58 research outputs found
On a class of constacyclic codes over the non-principal ideal ring
-constacyclic codes of arbitrary length over the non-principal ideal
ring are studied, where is a prime,
and an integer satisfying .
First, the structure of any -constacyclic code over are presented. Then enumerations for the number of all
codes and the number of codewords in each code, and the structure of dual codes
for these codes are given, respectively. Then self-dual -constacyclic
codes over are investigated, where
or if , and if
On a class of -constacyclic codes over
Let be a finite field of cardinality ,
which is a finite chain ring, and be a positive integer
satisfying . For any , an explicit representation for all distinct
-constacyclic codes over of length is given, and
the dual code for each of these codes is determined. For the case of
and , all self-dual -constacyclic codes over of
odd length are provided
Matrix-product structure of repeated-root constacyclic codes over finite fields
For any prime number , positive integers satisfying and , we prove that any
-constacyclic code of length over the finite field
is monomially equivalent to a matrix-product code of a
nested sequence of -constacyclic codes with length over
Cyclic codes over of oddly even length
Let be a finite field of characteristic and
() where satisfies . For any odd positive integer , it is
known that cyclic codes over of length are identified with ideals of
the ring . In this paper, an explicit
representation for each cyclic code over of length is provided and a
formula to count the number of codewords in each code is given. Then a formula
to calculate the number of cyclic codes over of length is obtained.
Moreover, the dual code of each cyclic code and self-dual cyclic codes over
of length are investigated. (AAECC-1522)Comment: AAECC-152
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
Constacyclic codes of length over
Let be a finite field of cardinality and
, where is a prime and is a positive integer. For any
, an explicit representation for all
distinct -constacyclic codes over of length is given by a
canonical form decomposition for each code, where and are positive
integers satisfying . For any such code, using its canonical
form decomposition the representation for the dual code of the code is
provided. Moreover, representations for all distinct negacyclic codes and their
dual codes of length over are obtained, and self-duality for these
codes are determined. Finally, all distinct self-dual negacyclic codes over
of length are listed for any
positive integer
On a class of left metacyclic codes
Let be a metacyclic
group of order , where , and (mod
). Then left ideals of the group algebra are
called left metacyclic codes over of length , and
abbreviated as left -codes. A system theory for left
-codes is developed for the case of and
for some positive integer , only using finite
field theory and basic theory of cyclic codes and skew cyclic codes. The fact
that any left -code is a direct sum of concatenated codes with
inner codes and outer codes is proved, where is
a minimal cyclic code over of length and is a skew
cyclic code of length over an extension field of . Then an
explicit expression for each outer code in any concatenated code is provided.
Moreover, the dual code of each left -code is given and
self-orthogonal left -codes are determined
Negacyclic codes over the local ring of oddly even length and their Gray images
Let () and be an odd
positive integer. Then is a local non-principal ideal ring of elements
and there is a -linear Gray map from onto
which preserves Lee distance and orthogonality. First, a
canonical form decomposition and the structure for any negacyclic code over
of length are presented. From this decomposition, a complete
classification of all these codes is obtained. Then the cardinality and the
dual code for each of these codes are given, and self-dual negacyclic codes
over of length are presented. Moreover, all negacyclic codes over of length and all
self-dual codes among them are
presented precisely, where is a Mersenne prime. Finally, new
and good self-dual -quasi-twisted linear codes over with
basic parameters and of type and basic
parameters and of type which are Gray
images of self-dual negacyclic codes over of length are listed.Comment: arXiv admin note: text overlap with arXiv:1710.0923
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
Matrix-product structure of constacyclic codes over finite chain rings
Let be positive integers, a prime number, be a
finite field of elements and
which is a finite chain ring. For any and positive
integers satisfying , we prove that any -constacyclic code of length over is monomially equivalent to a
matrix-product code of a nested sequence of cyclic codes with length
over and a matrix over . Using the
matrix-product structures, we give an iterative construction of every
-constacyclic code by -constacyclic codes of
shorter lengths over .Comment: arXiv admin note: text overlap with arXiv:1705.0881
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