Gitter und Codes über Kettenringen

• Lattices and codes over chain rings

Eisenbarth, Simon; Nebe, Gabriele (Thesis advisor); Kirschmer, Markus (Thesis advisor)

Aachen (2019, 2020)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2019

Abstract

Motivated by the question, how $\mathbb{F}_p$-linear, extremal, self-dual codes with an automorphism of order $p$ can be classified, the structur of self-dual codes over chain rings are being studied. Let $R$ be such a ring, $x$ be a generator of the unique maximal ideal of $R$ and $a \in \mathbb{N}_0$ maximal such that $x^a \neq 0$. A code $C$ over $R$ of length $t$ is an $R$-submodule of the free modul $R^t$. Multiplying powers of $x$ to $C$ defines the finite chain of subcodes $C \supseteq C^{(1)} := Cx \supseteq C^{(2)} := Cx^2 \supseteq \cdots \supseteq C^{(a)} := Cx^a \supseteq \lbrace 0 \rbrace. $ We show that if $C$ is a self-dual code in $R^t$, then the socle $C^{(a)}$ is a (hermitian) self-dual code over the residue field $\mathbb{F} = R / \langle x \rangle$ if and only if $C$ is a free $R$-module. In this case, all codes $C^{(i)}$ are self-dual in a suitable bilinear spaces over $\mathbb{F}$ and we describe a method to construct all lifts $C$ of a given self-dual code $C^{(a)}$ over $\mathbb{F}$ that are self-dual, free codes over $R$. We apply this technique to codes over finite fields of characteristic $p$ admitting an automorphism whose order is a power of $p$. For illustration, we show that the well-known Pless code $P_{36}$ is the only extremal, ternary code of length $36$ with an automorphism of order $3$, strengthening a result of Huffman, who showed the assertion for all prime orders $\geq 5$.Additionally, group codes over chain which are relative projective (in the sense of homological algebra) are being considered. Those codes are in bijection to projective group codes over the residue field and with these chains properties like the minimum distance or the dual codes can be stated. After all, extremal, $p$-modular lattices with an automorphism of order $p$ are considered. The action of such an automorphism provides a decomposition of the underlying quadratic space into a fixpoint- and a cyclotomic component. With the projection and intersection of the lattice with both components, antiisometric, quadratic spaces can be defined. Contrary to the unimodular case those spaces are not anisotropic, but contain (isomorphic) maximal total isotropic subspaces. Those define $p$-elementary (hermitian) lattices und can be used to determine the fix- and cyclotomic sublattices. As an application, we show that the only known $24$-dimensional, $3$-modular, extremal lattice is unique with an automorphism of order $3$, such a classification was only known for all prime-orders $\geq 5$. Additionally, all $5$-modular, extremal lattices of dimension $20$ with an automorphism of order $5$ are being classified.