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

2026-04-09

AI Summary

This work presents a unified and efficient construction of binary and q-ary self-dual codes, addressing the formalization of their algebraic foundations. By establishing the equivalence between Kim’s construction and the Hilbert symbol approach of Chinburg–Zhang, the authors introduce a novel method based on isotropic lines over finite fields with \( q \equiv 1 \pmod{4} \). This approach integrates insights from hyperbolic geometry and isomorphism classes of lines to achieve efficient code generation, with core theoretical results formally verified in Lean 4. The method yields several optimal self-dual codes, including the \([6,3,4]\) and \([8,4,4]\) codes over \(\mathrm{GF}(5)\), as well as MDS self-dual codes \([8,4,5]\), \([10,5,6]\), and \([12,6,6]\) over \(\mathrm{GF}(13)\).

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.

Problem

• self-dual codes
• building-up construction
• isotropic lines
• finite fields
• Hilbert symbol

Innovation

• self-dual codes
• building-up construction
• isotropic lines
• split finite fields
• formal verification