Table of Contents
Journal of Discrete Mathematics
Volume 2014, Article ID 985387, 7 pages
http://dx.doi.org/10.1155/2014/985387
Research Article

Hermitian Self-Orthogonal Constacyclic Codes over Finite Fields

Centre for Advanced Study in Mathematics, Panjab University, Chandigarh 160014, India

Received 21 July 2014; Accepted 23 October 2014; Published 12 November 2014

Academic Editor: Tamir Tassa

Copyright © 2014 Amita Sahni and Poonam Trama Sehgal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Necessary and sufficient conditions for the existence of Hermitian self-orthogonal constacyclic codes of length over a finite field , coprime to , are found. The defining sets and corresponding generator polynomials of these codes are also characterised. A formula for the number of Hermitian self-orthogonal constacyclic codes of length over a finite field is obtained. Conditions for the existence of numerous MDS Hermitian self-orthogonal constacyclic codes are obtained. The defining set and the number of such MDS codes are also found.

1. Introduction

Let denote a finite field with elements. An linear code of length and dimension over is a -dimensional subspace of the vector space . Elements of the subspace are called codewords and are written as row vectors . A linear code over is called -constacyclic if is in for every in . Let be the map given by . One can easily check that is an -module isomorphism. We can therefore identify -constacyclic codes of length over with ideals in . The Hamming weight of is the number of nonzero coordinates of . The minimum distance of is defined to be . An code, that is, a linear code with minimum distance , is said to be maximum distance separable (MDS) if . The Hermitian inner product of elements is defined as , for and . For a linear code of length over , the Hermitian dual code of is defined by . If , then is known as Hermitian self-dual and is Hermitian self-orthogonal if .

Aydin et al. [1] dealt with constacyclic codes and a constacyclic BCH bound was given. Gulliver et al. [2] showed that there exists Euclidean self-dual MDS code of length over when by using a Reed-Solomon (RS) code and its extension. They also constructed many new Euclidean and Hermitian self-dual MDS codes over finite fields. Blackford [3] studied negacyclic codes over finite fields by using multipliers. He gave conditions on the existence of Euclidean self-dual codes. Recently, Guenda [4] constructed MDS Euclidean and Hermitian self-dual codes from extended cyclic duadic or negacyclic codes and gave necessary and sufficient conditions on the existence of Hermitian self-dual negacyclic codes arising from negacyclic codes. In [5] the authors gave formulae to enumerate the number of Euclidean self-dual and self-orthogonal negacyclic codes of length over a finite field , where is coprime to . In [6] Yang and Cai gave the necessary and sufficient conditions for the existence of Hermitian self-dual constacyclic codes. They also gave some conditions under which Hermitian self-dual and self-orthogonal MDS codes exist. In this paper, we find necessary and sufficient conditions for the existence of Hermitian self-orthogonal constacyclic codes of length over a finite field , coprime to , and also give a characterization of their defining sets. We obtain a formula to calculate the number of these codes. We give conditions for the existence of some MDS Hermitian self-orthogonal constacyclic codes. We also found their number and defining sets (Table 1).

tab1
Table 1: Number of Hermitian self-orthogonal codes over .

2. Hermitian Self-Orthogonal Constacyclic Codes

Let be an odd prime power and a positive integer relatively prime to . Let be an   -constacyclic code over with , where denotes the order of in . Let be the generator polynomial of . Then divides . Write . The polynomial is called the check polynomial of . For , let be the -cyclotomic coset modulo containing , where is the least positive integer such that . Let be a primitive th root of unity in some extension field of such that . Then the polynomial is the minimal polynomial of over and where is the set of representatives of all the distinct -cyclotomic cosets modulo . As , one can check that the roots of are precisely , . Define Hence we have where .

Let be a -constacyclic code with defining set is a root of . Clearly is a union of some -cyclotomic cosets mod for . The Hermitian dual of the code is a -constacyclic code over with defining set mod (see Theorem 3.2 of [6]). Write the generator polynomial of the code as , where Then the generator polynomial of the Hermitian dual of is , where It can be easily verified that so that .

Lemma 1. Let be a -constacyclic code over with . If is a Hermitian self-orthogonal code, then .

Proof. The proof is similar to [6, Proposition 2.3].

Theorem 2. Nontrivial Hermitian self-orthogonal -constacyclic codes of length over exist if and only if for some .

Proof. Let be a nontrivial Hermitian self-orthogonal -constacyclic code of length over with defining set . Then there exists such that and . Hence giving us that . Thus, (as and ). Conversely, let be such that . Consider . The code is a nontrivial Hermitian self-orthogonal code since .

Define . The following theorem characterizes the defining set of a Hermitian self-orthogonal constacyclic code of length over .

Theorem 3. Let be a -constacyclic code of length over with the defining set . Then is Hermitian self-orthogonal if and only if (i) and (ii) for each , at least one of and belongs to .

Proof. Let be a Hermitian self-orthogonal constacyclic code. Let . Then . Suppose that . Then so that and , which contradicts the hypothesis that . Thus, . Now, let . Then either or , as required.
Conversely, let the defining set be such that and for each , at least one of and is in . Then , by condition (ii) so that the code having as a defining set is a Hermitian self-orthogonal code.

Corollary 4. A -constacyclic code of length over generated by is a Hermitian self-orthogonal if and only if for all .

Define and . Observe that .

Example 5. Let , , and ; then . We consider the -constacyclic code of length 15 over , where with order 7. Clearly . Let ; then . Hence, the code with defining set is a Hermitian self-orthogonal -constacyclic code.

Theorem 6. The number of Hermitian self-orthogonal -constacyclic codes of length over is .

Proof. Let be a Hermitian self-orthogonal -constacyclic code of length over generated by . Then . For , . However for , the pairs have three choices , , and . Hence, the number of Hermitian self-orthogonal -constacyclic codes of length over is .

In order to find the number of Hermitian self-orthogonal -constacyclic codes of length over , we need to compute the value of . Our aim is to prove the following.

Theorem 7. Let , , and be positive integers such that . Then the number of solutions for the linear congruence in the set and is exactly .

Since , the linear congruence has a unique solution modulo . Let it be . Then . We write . The solutions of (7) will be amongst . Clearly, the elements of are relatively prime to . We need to count the number of elements of which are coprime to . Also, . Therefore, the required number .

Lemma 8. Let be a prime divisor of such that . The number of multiples of in is .

Proof. Write , where for each , . Since each contains elements which are pairwise incongruent mod , each forms a complete residue system mod . Hence exactly one element in each is divisible by . Consequently, there exist elements in which are divisible by .

Lemma 9. Let and be two distinct prime divisors of with . Then the number of multiples of or in is .

Proof. The number of multiples of in equals , while the number of multiples of in is . By a similar argument as in Lemma 8, the number of multiples of both and equals . Therefore the required number is .

Theorem 10. Let be all the distinct prime divisors of which are relatively prime to . The number of elements in which are not coprime to is

Proof. The proof follows by induction on Lemmas 8 and 9.

In order to prove Theorem 7, it is enough to show that .

Now,

Let be all the distinct prime divisors of . Also, let be all the distinct prime divisors of . Then , which completes the proof of Theorem 7.

Pick . Let . Define .

Theorem 11. Consider where .

Proof. As , , so that . Since is a divisor of , we have that . Thus , whenever is not coprime to .
As , we write . Also . As , holds giving us that holds with and (). By Theorem 7, there is number of elements in satisfying , , and . However, we have to calculate the number of such in . Now, for , , Consequently, , whenever .

Define Observe that, for , holds if and only if , where , as defined earlier. Recall that, for , .

Theorem 12. One has

Proof. The proof follows from Theorem 11 and the above definition.

Example 13 (let , , and ). Then , , , , and . Take . Thus (as and so that . By Theorem 12, are possible values of on right hand side. Now, as . However, as there does not exist any odd integer such that , so that

We will now investigate the behavior of the function .

Lemma 14. Let and be two integers coprime to such that and for some odd integers and . If , where and , then there exists an odd integer such that .

Proof. Write , being odd distinct primes, . Let be an odd prime divisor of , so that there exists an odd integer such that . Therefore, . Consequently, but , giving us that . For we have , since for some positive integer . Hence , where is odd. Thus, where is odd.

Lemma 15. Let . Then holds for some integer if and only if . In fact, such a is odd.

Proof. Proof is trivial.

Theorem 16. if and only if and for all odd prime divisors of , where and .

Proof. If , then there exists an odd integer such that . Thus so that, by Lemma 15, . Also for every odd prime divisor of . Thus, showing that and . Therefore for all odd prime divisors of .
Conversely, let and for all odd prime divisors of . To prove , we need to find an odd integer such that . For any odd prime divisor of , as , , where is odd. As in the proof of Lemma 14, there exists an odd integer such that , where . Also, . Therefore, with odd. By Lemma 1, . Using Lemma 14, we get that , for some odd integer .

Proposition 17. If and are coprime, then for all divisors of .

Proof. Since and , for some odd integer , if and only if .

Corollary 18. There does not exist any nontrivial Hermitian self-orthogonal -constacyclic code of length over if and only if and for all odd prime divisors of , where and , for .

Proof. The proof follows easily from Theorems 12 and 16.

For odd, we have . The condition reads as , which is always true as . Hence, we have the following.

Corollary 19. There does not exist any nontrivial Hermitian self-orthogonal -constacyclic code of odd length over if and only if for all prime divisors of .

3. MDS Hermitian Self-Orthogonal Constacyclic Codes Over

Let be a -constacyclic code of length over and . Let be a primitive th root of unity in some extension field of such that . Then roots of are of the form , . Put .

Theorem 20. Let the generator polynomial of have roots that include the set . Then the minimum distance of is at least .

Proof. See [1, Theorem 2.2]

By Lemma 1, . Write .

Theorem 21. Let be a divisor of . Let , where for each and each , with Then the code with defining set is a Hermitian self-orthogonal -constacyclic MDS code with parameters .

Proof. Let , be as above and Each has elements. If denotes the code with defining set , then the dimension of , . Let . Then the set has consecutive elements modulo . By Theorem 20, the minimum distance of is at least . However, using the singleton bound, the minimum distance is at most . Consequently, the minimum distance of equals , proving that is an MDS code.
In order to prove that is self-orthogonal, it is enough to prove that . We have . Let and . Then if and only if . Let , then for some so that As , , . Thus so that Consequently, there does not exist any , such that both (18) and (19) hold, thereby showing that is a Hermitian self-orthogonal constacyclic MDS code.

Remark 22. For even, and , the codes obtained from above theorem are the same as given by Theorem 4.3 of [6].

Proposition 23. Let be the number of Hermitian self-orthogonal -constacyclic MDS codes of length which can be obtained from above theorem. Then is given by

Proof. The required number equals the number of ; that is, we need to select consecutive integers from the set . The number of ways of selecting consecutive integers from the set is . Thus, the number of such codes is . Hence the result follows.

Tables 2 and 3 list Hermitian self-orthogonal MDS codes over for . Here , , and denote, respectively, the length, dimension, and minimum distance of the code while denotes the defining set. denotes the number of such Hermitian self-orthogonal MDS codes.

tab2
Table 2: Hermitian self-orthogonal MDS codes over .
tab3
Table 3: Hermitian self-orthogonal MDS codes over .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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