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Lattices of type $A_n$, $D_n$, $E_n$ and codes

Riku Higa

TL;DR

The paper develops a unified construction that generalizes Construction A to lattices of type $A_n$, $D_n$, and $E_n$ by pairing them with suitable codes over rings such as $ ext{Z}/(n+1) ext{Z}$, $ ext{Z}/4 ext{Z}$, $ ext{F}_2+u ext{F}_2$, $ ext{F}_4$, and $ ext{F}_2 imes ext{F}_2$. It proves a central theorem: given a lattice type $ ext{Λ}$ and a code $C$, there is a lattice $ ext{Γ}_C$ between $ ext{Λ}^{boplus m}$ and $( ext{Λ}^*)^{boplus m}$ whose integrality, unimodularity, and evenness are exactly characterized by duality properties of $C$ (Euclidean/Hermitian self-duality) and weight criteria (Euclidean, Lee, Bachoc) depending on the root type and parity of $n$. The framework yields numerous even unimodular lattices from codes and links these lattices to Hilbert modular forms via theta functions, with explicit $K$-lattice realizations (e.g., a $D_4$-type lattice over $ ext{Q}( ext{e}^{2 ext{i}\pi/4})$ and an $E_6$-type lattice over $ ext{Q}( ext{e}^{2 ext{i}\pi/9})$) where theta series align with weight enumerators. This work provides both structural lattice-code correspondences and concrete modular-form applications, expanding the toolkit for constructing and analyzing lattices linked to automorphic forms.

Abstract

We propose a construction of lattices from codes corresponding to lattices of type $A_n$, $D_n$ and $E_n$. This construction is a generalization of construction A of lattices from $p$-ary codes corresponding to a lattice of type $A_{p-1}$. Moreover, we introduce some examples of application of lattices from the construction to Hilbert modular form.

Lattices of type $A_n$, $D_n$, $E_n$ and codes

TL;DR

The paper develops a unified construction that generalizes Construction A to lattices of type , , and by pairing them with suitable codes over rings such as , , , , and . It proves a central theorem: given a lattice type and a code , there is a lattice between and whose integrality, unimodularity, and evenness are exactly characterized by duality properties of (Euclidean/Hermitian self-duality) and weight criteria (Euclidean, Lee, Bachoc) depending on the root type and parity of . The framework yields numerous even unimodular lattices from codes and links these lattices to Hilbert modular forms via theta functions, with explicit -lattice realizations (e.g., a -type lattice over and an -type lattice over ) where theta series align with weight enumerators. This work provides both structural lattice-code correspondences and concrete modular-form applications, expanding the toolkit for constructing and analyzing lattices linked to automorphic forms.

Abstract

We propose a construction of lattices from codes corresponding to lattices of type , and . This construction is a generalization of construction A of lattices from -ary codes corresponding to a lattice of type . Moreover, we introduce some examples of application of lattices from the construction to Hilbert modular form.

Paper Structure

This paper contains 14 sections, 36 theorems, 188 equations, 18 tables.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Lattices and codes
  4. Lattices of type $A_n$ and $\mathbb{Z}/(n+1)\mathbb{Z}$-codes
  5. Lattices of type $D_n$ and $\mathbb{Z}/4\mathbb{Z}$-codes
  6. Lattices of type $D_n$ and $(\mathbb{F}_2+u\mathbb{F}_2)$-codes
  7. Lattices of type $D_n$ and $\mathbb{F}_4$-codes
  8. Lattices of type $D_n$ and $(\mathbb{F}_2\times \mathbb{F}_2)$-codes
  9. Lattices of type $E_6$ and $\mathbb{Z}/3\mathbb{Z}$-codes
  10. Lattices of type $E_7$ and $\mathbb{Z}/2\mathbb{Z}-codes$
  11. Application to Hilbert modular form
  12. $K$-lattice
  13. Lattice of type $D_4$ over integers of $\mathbb{Q}(\zeta_8)$
  14. Lattice of type $E_6$ over integers of $\mathbb{Q}(\zeta_9)$

Key Result

Theorem 1.1

Let $n$ and $m$ be a positive integer, $\Lambda$ be a lattice of type $A_n$, $D_n$ or $E_n$ and $C$ be an $R$-code of length $m$, where There exists a mapping $\rho^{\oplus m}:~(\Lambda^*)^{\oplus m}\rightarrow R^{\oplus m}$ such that

Theorems & Definitions (67)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 57 more