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Constacyclic codes of length
4
p
s
4p^s
4
p
s
over the Galois ring
G
R
(
p
a
,
m
)
GR(p^a,m)
GR
(
p
a
,
m
)
Authors
Habibul Islam
Om Prakash
Ram Krishna Verma
Publication date
8 November 2019
Publisher
View
on
arXiv
Abstract
For prime
p
p
p
,
G
R
(
p
a
,
m
)
GR(p^a,m)
GR
(
p
a
,
m
)
represents the Galois ring of order
p
a
m
p^{am}
p
am
and characterise
p
p
p
, where
a
a
a
is any positive integer. In this article, we study the Type (1)
λ
\lambda
λ
-constacyclic codes of length
4
p
s
4p^s
4
p
s
over the ring
G
R
(
p
a
,
m
)
GR(p^a,m)
GR
(
p
a
,
m
)
, where
λ
=
ξ
0
+
p
ξ
1
+
p
2
z
\lambda=\xi_0+p\xi_1+p^2z
λ
=
ξ
0
​
+
p
ξ
1
​
+
p
2
z
,
ξ
0
,
ξ
1
∈
T
(
p
,
m
)
\xi_0,\xi_1\in T(p,m)
ξ
0
​
,
ξ
1
​
∈
T
(
p
,
m
)
are nonzero elements and
z
∈
G
R
(
p
a
,
m
)
z\in GR(p^a,m)
z
∈
GR
(
p
a
,
m
)
. In first case, when
λ
\lambda
λ
is a square, we show that any ideal of
R
p
(
a
,
m
,
λ
)
=
G
R
(
p
a
,
m
)
[
x
]
⟨
x
4
p
s
−
λ
⟩
\mathcal{R}_p(a,m,\lambda)=\frac{GR(p^a,m)[x]}{\langle x^{4p^s}-\lambda\rangle}
R
p
​
(
a
,
m
,
λ
)
=
⟨
x
4
p
s
−
λ
⟩
GR
(
p
a
,
m
)
[
x
]
​
is the direct sum of the ideals of
G
R
(
p
a
,
m
)
[
x
]
⟨
x
2
p
s
−
δ
⟩
\frac{GR(p^a,m)[x]}{\langle x^{2p^s}-\delta\rangle}
⟨
x
2
p
s
−
δ
⟩
GR
(
p
a
,
m
)
[
x
]
​
and
G
R
(
p
a
,
m
)
[
x
]
⟨
x
2
p
s
+
δ
⟩
\frac{GR(p^a,m)[x]}{\langle x^{2p^s}+\delta\rangle}
⟨
x
2
p
s
+
δ
⟩
GR
(
p
a
,
m
)
[
x
]
​
. In second, when
λ
\lambda
λ
is not a square, we show that
R
p
(
a
,
m
,
λ
)
\mathcal{R}_p(a,m,\lambda)
R
p
​
(
a
,
m
,
λ
)
is a chain ring whose ideals are
⟨
(
x
4
−
α
)
i
⟩
⊆
R
p
(
a
,
m
,
λ
)
\langle (x^4-\alpha)^i\rangle\subseteq \mathcal{R}_p(a,m,\lambda)
⟨(
x
4
−
α
)
i
⟩
⊆
R
p
​
(
a
,
m
,
λ
)
, for
0
≤
i
≤
a
p
s
0\leq i\leq ap^s
0
≤
i
≤
a
p
s
where
α
p
s
=
ξ
0
\alpha^{p^s}=\xi_0
α
p
s
=
ξ
0
​
. Also, we prove the dual of the above code is
⟨
(
x
4
−
α
−
1
)
a
p
s
−
i
⟩
⊆
R
p
(
a
,
m
,
λ
−
1
)
\langle (x^4-\alpha^{-1})^{ap^s-i}\rangle\subseteq \mathcal{R}_p(a,m,\lambda^{-1})
⟨(
x
4
−
α
−
1
)
a
p
s
−
i
⟩
⊆
R
p
​
(
a
,
m
,
λ
−
1
)
and present the necessary and sufficient condition for these codes to be self-orthogonal and self-dual, respectively. Moreover, the Rosenbloom-Tsfasman (RT) distance, Hamming distance and weight distribution of Type (1)
λ
\lambda
λ
-constacyclic codes of length
4
p
s
4p^s
4
p
s
are obtained when
λ
\lambda
λ
is not a square.Comment: This article has 18 pages and ready to submit in a journa
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Last time updated on 12/10/2020