Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Codes from Zero-divisors and Units in Group Rings   We describe and present a new construction method for codes using encodings
from group rings. They consist primarily of two types: zero-divisor and
unit-derived codes. Previous codes from group rings focused on ideals; for
example cyclic codes are ideals in the group ring over a cyclic group. The
fresh focus is on the encodings themselves, which only under very limited
conditions result in ideals. We use the result that a group ring is isomorphic
to a certain well-defined ring of matrices, and thus every group ring element
has an associated matrix. This allows matrix algebra to be used as needed in
the study and production of codes, enabling the creation of standard generator
and check matrices. Group rings are a fruitful source of units and
zero-divisors from which new codes result. Many code properties, such as being
LDPC or self-dual, may be expressed as properties within the group ring thus
enabling the construction of codes with these properties. The methods are
general enabling the construction of codes with many types of group rings.
There is no restriction on the ring and thus codes over the integers, over
matrix rings or even over group rings themselves are possible and fruitful.

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The categories of this paper are: funct-an
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: patt-sol
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The categories of this paper are: stat
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The categories of this paper are: alg-geom
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The categories of this paper are: comp-gas
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The categories of this paper are: cond-mat
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The categories of this paper are: atom-ph
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The categories of this paper are: q-bio
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The categories of this paper are: mtrl-th
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The categories of this paper are: math-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: econ