50 research outputs found
Self-Dual and Complementary Dual Abelian Codes over Galois Rings
Self-dual and complementary dual cyclic/abelian codes over finite fields form
important classes of linear codes that have been extensively studied due to
their rich algebraic structures and wide applications. In this paper, abelian
codes over Galois rings are studied in terms of the ideals in the group ring
, where is a finite abelian group and
is a Galois ring. Characterizations of self-dual abelian codes have been given
together with necessary and sufficient conditions for the existence of a
self-dual abelian code in . A general formula for the
number of such self-dual codes is established. In the case where
, the number of self-dual abelian codes in
is completely and explicitly determined. Applying known results on cyclic codes
of length over , an explicit formula for the number of
self-dual abelian codes in are given, where the Sylow
-subgroup of is cyclic. Subsequently, the characterization and
enumeration of complementary dual abelian codes in are
established. The analogous results for self-dual and complementary dual cyclic
codes over Galois rings are therefore obtained as corollaries.Comment: 22 page
On Linear Codes Over a Non-Chain Ring
In this paper, we study skew cyclic and quasi cyclic codes over the ring S = F2 +uF2 + vF2 where u2 = u, v2 = v, uv = vu = 0.We investigate the structural properties of them. Using a Gray map on S we obtain the MacWilliams identities for codes over S. The relationships between Symmetrized, Lee and Hamming weight enumerator are determined
Reversible cyclic codes over finite chain rings
In this paper, necessary and sufficient conditions for the reversibility of a
cyclic code of arbitrary length over a finite commutative chain ring have been
derived. MDS reversible cyclic codes having length p^s over a finite chain ring
with nilpotency index 2 have been characterized and a few examples of MDS
reversible cyclic codes have been presented. Further, it is shown that the
torsion codes of a reversible cyclic code over a finite chain ring are
reversible. Also, an example of a non-reversible cyclic code for which all its
torsion codes are reversible has been presented to show that the converse of
this statement is not true. The cardinality and Hamming distance of a cyclic
code over a finite commutative chain ring have also been determined
Self-dual cyclic codes over finite chain rings
Let be a finite commutative chain ring with unique maximal ideal , and let be a positive integer coprime with the
characteristic of . In this paper, the algebraic
structure of cyclic codes of length over is investigated. Some new
necessary and sufficient conditions for the existence of nontrivial self-dual
cyclic codes are provided. An enumeration formula for the self-dual cyclic
codes is also studied.Comment: 15 page
p-Adic estimates of Hamming weights in Abelian codes over Galois rings
A generalization of McEliece's theorem on the p-adic valuation of Hamming weights of words in cyclic codes is proved in this paper by means of counting polynomial techniques introduced by Wilson along with a technique known as trace-averaging introduced here. The original theorem of McEliece concerned cyclic codes over prime fields. Delsarte and McEliece later extended this to Abelian codes over finite fields. Calderbank, Li, and Poonen extended McEliece's original theorem to cover cyclic codes over the rings /spl Zopf//sub 2//sup d/, Wilson strengthened their results and extended them to cyclic codes over /spl Zopf//sub p//sup d/, and Katz strengthened Wilson's results and extended them to Abelian codes over /spl Zopf//sub p//sup d/. It is natural to ask whether there is a single analogue of McEliece's theorem which correctly captures the behavior of codes over all finite fields and all rings of integers modulo prime powers. In this paper, this question is answered affirmatively: a single theorem for Abelian codes over Galois rings is presented. This theorem contains all previously mentioned results and more
MDS and MHDR cyclic codes over finite chain rings
In this work, a unique set of generators for a cyclic code over a finite
chain ring has been established. The minimal spanning set and rank of the code
have also been determined. Further, sufficient as well as necessary conditions
for a cyclic code to be an MDS code and for a cyclic code to be an MHDR code
have been obtained. Some examples of optimal cyclic codes have also been
presented