504 research outputs found
Constacyclic Codes over Finite Fields
An equivalence relation called isometry is introduced to classify
constacyclic codes over a finite field; the polynomial generators of
constacyclic codes of length are characterized, where is the
characteristic of the finite field and is a prime different from
Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes
Cyclic, negacyclic and constacyclic codes are part of a larger class of codes
called polycyclic codes; namely, those codes which can be viewed as ideals of a
factor ring of a polynomial ring. The structure of the ambient ring of
polycyclic codes over GR(p^a,m) and generating sets for its ideals are
considered. Along with some structure details of the ambient ring, the
existance of a certain type of generating set for an ideal is proven.Comment: arXiv admin note: text overlap with arXiv:0906.400
Coding Theory and Algebraic Combinatorics
This chapter introduces and elaborates on the fruitful interplay of coding
theory and algebraic combinatorics, with most of the focus on the interaction
of codes with combinatorial designs, finite geometries, simple groups, sphere
packings, kissing numbers, lattices, and association schemes. In particular,
special interest is devoted to the relationship between codes and combinatorial
designs. We describe and recapitulate important results in the development of
the state of the art. In addition, we give illustrative examples and
constructions, and highlight recent advances. Finally, we provide a collection
of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in
Information and Coding Theory", ed. by I. Woungang et al., World Scientific,
Singapore, 201
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
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
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