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.

翻译：本文目的有二。首先，我们证明Kim的二元自对偶码构建方法与Chinburg-Zhang的希尔伯特符号构造等价。其次，引入Chinburg-Zhang构造的q元版本，以高效构建q元自对偶码。针对后者，我们从三个互补视角研究分裂有限域\(\F_q\)（其中\(q \equiv 1 \pmod{4}\)）上的自对偶码：构建方法、Chinburg-Zhang的二元算术约化以及欧几里得平面的双曲几何。条件“-1为平方数”是连接这些视角的公共代数输入：在二元情形中，该条件奠定拉格朗日约化图景的基础；而在分裂的q元情形中，它产生支配扩增公式中修正项的各向同性线。作为生成矩阵高效形式的应用，我们通过分裂箱式构造构建了最优自对偶码，包括\(\GF{5}\)上的自对偶\([6,3,4]\)码与\([8,4,4]\)码、\(\GF{13}\)上的MDS自对偶\([8,4,5]\)码与\([10,5,6]\)码，以及\(\GF{13}\)上的自对偶\([12,6,6]\)码。这些结构结论伴随有代数核心的Lean~4形式化实现。