Abstract
The polynomial over finite fields has been of interest due to its applications in the study of negacyclic codes over finite fields. In this paper, a rigorous treatment of the factorization of over finite fields is given as well as its applications. Explicit and recursive methods for factorizing over finite fields are provided together with the enumeration formula. As applications, some families of negacyclic codes are revisited with more clear and simpler forms.
1. Introduction
In coding theory, the polynomial over finite fields plays an important role in the study of negacyclic codes (see [1–5] and references therein). Precisely, a negacyclic code of length over can be uniquely determined by an ideal in the principal ring generated by a monic divisor of . A brief discussion on the factorization of over finite fields has been given in [3, 4]. In the case where the characteristic of is even, the factorization of over has been given and applied in the study of cyclic codes over finite fields in [6]. In [7, 8], an explicit form of the factorization of over finite fields of odd characteristic has been established. Some results on the factorization of over finite fields have been presented in [5].
In this paper, we focus on the factorization of over finite fields for arbitrary positive integers and all odd prime powers . If the characteristic of is , we have for all integers and . It is therefore sufficient to study the factorization of over such that is coprime to . Here, we write for some integer and odd positive integer such that .
Before proceeding to the general results, we consider a pattern on the factorization of over . We havefor all , where , , , , , and . It is easily seen that the factorization can be determined recursively on the exponent of 2 and the number of monic irreducible factors of is a constant independent of .
In this paper, a complete study on the above pattern of the factorization of over is given. Precisely, we prove that there exists a positive integer such that the number of monic irreducible factors of over becomes a constant for all positive integers . In the cases where is odd, a complete recursive factorization of over is provided together with a recursive formula for the number of its monic irreducible factors for all positive integers . In the cases where is even, a recursive factorization of over is given for all positive integers . As applications, constructions and enumerations of some negacyclic codes of lengths over are given based on the above results.
The paper is organized as follows. Preliminary concepts and results on the factorization of over finite fields are recalled in Section 2. In Section 3, the number theoretical results and properties of -cyclotomic cosets required in the study of the factorization of are established. Recursive methods for factorizing and enumerating its monic irreducible factors are given in Section 4. Applications in the study of negacyclic codes over finite fields are revisited in Section 5.
2. Preliminary
In this section, basic concepts and tools used in the study of the factorization of over finite fields and the enumeration of its monic irreducible factors are recalled.
For a positive integer and an integer , the notation is used whenever is the largest integer such that is divisible by , or equivalently, but . For an integer and a positive integer , denote by the additive order of modulo . In the case where , denote by the multiplicative order of modulo . By abuse of notation, we write .
For a prime power , a positive integer coprime to , and an integer , the -cyclotomic coset modulocontaining is defined to be
It is not difficult to see that and . Moreover, for all . Let denote a complete set of representatives of the -cyclotomic cosets modulo , and let be a primitive th root of unity in some extension field of . It is well known (see [9]) thatwhereis the minimal polynomial of over referred as the irreducible polynomial induced by .
In [10], a basic idea for the factorization of is given using (3) and the following lemmas.
Lemma 1 (see [10], Lemma 2). Let be an odd prime power, and let be an odd positive integer such that . Let and be integers. Then, the elements in have the same parity.
Lemma 2 (see [10], Lemma 3). Let be an odd prime power, and let be an odd positive integer such that . Let and be integers. Then, the polynomial induced by is a divisor of if and only if is odd.
From Lemma 1, the parity of a representative of is independent of its choices. By Lemma 2, the monic irreducible divisors of are induced by the -cyclotomic cosets modulo containing odd integers. Let (resp., ) denote a complete set of representatives of the -cyclotomic cosets containing odd integers (resp., even integers) modulo . It follows thatfor all .
For a positive integer and a prime power , let denote the number of monic irreducible factors of over . Based on ([3], equation 3.1), it can be deduced that
As discussed above, the -cyclotomic cosets modulo containing odd integers are key to determine the factorization of over and the enumeration of its monic irreducible factors. Properties of these cosets are studied in Section 3.
3. Number Theoretical Results and Cyclotomic Cosets
In this section, number theoretical results required in the factorization of are derived. Subsequently, properties of -cyclotomic cosets modulo containing odd integers are established for all positive integers and odd positive integers . These results are key in the study of the factorization of in Section 4.
A relation on the carnality of the -cyclotomic costs containing odd integers and modulo is given in the following lemma.
Lemma 3. Let be an odd prime power, and let be an odd positive integer such that . Then, for all odd integers and for all positive integers .
Proof. Let be an odd integer, and let be a positive integer. Then,Hence,as desired.
Properties of -cyclotomic cosets with and are given separately in the following sections.
3.1.
In this section, we focus on properties of -cyclotomic cosets in the case where .
First, an explicit formula for is recalled for all odd prime powers and positive integers . This result can be derived from ([11], Proposition 1). For completeness, a detailed proof is given.
Lemma 4. Let be an odd prime power, and let be the positive integer such that . Let be a positive integer. If , then
Proof. Assume that . Then, and , for all . Since and is odd, we have . Hence, and , for all .
Assume that . Since , it follows that , for all . Hence, , for all . Since , we have and , for all . Hence, , for all .
Properties of -cyclotomic cosets modulo containing odd integers are established in Proposition 1.
Proposition 1. Let be a prime power such that , and let be an odd positive integer such that . Let be the integer such that , and let be the positive integer such that . Then, the following statements hold:
(i)If , then the following statements hold:(a) all odd integers and integers (b) for all odd integers and integers or (ii)If , then the following statements hold:(a)(b) for all odd integers and integers
Proof. First, we observe that , and .
To prove (i), assume that . In this case, is odd which implies that is odd for all odd positive integers .
To prove (a), let be an odd integer, and let be an integer such that . By Lemma 4, it follows that . Since is odd, it can be deduced that . Suppose that . Since , there exists such that . Hence, we have which implies that , a contradiction. Therefore, , as desired.
To prove (b), let be an odd integer, and let be an integer such that or . By Lemma 4, we have . Since is odd, we have which implies that . Since , we have . Hence, which implies that . This proves the first equality.
For the second equality, let . Then, for some . It follows that . If , then . Otherwise, which implies that . Hence, . Since and are disjoint sets of the same size , we have . Therefore, as desired.
To prove (ii), assume that . For (a), suppose that . If , then , we have by Lemma 4. Since , we have is odd and it follows that . Assume that . Since , we have and by Lemma 4. Since , it follows thatSince , is odd. Hence, . Since , we have for some . It follows that which implies that , a contradiction. Therefore, , as desired.
To prove (b), let be an odd integer, and let be an integer such that . Then, which implies that and by Lemma 4. Since , is odd andwhich implies that . Since , we have . Hence, which implies that . The first equality holds.
For the second equality, let . Then, for some . It follows that . If , then . Otherwise, which implies that . Hence, . Since and are disjoint sets of the same size , we have . Therefore, as desired.
3.2.
Here, we investigate properties of -cyclotomic cosets in the case where . We begin with an explicit formula for which can be derived from ([11], Proposition 1). For completeness, a rigorous proof is provided.
Lemma 5. Let be an odd prime power, and let be the positive integer such that . Let be a positive integer. If , then
Proof. Assume that . Then, which implies that , for all . Next, assume that . Since , it follows that for all . Hence, for all . Since , it can be concluded that and , for all . As desired, we have for all .
Proposition 2. Let be a prime power such that , and let be an odd positive integer such that . Let be the integer such that , and let be the positive integer such that . Then, the following statements hold:
(i)If , then(a) for all odd integers and integers (b) for all odd integers and integers (ii)If , then(a)(b) for odd integers and integers
Proof. First, we observe that , and . Using Lemma 5 and arguments similar to those in the proof of Proposition 1, the following key results can be deduced:(1)If , then , for all odd integers and integers , and , for all integers .(2)If , then and , for all integers .The complete proof can be obtained using arguments similar to those in Proposition 1, while the above discussion and Lemma 5 is applied instead of Lemma 4.
4. Factorization of over Finite Fields
In this section, the factorization of over is established. First, we prove that there exists a positive integer such that the number of monic irreducible factors of over becomes a constant for all integers . In the case where is odd, a complete recursive factorization of over is given together with a recursive formula for the number of its monic irreducible factors for all positive integers in Section 4.1. In the case where is even, a recursive factorization of over is given as well as a recursive formula for the number of its monic irreducible factors for all integers in Section 4.2.
4.1. Recursive Factorization of over with Odd
In this section, we establish a complete recursive factorization of over in the case where is odd. Subsequently, a formula for the number of monic irreducible factors of over is given recursively on .
4.1.1.
We begin with useful relations between -cyclotomic cosets and their induced polynomials for the case .
Lemma 6. Let be a prime power such that , and let be an odd positive integer such that and is odd. Let be the positive integer such that . Let be a positive integer, and let be an odd integer. Then, one of the following statements holds:
(i) and induce distinct monic irreducible polynomials of degree , for all .(ii)For each or , if is induced by , then induces .
Proof. To prove (i), assume that . By Proposition 1 ((a) in (i)), we have . From Lemma 3, it follows that which equals to by the proof of Proposition 1 ((a) in (i)). Hence, and induce distinct monic irreducible polynomials of degree .
To prove (ii), assume that or . Assume that is induced by . Let be a th root of unity. Then, is a th root of unity and . From Proposition 1 ((b) in (i)), we have . It follows thatTherefore, induces as desired.
The next corollary can be deduced directly from the above lemma.
Corollary 1. Assume the notations as in Lemma 6 with . If is induced by , then is irreducible for all .
In order to simplify the notations in the following theorem, let and be th and th roots of unity, respectively. For each , letbe the irreducible polynomials induced by and , respectively. Using these notations, a recursive factorization of is given as follows.
Theorem 1. Let be a prime power such that , and let be an odd positive integer such that and is odd. Let be the positive integer such that . Then, the following statements hold:
(i)If , then(ii)If , thenwhere and are given in (14).
In this case, we havefor all .
Proof. From (5), we note thatThe first statement is the special case where . From Proposition 1 (i), it can be deduced thatwhere the union is disjoint. The results therefore follow from Lemma 6.
A recursive formula for the number of monic irreducible factors of over follows immediately from the theorem.
Corollary 2. Let be a prime power such that , and let be an odd positive integer such that and is odd. Let be an integer, and let be the positive integer such that . Then,
Proof. Equation (20) is a special case of (6). Equation (21) follows immediately from Theorem 1.
4.1.2.
Here, we focus on . First, some useful relations between the -cyclotomic coset and its induced polynomial are established.
Lemma 7. Let be a prime power such that , and let be an odd positive integer such that and is odd. Let be the positive integer such that . Let be a positive integer, and let be an odd integer. Then, one of the following statements holds:
(i) and induce distinct monic irreducible polynomials of the same degree for all (ii)For each , if is induced by , then induces
Proof. The proof can be obtained using arguments similar to those in the proof of Lemma 6, while Proposition 2 (i) is applied instead of Proposition 1 (i).
Corollary 3. Assume the notations as in Lemma 7 with . If is induced by , then is irreducible for all .
The factorization of is given in the following theorem.
Theorem 2. Let be a prime power such that , and let be an odd positive integer such that and is odd. Let be the positive integer such that . Then, the following statements hold:
(i)If , then(ii)If , thenwhere and are given in (14).
In this case, we havefor all .
Proof. The proof can be obtained using arguments similar to those in the proof of Theorem 1, while Proposition 2 (i) and Lemma 7 are applied instead of Proposition 1 (i) and Lemma 6.
From the theorem, the enumeration of monic irreducible factors of over can be concluded in the following corollary.
Corollary 4. Let be a prime power such that , and let be an odd positive integer such that and is odd. Let be an integer, and let be the positive integer such that . Then,
Proof. Equation (25) is given in (6). Equation (26) follows immediately from Theorem 2.
4.2. Factorization of over with Even
In this section, we focus on the case where is even, i.e., for some positive integer . The results are not strong as the previous section. Precisely, a recursive factorization of over is given only for all sufficiently large positive integers .
In general, the factorization of over is given in (3). For , a simpler recursive method for the factorization is given in the following theorem.
Theorem 3. Let be an odd prime power, and let be an odd positive integer such that . Let be the positive integer such that , and let be the positive integer such that . Then,for all .
Proof. The proof can be obtained using arguments similar to those in the proof of Theorem 1, while Proposition 2 (ii) and Proposition 1 (ii) are applied instead of Proposition 2 (i) and Proposition 1 (i).
Corollary 5 follows immediately.
Corollary 5. Let be an odd prime power, and let be an odd positive integer such that . Let be the positive integer such that , and let be the positive integer such that . Then,for all .
4.3. Algorithm and Examples
In this section, the above results are summarized as an algorithm for factorizing over . Some illustrative examples are given as well. An algorithm for the factorization of over is given in Algorithm 1.
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Note that and are given in (14).
For the enumeration of monic irreducible factors of over , it can be calculated using (6). With more information on , , and , the formula can be simplified using Corollaries 2, 4, and 5 of the formwhere is the positive integer such that , is the positive integer such that , and
From (29), the number of monic irreducible factors of over becomes a constant independent of for all if and and for all otherwise. Illustrative examples for the number of monic irreducible factors of over with odd and even are given in Tables 1 and 2, respectively.
In Table 1, the results for and are obtained from Corollaries 2 and 4, respectively.
In Table 2, the last row of each is obtained from Corollary 5. Otherwise, it is computed using (6).
5. Applications
In this section, the factorization of over obtained in Section 4 is applied in the study of negacyclic codes. Some known results are revisited in simpler forms.
A linear code of length over is defined to be a subspace of the -vector space . The dual of a linear code of length over is defined to be
A linear code is said to be self-dual if and it is said to be complementary dual if .
A linear code of length over is said to be negacyclic if it is closed under the negacyclic shift. Precisely, , for every . Under the map defined byit is well known (see [4]) that a linear code of length over is negacyclic if and only if is an ideal in the principal ideal ring . The map induces a one-to-one correspondence between negacyclic codes of length over and ideas in . In this case, is uniquely generated by the monic divisor of of minimal degree in . Such polynomial is called the generator polynomial of .
Let be an odd prime power, and let be an odd positive integer such that . Let be the positive integer such that , and let be the positive integer such that . Let
In general, negacyclic codes have been studied in [3, 4, 10]. Here, we focus on negacyclic codes of length with , where is the characteristic of . The construction and enumeration of such negacyclic codes are simplified using the results from Section 4.
From (5), we have
Based on Theorems 1–3, it follows thatand is irreducible for all .
The following characterization and enumeration of negacyclic codes of length with are straightforward. The proof is committed.
Theorem 4. Assume the notations above. The following statements hold:
(1)The map , defined by , is a ring isomorphism for all integers (2)For each integer , is the generator polynomial of a negacyclic code of length over if and only if is the generator polynomial of a negacyclic code of length over (3)The number of negacyclic codes of length over is , for all
From the theorem, all negacyclic codes of length over with can be determined using the negacyclic codes of length over .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
S. Jitman was supported by the Thailand Research Fund under Research Grant RSA6280042.