Let p be an odd prime and consider the finite field ${\mathbf{F}}_{p^2}$. Given a linear code $\mathcal{C}\subset {\mathbf{F}}_{p^2}^n$, we use algebraic number theory to construct an associated lattice ${\mathrm{\Lambda }}_{\mathcal{C}}\subset {\mathcal{O}}_L^n$ for L an algebraic number field and ${\mathcal{O}}_L$ the ring of integers of L. We attach a theta series ${\theta }_{{\mathrm{\Lambda }}_{\mathcal{C}}}$ to the lattice ${\mathrm{\Lambda }}_{\mathcal{C}}$ and prove a relation between ${\theta }_{{\mathrm{\Lambda }}_{\mathcal{C}}}$ and the complete weight enumerator evaluated on weight one theta series.

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希尔伯特模块化表格和代码 ${\mathbf{F}}_{p^2}$

设p为奇数素数并考虑有限域${\mathbf{F}}_{p^2}$。给定线性代码$\mathcal{C}\subset {\mathbf{F}}_{p^2}^{ñ}$，我们使用代数数论来构造一个相关的格 ${\mathrm{\Lambda }}_{\mathcal{C}}\subset {\mathcal{Ø}}_{\mathrm{大号}}^{ñ}$对于L的代数数域和${\mathcal{Ø}}_{\mathrm{大号}}$L的整数环。我们附上theta系列${\theta }_{{\mathrm{\Lambda }}_{\mathcal{C}}}$ 到格子 ${\mathrm{\Lambda }}_{\mathcal{C}}$ 并证明之间的关系 ${\theta }_{{\mathrm{\Lambda }}_{\mathcal{C}}}$ 完整的重量枚举器将按权重theta系列进行评估。