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@PHDTHESIS{Eisenbarth:788230,
author = {Eisenbarth, Simon},
othercontributors = {Nebe, Gabriele and Kirschmer, Markus},
title = {{G}itter und {C}odes über {K}ettenringen},
school = {RWTH Aachen University},
type = {Dissertation},
address = {Aachen},
reportid = {RWTH-2020-04503},
pages = {1 Online-Ressource (127 Seiten) : Illustrationen},
year = {2019},
note = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
University 2020; 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.},
cin = {114710 / 114820 / 110000},
ddc = {510},
cid = {$I:(DE-82)114710_20140620$ / $I:(DE-82)114820_20140620$ /
$I:(DE-82)110000_20140620$},
typ = {PUB:(DE-HGF)11},
doi = {10.18154/RWTH-2020-04503},
url = {https://publications.rwth-aachen.de/record/788230},
}
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