1. Introduction
Information transmission is an important means of human communication, and with the development of technology, coding theory has also been established. In the information age, cyberspace security is a very important issue, and cryptography and encoding play important roles in it. Coding theory is a technique for encoding information. During the process of transmitting information, it is inevitable that information may be distorted due to some reasons. In this process, information cannot correct errors on its own. Therefore, a self-correcting code space has been studied, which is called the error-correcting-codes.
Among error-correction-codes, linear codes are widely studied due to their excellent algebraic structure and other characteristics, and cyclic codes are the most important among them. Due to their excellent algebraic structure and cyclic properties, they can be easily studied and obtained through algebraic methods, and are widely used in various information security systems.
Let
be a finite field with
elements, where p is a prime. A linear code
with parameters
over
is a linear subspace of
, which has the length n, dimension k and minimum Hamming distance d. We say the linear code
is a cyclic code if for any codewords
, the cyclic shift of the codeword
. Now we use the polynomial ring
and the quotient ring
to describe the cyclic code. We define a linear code as a cyclic code if for any codeword
, which can be identified with a polynomial
the codeword
. So we can easily know that an nonempty set
in
is a cyclic code if and only if
is a principal ring in
. We denote any cyclic code
as
, and
is called the generator polynomial of
.
The study of cyclic codes has been the focus of attention in recent years. Because of its excellent characteristics, it has been widely used in lots of fields. We always hope that a cyclic code has better error correction ability. The error correction ability is closely related to the minimum distance. The larger minimum distance it has, the better error correction ability it gets. Therefore, we are very interested in the minimum distance of a code.
Let
, we consider the cyclic code
over the finite field
. Ding and Helleseth [1] state the theory of the APN monomials and used some of these to construct many classes of optimal ternary cyclic codes in 2013. In 2019, by giving a new ternary power mapping, Yan and Han [2] considered a related optimal ternary cyclic code which
in some conditions. Zha and Hu [3] proved some new classes of optimal ternary cyclic codes with minimum distance 4 for some given parameters v,
in 2020. For the given
, Ding and Zhou [4] studied the cyclic code is optimal when
in some conditions. Similarly, Fan, Zhou and Li [5] proved that
the cyclic code
is optimal when m is odd in 2016. They also
discussed the weight distribution of the dual of this code. In 2020, Liu, Cao and Lu [6] studied the code
, which is constructed by using monomials
and
. For
,
is optimal by choosing suitable m and k. Recently, by choosing proper u and v, Zha, Hu, Liu and Cao [7] show
that
and
have the same optimality.
In previous studies, there are not many studies on cyclic codes with parameter
,
. In this paper, we study the cyclic code
with the parameters which is
. We show that the minimum distance of this cyclic
code is equal to 4 for the given
and
, according to the Sphere Packing bound. It is optimal. Therefore, in the coding theory, we can obtain a new class of ternary cyclic codes whose minimum distance can reach the theoretical maximum for the given length and dimension. It can achieve the best error correction effect and ensure that the information is not distorted as much as possible in the transmission process. These cyclic codes will have important applications in radar, satellite communications and other communication fields.
2. Preliminaries
● The notation we use in this paper
(1) p is prime, and is an odd. Let
.
(2)
are positive integers, m is odd.
(3) Let SQ be the set of square in
, NSQ be the set of the nonsquare in
.
(4) In
, we have
if
and
if
.
● The p-cyclotomic cosets modulo n,
We define the p-cyclotomic coset modulo n containing j as
and k is the smallest positive integer such that
. In this paper, let
. The cyclic code of length
. The dimension of this code
is determined by k, where
. The dimension of
is equal to
. We now consider the case that
and
, so the dimension is equal to
.
Theorem 2.1. [8] (Sphere Packing Bound)
is the maximum number of codewords in a code over
of length n and minimum distance at least d, or we use
to represent it. Then
where
.
We can see that by using Sphere Packing Bound, we can get a bound of the minimum distance of a cyclic code. Taking the cyclic code to be studied in this paper as an example, let
, and when
,
, the minimum distance of this cyclic codes can be obtained no more than 4. We obtain the upper bound of the minimum distance of this cyclic code. Therefore, we only need to prove that the minimum distance of this cyclic code can reach this upper bound, and it can be shown that it is optimal.
The distance d between two codewords
is defined to be the number of coordinates in which
are different. The minimum distance of a code
is the smallest distance between distinct codewords. The weight wt(c) of a codeword c is the number of the nonzero coordinates in c. It has
[9] . If
is a linear code, the minimum distance equal to the minimum weight of the nonzero codewords of
. The parity check matrices of a code is a matrices H which satisfied
. The parity check matrices of a code are the generator matrices of its dual code. From the definition of dual codes, the parity check matrices of the code
is define as
is a generator of
.
If a linear code
has minimum distance d, there exist two distinct codewords
,
,
, satisfied
is the coordinates of the codeword
, respectively.
,
,
,
,
.
If the code has minimum distance d, the equations above has solution, if the code has not minimum distance d, the equations above has not solution. So we can discuss the solution of the equations to find if the code has the codeword of weight d. According to the minimum distance
given by the sphere packing bound, we can prove that
.
Lemma 2.2. Let
, v be an odd,
, and
. Cyclic code
has parameters
if and only if the following equations:
(1)
(2)
(3)
and the equation
(4)
have no solution in
.
Proof. It is clear that the distance of the code cannot be 1. The code
has a codeword of Hamming weight 2 if and only if there exist two elements
and two distinct elements
such that
Case 1:
If
,
, the first equation becomes to
, which is impossible because
are all SQ. If
,
, the first equation becomes to
, let
, then we have
,
but
is a NSQ, which is also impossible. If
,
or
,
, the first equation becomes to
, which is also a contradiction..
Case 2:
If
,
or
,
, the first equation becomes to
, which is a contradiction. If
,
or
,
, the first equation becomes to
. Taking it into the second equation we will get
, which is a contradiction.
Thus it does not have a codeword of Hamming weight 2.
The code
has a codeword of Hamming weight 3 if and only if there exist three elements
and three distinct elements
such that
(5)
Case 1:
. In this case, let
. It follows from (5) that
(6)
. If
or
, (6) becomes to
If
or
, (6) becomes to
Case 2:
. Similarly, we arrive at
(7)
. If
or
, (7) becomes to
If
or
, (7) becomes to
So if the four equations have no solutions in
, we get
, according to the Sphere Packing bound, the minimal distance of any linear code with length
and the dimension
should be less than or equal 4. Hence
. □
The following Lemma will be used in the sequel of the proof.
Lemma 2.3. [10] Let
be a irreducible polynomial with degree r over
. If
has a root in
, then
.
3. A Class of Optimal Ternary Cyclic Codes
In this section, we construct a class of optimal ternary cyclic codes
with parameters
.
Theorem 3.1. Let m is odd,
,
,
.
,
. The cyclic code
is an optimal ternary cyclic code with parameters
.
Proof. It is easy to prove that the minimal distance of the code
. By lemma 2.2 we can know that it does not have a codeword of Hamming weight 2, which means
. Now we prove that the minimal distance of the code
.
Now, we prove that the code has no codewords of Hamming weight 3. It has a codeword of Hamming weight 3 if and only if there exist three elements
and three distinct elements
such that
(8)
Case 1:
. In this case, let
,
It follows from (8) that
(9)
.
Now we consider the following four cases.
Case 1.1: When
, (9) follows to
(10)
The Equation (10) leads to
Let
, Then we have
By taking the eight power of both sides of the equation, we can get
If
, let
, and if t also
, we have
It becomes to
Expand it, we can get
Because
, we have
It can be factorized over
by Magma to
if
, it means
, is impossible, and by lemma 2.3, we get it has no root in
.
If
, let
, and if
, we have
By following the same steps, we get
It can be factorized over
by Magma to
By the same reason, it has no root in
.
If
,
and
or
, it is similar to the above case, we omit it here.
Case 1.2: When
, (8) follows to
(11)
The Equation (11) leads to
Let
, Then we have
By taking the eight power of both sides of the equation, we can get
If
, let
, and if t also
, by the similar steps, we directly obtain
As before, with the help of Magma, we get the same equation
if
, it means
, but
is not the solution of
, and by lemma 2.3, we get it has no root in
If
, let
, and if
, we have
By following the same steps, we get
By the same reason as before, it has no root in
.
If
,
and
or
, it is similar to the above case, we omit it here.
Case 1.3: When
. It is similar as before, we omit it here.
Case 1.4: When
. It is similar as before, we omit it here.
Case 2:
. By the similar calculation as Case 1, we can prove that the equation
(12)
also has no solution in
. We omit the details of the proof.
By the Lemma 2.2, we have finished the proof. □
Example
Let
,
,
. Let
be the generator of
with
. Then
is a ternary cyclic code with parameters [26, 20, 4] and generator polynomial
.
4. Conclusions
In this paper, based on the Sphere Packing Bound, we show that for the fixed length and dimension, with the help of factorization by Magma, by discussing the solutions of some correlative equations on
, the ternary cyclic code
has the minimum distance 4, according to the Sphere Packing bound. It is optimal.