1. Introduction
High-density data storage systems are very popular nowadays due to their great advantages in simplifying and improving storage, saving money, and saving time. However, these systems still have restrictions on the combination of high density, rewriting capability, fast response, and long retention time. To overcome these restrictions in high-density data storage systems, a new metric, called symbol-pair metric, was introduced by Cassuto and Blaum in [
1,
2]. This metric is reasonable for channels for which outputs are overlapping pairs of symbols. After that, for the read channel output larger than 2,
, Yaakobi et al. [
3] introduced
b-symbol metrics, which generalize the symbol-pair metric, and provided extensions of many results and code constructions. By putting
, we obtained the symbol-triple metric.
Consider R a finite commutative ring with unity. For a positive integer n, the number of nonzero entries of a codeword is the Hamming weight of , denoted by . Let and be two codewords, and their Hamming distance denoted by is the number of their components which differ.
A codeword
is represented in symbol-triple read channels as
The Hamming weight of the symbol-triple vector
is called the symbol-triple weight of the vector
:
For two vectors
and
in
, the symbol-triple distance between them is:
Then, for a code
C, the symbol-triple distance is defined as
For a linear code
C, the symbol-triple distance and symbol-triple weight are coincided. They are defined as:
Constacyclic codes are linear codes that have an important role in error-correcting codes theory, which can be viewed as a generalization of cyclic codes. They are preferred in engineering because of their rich algebraic structures that make them more practical, since they are efficiently encoded, and they provide efficient error detection and correction. Castagnoli et al. [
4] and Van Lint [
5] were the first to investigate constacyclic codes. For a unit
of
R,
-constacyclic codes of length
n over
R are in one-to-one correspondence with ideals of the polynomial ring
. If
n is divisible by the characteristic
p of
R, then we obtain the so-called repeated-root codes.
Let
be a finite field of
elements, where
p is a prime,
m is a positive integer and let
be an integer. Then, the ring
is a finite commutative chain ring. Many authors [
6,
7,
8] studied algebraic structures of constacyclic codes over
R:
For
, many authors studied them (see, e.g., [
9,
10,
11,
12,
13,
14,
15,
16,
17,
18]). In particular, for a prime power length, their structure and their symbol-pair distance were completely established in [
12,
19].
For
, DNA cyclic codes were studied in [
20], which are of deep importance for biology. In [
21], Laaouine et al. described the structure of all
-constacyclic codes of length
over
for a nonzero element
of
and classified them into eight types. Then, their Hamming distances have been computed by Dinh et al. [
22]. In [
23], Charkani et al. computed their symbol-pair distances.
Accordingly, we resolve to compute the symbol-triple distance of all
-constacyclic codes of length
over
. After giving some preliminaries and notations in
Section 2,
Section 3 establishes their symbol-triple distance.
Section 4 contains some examples by fixing some values for
p and
. Finally, a conclusion is given in
Section 5.
2. Some Preliminaries
From now on in this paper,
is the field of order
q with
where
p is a prime positive integer,
m is a positive integer, and we denote
is a finite chain ring,
is its maximal ideal, and its invertible elements in are of the form:
where
and
.
Linear codes
of length
n over
are defined as
-submodules of
. For a unit
in
,
-constacyclic codes
of length
n over
are the linear codes of length
n over
, which satisfy the following property:
implies that
. Then, in order to study them, it suffices to study ideals of the ring
(cf. [
24,
25]).
So, for a unit
of
,
-constacyclic codes of length
over
are exactly the ideals of
In [
21], Laaouine et al. classified and determined their structure:
Theorem 1 (cf. [
21]).
The ring is local finite non-chain ring, its maximal ideal is , where such that . ψ-constacyclic codes of length over (i.e., ideals of the ring ) arewhere . where , , either is 0 or is a unit in . Here, is the smallest integer such that . where , , either is 0 or is a unit in , and is the same as in . where , , and are 0 or are units in . Here, being the smallest integer such that , for some and is the smallest integer satisfying . where , , and are 0 or are units in . Here, , are the same as in . where , , , and are 0 or are units in . Here, is the smallest integer such that and is the same as in . where , , , and are 0 or are units in . Here, as in and is the same as in . 3. Symbol-Triple Distance
This section will be devoted to determining the symbol-triple distances of all types of -constacyclic codes of length over . Therefore, we recall the following result.
Theorem 2 (cf. [
26]).
Let be a ψ-constacyclic code of length over . Then, , , and its symbol-triple distance is determined by: For a code over , the symbol-triple distance is denoted by .
Now, for each type of -constacyclic codes of length over , we compute the symbol-triple distance one by one.
consists only of the trivial ideals , . Hence, they have symbol-triple distances of 0 and 3, respectively.
The codewords of a code of with are exactly those of the -constacyclic codes in multiplied by . Thus, we obtain and Theorem 2 gives it.
Theorem 3. For a ψ-constacyclic code , of . The symbol-triple distance is given by Now, we are going to calculate the symbol-triple distances of codes of
,
,
,
,
and
. To do this, notice that
where
.
The symbol-triple distance -constacyclic codes of can be calculated by the next theorem:
Theorem 4. Let be of . The symbol-triple distance of is given by Proof. Let be an arbitrary nonzero element of . That means that there exists
Since
, we have
It follows that
for any nonzero element
of
. Then,
Moreover, we have
it follows that
Then, by combining (
2) and (
3), we obtain
This proves the theorem. □
Now, let us show the symbol-triple distance for .
Theorem 5. For ψ-constacyclic code of , the symbol-triple distance is given by Proof. At first, since
, we obtain
To prove that , let , we should prove that
Hence,
, forcing
This proves the theorem. □
Now, we calculate the symbol-triple distance for :
Theorem 6. For a ψ-constacyclic code of , the symbol-triple distance is given by Proof. Let
be of
. Now, for each nonzero
, there exist
such that
The fact that
follows that
This implies that .
Moreover, we have that
then
and we obtain
This proves the theorem. □
The symbol-triple distance for can be established by the next theorem:
Theorem 7. For a ψ-constacyclic codeof, the symbol-triple distanceis given by Proof. At first, since
, we obtain
Now, let
. Thus, by (
1), we find that
Hence,
, forcing
This proves the theorem. □
Now, we show the symbol-triple distance for .
Theorem 8. For a ψ-constacyclic code of , the symbol-triple distance is given by Proof. At first, since
, we obtain
Now, let . Then, we distinguish two different cases.
: if
, then by (
1), we obtain
: if
, then by (
1), we obtain
We have
, then, we obtain
Therefore,
, forcing
This completes the proof. □
Finally, we give the symbol-triple distance for .
Theorem 9. For a ψ-constacyclic code of , the symbol-triple distance is given by Proof. Let be of . In addition, consider an arbitrary polynomial . Now, we distinguish two cases:
: if
then by (
1), we obtain
: if
, then by (
1), we obtain
This implies that .
Moreover, we have that
then
and we obtain
This completes the proof. □