6 research outputs found
An explicit representation and enumeration for self-dual cyclic codes over of length
Let be a finite field of cardinality and a
positive integer. Using properties for Kronecker product of matrices and
calculation for linear equations over , an efficient method
for the construction of all distinct self-dual cyclic codes with length
over the finite chain ring is
provided. On that basis, an explicit representation for every self-dual cyclic
code of length over and an exact
formula to count the number of all these self-dual cyclic codes are given
Construction and enumeration for self-dual cyclic codes of even length over
Let be a finite field of cardinality ,
and be positive integers
such that is odd. In this paper, we give an explicit representation for
every self-dual cyclic code over the finite chain ring of length and
provide a calculation method to obtain all distinct codes. Moreover, we obtain
a clear formula to count the number of all these self-dual cyclic codes. As an
application, self-dual and -quasi-cyclic codes over of
length can be obtained from self-dual cyclic code over of length
and by a Gray map preserving orthogonality and distances from onto
.Comment: arXiv admin note: text overlap with arXiv:1811.11018,
arXiv:1907.0710
On self-duality and hulls of cyclic codes over with oddly even length
Let be a finite field of elements, and
() where is an integer satisfying . For any odd positive
integer , an explicit representation for every self-dual cyclic code over
of length and a mass formula to count the number of these codes are
given first. Then a generator matrix is provided for the self-dual and
-quasi-cyclic code of length over derived by every
self-dual cyclic code of length over
and a Gray map from onto
. Finally, the hull of each cyclic code with length
over is determined and all distinct
self-orthogonal cyclic codes of length over
are listed
Explicit representation for a class of Type 2 constacyclic codes over the ring with even length
Let be a finite field of cardinality , and
be integers satisfying and denote
. Let . For any odd positive integer , we give an
explicit representation and enumeration for all distinct -constacyclic codes over of length , and provide a clear formula
to count the number of all these codes. As a corollary, we conclude that every
-constacyclic code over of length is an ideal
generated by at most polynomials in the residue class ring .Comment: arXiv admin note: text overlap with arXiv:1805.0559
An explicit expression for Euclidean self-dual cyclic codes of length over Galois ring
For any positive integers and , existing literature only determines
the number of all Euclidean self-dual cyclic codes of length over the
Galois ring , such as in [Des. Codes Cryptogr. (2012)
63:105--112]. Using properties for Kronecker products of matrices of a specific
type and column vectors of these matrices, we give a simple and efficient
method to construct all these self-dual cyclic codes precisely. On this basis,
we provide an explicit expression to accurately represent all distinct
Euclidean self-dual cyclic codes of length over , using
combination numbers. As an application, we list all distinct Euclidean
self-dual cyclic codes over of length explicitly, for
Non-Invertible-Element Constacyclic Codes over Finite PIRs
In this paper we introduce the notion of -constacyclic codes over
finite rings for arbitary element of . We study the
non-invertible-element constacyclic codes (NIE-constacyclic codes) over finite
principal ideal rings (PIRs). We determine the algebraic structures of all
NIE-constacyclic codes over finite chain rings, give the unique form of the
sets of the defining polynomials and obtain their minimum Hamming distances. A
general form of the duals of NIE-constacyclic codes over finite chain rings is
also provided. In particular, we give a necessary and sufficient condition for
the dual of an NIE-constacyclic code to be an NIE-constacyclic code. Using the
Chinese Remainder Theorem, we study the NIE-constacyclic codes over finite
PIRs. Furthermore, we construct some optimal NIE-constacyclic codes over finite
PIRs in the sense that they achieve the maximum possible minimum Hamming
distances for some given lengths and cardinalities