98 research outputs found
The -intersection Pairs of Constacyclic and Conjucyclic Codes
A pair of linear codes whose intersection is of dimension , where
is a non-negetive integer, is called an -intersection pair of
codes. This paper focuses on studying -intersection pairs of
-constacyclic, and conjucyclic codes. We first characterize
an -intersection pair of -constacyclic codes. A formula for
has been established in terms of the degrees of the generator
polynomials of -constacyclic codes. This allows obtaining a
condition for -linear complementary pairs (LPC) of constacyclic codes.
Later, we introduce and characterize the -intersection pair of
conjucyclic codes over . The first observation in the process
is that there are no non-trivial linear conjucyclic codes over finite fields.
So focus on the characterization of additive conjucyclic (ACC) codes. We show
that the largest -subcode of an ACC code over
is cyclic and obtain its generating polynomial. This enables us to find the
size of an ACC code. Furthermore, we discuss the trace code of an ACC code and
show that it is cyclic. Finally, we determine -intersection pairs of
trace codes of ACC codes over
Asymptotically Good Additive Cyclic Codes Exist
Long quasi-cyclic codes of any fixed index have been shown to be
asymptotically good, depending on Artin primitive root conjecture in (A.
Alahmadi, C. G\"uneri, H. Shoaib, P. Sol\'e, 2017). We use this recent result
to construct good long additive cyclic codes on any extension of fixed degree
of the base field. Similarly self-dual double circulant codes, and self-dual
four circulant codes, have been shown to be good, also depending on Artin
primitive root conjecture in (A. Alahmadi, F. \"Ozdemir, P. Sol\'e, 2017) and (
M. Shi, H. Zhu, P. Sol\'e, 2017) respectively. Building on these recent
results, we can show that long cyclic codes are good over \F_q, for many
classes of 's. This is a partial solution to a fifty year old open problem
On ZpZp[u, v]-additive cyclic and constacyclic codes
Let be the ring of residue classes modulo a prime . The
-additive cyclic codes of length
is identify as -submodule of
where
with
. In this article, we obtain the complete sets of
generator polynomials, minimal generating sets for cyclic codes with length
over and
-additive cyclic codes with length
respectively. We show that the Gray image of
-additive cyclic code with length
is either a QC code of length with index or a
generalized QC code of length over .
Moreover, some structural properties like generating polynomials, minimal
generating sets of -additive constacyclic
code with length are determined.Comment: It is submitted to the journa
Quantum Codes from additive constacyclic codes over a mixed alphabet and the MacWilliams identities
Let be the ring of integers modulo a prime number where
is a quadratic residue modulo . This paper presents the study of
constacyclic codes over chain rings and . We
also study additive constacyclic codes over and
using the generator polynomials over the
rings and respectively. Further, by defining Gray
maps on , and
we obtain some results on the Gray images of additive codes. Then we give the
weight enumeration and MacWilliams identities corresponding to the additive
codes over . Finally, as an application of
the obtained codes, we give quantum codes using the CSS construction.Comment: 22 page
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