewpoints: in the binary case it underlies the Lagrangian reduction picture, while in the split (q)-ary case it produces the isotropic line governing the correction terms in the extension formulas. As an application of our efficient form of generator matrices, we construct optimal self-dual codes from the split boxed construction, including self-dual ([6,3,4]) and ([8,4,4]) codes over (\GF{5}), MDS self-dual ([8,4,5]) and ([10,5,6]) codes over (\GF{13}), and a self-dual ([12,6,6]) code over (\GF{13}). These structural statements are accompanied by a Lean~4 formalization of the algebraic core.

• The paper establishes the equivalence between Kim's building-up construction and Chinburg-Zhang's cohomological method for self-dual code construction.
• It introduces a generalized q-ary framework via isotropic lines, yielding explicit optimal self-dual codes over GF(5) and GF(13).
• The work is fully formalized in Lean 4, certifying 256 theorems and ensuring rigorous correctness through formal verification.

Formalization and Structure of Self-Dual Code Building-Up Constructions via Isotropic Lines in Lean

Introduction

The paper "Formalizing building-up constructions of self-dual codes through isotropic lines in Lean" (2604.08485) systematically unifies, generalizes, and formalizes the construction of self-dual codes using isotropic lines in finite fields, with explicit constructive perspectives and formal verification in Lean 4. It delivers both theoretical synthesis—showing the equivalence between Kim's building-up construction and Chinburg-Zhang's Hilbert symbol-based approach—and pragmatic advances in algorithmically constructing optimal self-dual codes over fields where $q = 1 \pmod{4}$, with formal correctness proofs.

Summary of Main Results

Equivalence of Kim and Chinburg-Zhang Constructions

The paper rigorously identifies Kim's building-up construction with the cohomological construction of Chinburg and Zhang for binary self-dual codes. Specifically, Theorem 3.4 establishes that Kim's bottom-up approach and the Chinburg-Zhang top-down boxed reduction are inverse mechanisms: each step in one construction corresponds to a complementary operation in the other, both defining identical recursive predecessor-successor relationships but in reverse order. This bridges coding theory, arithmetic geometry, and cohomology, linking explicit code extension strategies with deep structural properties of maximal isotropic subspaces (Lagrangians) in bilinear spaces.

Generalization to $q$-Ary Self-Dual Codes via Isotropic Lines

For finite fields $\mathbb{F}_q$ with $q = 1\pmod 4$, the authors develop a $q$-ary version of the building-up construction, extending the boxed formalism to arbitrary $q$ via the algebraic property that $-1$ is a quadratic residue. Geometrically, this enables the use of isotropic lines in a distinguished split hyperbolic plane to organize the correction vectors in code extension. The core algebraic identity $c^2=-1$ underlies the construction, yielding explicit generator matrix forms and extension criteria (Theorems 3.9 and 3.12), ensuring that the newly constructed codes are always self-dual.

Explicit Optimal Self-Dual Codes Over GF(5) and GF(13)

The formalism is leveraged to construct explicit generator matrices for small optimal self-dual codes over $\mathrm{GF}(5)$ and $\mathrm{GF}(13)$. These include:

• Self-dual $q$0 and $q$1 codes over $q$2
• MDS self-dual $q$3 and $q$4 codes over $q$5
• A self-dual $q$6 code over $q$7

All these codes meet established bounds (Singleton, Griesmer) for their lengths and dimensions. The construction method is not only theoretically justified but also validated by exhaustive distance enumeration confirming their optimality.

Lean 4 Formalization

The work is formalized in Lean 4 with a single artifact certifying 256 theorems with zero sorry. The formalization covers:

• Definitions of self-dual codes and Lagrangian subspaces;
• Building-up constructions and their inverses;
• Equivalences between different perspectives (cohomological, matrix-normal, split-boxed);
• Forward and reverse theorems for constructing and deconstructing generator matrices for self-dual codes.

This offers a reusable and verified framework for further mechanized research in coding theory and the interaction between combinatorics, algebraic geometry, and formal mathematics.

Technical Discussion

The paper constructs a uniform algebraic interface for code extension based on isotropic lines in hyperbolic planes. In the binary case, the connection to étale cohomology and Hilbert symbols provides an arithmetic-geometric underpinning. The split $q$8-ary regime exploits the norm-splitting $q$9 afforded by $\mathbb{F}_q$0. Generator matrices are organized in split boxed block forms, with isolated equations in the isotropic parameters ensuring self-duality.

A notable technical advance is the forward and reverse boxed normal forms. In the forward direction (Theorem 3.12), explicit generator matrices are built recursively, with correction terms and cross-block symmetry enforced by the split geometry. In the reverse direction (Theorem 3.13), the form is shown to be canonical up to isometry given the presence of a distinguished isotropic-line row, yielding a structural classification of self-dual codes in this context.

The exhaustive formalization in Lean abstracts the essential algebraic content (orthogonality, dimension-counting, norm forms) away from field-dependent details, capturing the universality of the construction as a family of theorems and algorithmic interfaces.

Implications and Future Directions

The constructive formalism connects classical results on the enumeration and structure of self-dual codes with explicit, verifiable, and machine-checkable algorithms for code construction. This is of direct importance for reliable construction of quantum codes, as self-duality in classical codes often translates into stabilizer code conditions in the quantum domain. The framework can, in principle, be extended and adapted to other classes of codes (over rings, modules, etc.) and other bilinear forms, and may facilitate the automated discovery and certification of new extremal codes.

The integrated Lean 4 development enables the leverage of automated theorem proving in coding theory, opening the path to larger-scale formal verification projects, including classification efforts and the computer-aided search for new optimal codes. Additional directions include formalizing the connections to cohomology more generally (beyond the split case), incorporating automorphism group actions, and expanding algorithmic searches with formal correctness guarantees.

Conclusion

This paper achieves a fully formalized, algebraically explicit, and geometrically motivated synthesis of the building-up constructions of self-dual codes over split fields, connecting classical extension theorems, cohomological reduction, and modern formal methods. It delivers new optimal codes, forges a direct equivalence between key constructions, and provides a reusable, certified Lean 4 foundation for subsequent research in the area, underscoring the feasibility and value of full formalization in advanced algebraic coding theory.