Contemporary Mathematics
Volume 376, 2005
Recent Results on Ring Constructions for Error-Correcting
Codes
Ricardo Alfaro and Andrei Kelarev
ABSTRACT. We give a survey of new recent results that have appeared in the
literature on error-correcting codes defined with ring constructions.
1.
Introduction
The structure of a ring is well known as an efficient tool that allows for com-
pact storage and manipulation of error-correcting codes, and makes it possible to
develop fast encoding and decoding algorithms. To illustrate its role it suffices to
mention that all classical cyclic codes can be regarded as ideals in several classes
of group rings. For example, Berman
[Ber],
in the case of characteristic two, and
Charpin
[Ch88],
in the general case, proved that all generalized Reed-Muller codes
coincide with powers of the radical of the algebra
where F
q
is a finite field, p
=
char F
q
0
ci
is a positive integer and
Qi
=
pc', for
i
=
1, ... ,
n,
and gave formulas for their Hamming weights. These codes form an
important class containing many codes of practical value. Berman
[Ber]
showed
that in certain cases abelian group codes enjoy better correcting properties than
cyclic codes. Using the underlying algebraic structure, a new fast decoding algo-
rithm for Reed-Muller codes was developed by Landrock and Manz
[LaMa].
1991
Mathematics Subject Classification.
Primary: 94B60, 94B65, Secondary: 16850.
Key words and phrases.
Error-correcting codes, ring constructions.
The first author was supported by UM-Flint Office of Research Grant.
The second author has been supported by ARC Discovery grant DP0449469.
©
2005 American Mathematical Society
http://dx.doi.org/10.1090/conm/376/06948
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