WHILE the algebraic theory of error-correcting codes

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1 1728 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004 Cyclic Negacyclic Codes Over Finite Chain Rings Hai Quang Dinh Sergio R. López-Permouth Abstract The structures of cyclic negacyclic codes of length their duals over a finite chain ring are established when is not divisible by the characteristic of the residue field. Some cases where is divisible by the characteristic of the residue field are also considered. Namely, the structure of negacyclic codes of length 2 over 2 that of their duals are derived. Index Terms Chain rings, cyclic codes, dual codes, linear codes, negacyclic codes, repeated-root negacyclic codes, self-dual codes. I. INTRODUCTION WHILE the algebraic theory of error-correcting codes originally took place in the setting of vector spaces over finite fields, the study of linear codes over finite rings has continued to be increasingly more important since the realization, some years ago, that many seemingly nonlinear codes are actually related to linear codes over the ring of integers modulo four (cf. [7], [11], [10], [18], [19], [22], [23], [27]). The theory of error-correcting codes over finite rings has experienced tremendous growth since its inception. Progress has been attained in the direction of determining the structural properties of codes over large families of rings. This paper is a contribution along those lines as we focus on codes over finite rings with a linear lattice of ideals (the so-called chain rings). Finite chain rings are abundant. For a prime an integer, the ring of integers modulo is a chain ring. Galois rings are also chain rings. The structure of cyclic codes over was obtained by Calderbank Sloane in [8] later on, with a dferent proof, by Kanwar López-Permouth in [14]. Using the techniques presented in [14], Wan [28] extended Kanwar López-Permouth s results to cyclic codes over Galois rings. In 1999, with a dferent technique, Norton Sălăgean-Mache extended the structure theorems given in [8] [14] to cyclic codes over finite chain rings (cf. [20]). That paper provided an alternative approach which did not require commutative algebra the way [8] [14] did. A common pattern in [8], [14], [28], [20] is that they were all concerned with cyclic codes of length over a certain class of finite rings, with the critical condition that is not divisible by the characteristic of the residue field. Not much work has been done for cyclic codes of length over a finite chain ring when is divisible by the character- Manuscript received September 26, 2003; revised February 29, The material in this paper was presented in part at the 995th AMS Meeting on Algebraic Coding, Ohio University, Athens, OH, March H. Q. Dinh was with the Department of Mathematics, North Dakota State University, Fargo, ND USA. He is now with the Department of Mathematical Sciences, Kent State University Trumbull, Warren, OH USA ( hai.dinh@ndsu.nodak.edu; dinhh@trumbull.kent.edu). S. R. López-Permouth is with the Department of Mathematics, Ohio University, Athens, OH 45701, USA ( slopez@math.ohiou.edu). Communicated by R. J. McEliece, Associate Editor for Coding Theory. Digital Object Identier /TIT istic of the residue field. When is a field, such cyclic codes are called repeated-root cyclic codes, have been studied by Castagnoli, Massey, Schoeller, von Seemann (cf. [9]), by van Lint (cf. [25]). Recently, cyclic codes over with length a power of were discussed by Abualrub Oehmke in [1], where -cyclic codes were characterized in terms of their sets of generators. The purpose of this paper is to obtain structure theorems for cyclic negacyclic codes their duals in a more general setting. On the one h, we will generalize the methods of [8], [14] to obtain cyclic self-dual cyclic codes over finite chain rings with the condition that the length of the code is not divisible by the characteristic of the residue field. Our strategy will be independent from the approach in [20] the results are more detailed. For instance, we fully describe the structure of cyclic codes their duals also provide necessary sufficient conditions for the existence of nontrivial cyclic dual codes. We will then apply similar strategies to characterize the structure of negacyclic codes their duals under the additional condition that the length is odd. Finally, we will consider cases when the length of the code is divisible by the characteristic of the residue field. In particular, the structure of negacyclic codes of length over will be obtained. After presenting preliminary concepts results in Section II, we will generalize the techniques used in [8], [14] (for ) to obtain structure theorems for cyclic codes over finite chain rings with the condition that the length of the code is not divisible by the characteristic of the residue field in Sections III IV. A necessary sufficient condition for the existence of self-dual cylic codes over will be proved, extending the cases for codes over ([23]) ([14]). The structure of negacyclic self-dual negacyclic codes of odd length over finite chain rings is obtained in Section V. We then consider cases where the length is divisible by the characteristic of the residue field. Section VI gives the structure of negacyclic codes of length over. That section also contains a discussion of self-dual -codes. We close with an Appendix on the theory of quadratic residues. II. PRELIMINARIES All rings are associative rings with identity. An ideal of a ring is called principal it is generated by one element. A ring is a principal ideal ring its ideals are principal. is called a local ring is a division ring (or, equivalently, has a unique maximal right (left) ideal). Furthermore, a ring is called a right (left) chain ring the set of all right (left) ideals of is a chain under set-theoretic inclusion. If is both a right a left chain ring, we simply call a chain ring. For the class of finite commutative chain rings, we have the following equivalent conditions /04$ IEEE

2 DINH AND LÓPEZ-PERMOUTH: CYCLIC AND NEGACYCLIC CODES OVER FINITE CHAIN RINGS Proposition: For a finite commutative ring the following conditions are equivalent: i) is a local ring the maximal ideal of is principal; ii) is a local principal ideal ring; iii) is a chain ring. Proof: i) ii). Let be an ideal of.if then is generated by the identity.if, then. By i), is generated by an element, say. Therefore,, for some integer. Hence, is a local principal ideal ring. ii) iii). Let be a local principal ideal ring with the maximal ideal, be proper ideals of. Then, whence there exist integers ( the nilpotency of ). Hence, either,or. Thus, is a chain ring. iii) i). Assume is a finite commutative chain ring, then clearly is local. To show the maximal ideal of is principal, suppose to the contrary that is generated by more than one element, say are in the generator set of,. Then, a contradiction with the assumption that is a chain ring. Thus, is principal, proving i). In this paper, we will consider finite commutative chain rings will make use of the equivalent conditions in Proposition 2.1. Let be a fixed generator of the maximal ideal. Then is nilpotent we denote its nilpotency index by. The ideals of form a chain Let. By, we denote the natural ring homomorphism that maps the variable to. The following is a well-known fact about finite commutative chain ring (cf. [16]). 2.2 Proposition: Let be a finite commutative chain ring, with maximal ideal, let be the nilpotency. Then we get the following statements. a) For some prime positive integers,, the characteristic of are powers of. b) For,. In particular,, i.e.,. Two polynomials are called coprime, or equivalently, there exist such that. A polynomial is called basic irreducible is irreducible in. A polynomial is called regular it is not a zero divisor. 2.3 Proposition: (cf. [16, Theorem XIII.2(c)]) Let be in, then the following are equivalent: i) is regular, ii), iii) is a unit for some,, iv). The following so-called Hensel s lemma guarantees that factorizations into product of pairwise coprime polynomials in lt to such factorizations over (cf. [16, Theorem XIII.4]). 2.4 Lemma: (Hensel s Lemma) Let be a polynomial over assume where are pairwise coprime polynomials over. Then there exist pairwise coprime polynomials over for. Let denote the set of all polynomials such that has distinct zeros in the algebraic closure of. The following proposition explores the relationships between irreducibility basic irreducibility for regular polynomials for elements of. 2.5 Proposition: (cf. [16, Theorem XIII.7]) Let be a regular polynomial. Then we have the following. i) If is basic irreducible then is irreducible. ii) If is irreducible then, where is monic irreducible in. iii) If is in then is irreducible only is basic irreducible. An ideal is primary whenever, then either or for some positive integer.a polynomial is called primary is a primary ideal of. 2.6 Proposition: (cf. [16, Theorem XIII.11]) Let be a regular polynomial in. Then where is a unit are regular primary coprime polynomials. Moreover, are unique in the sense that, where are units, are regular primary coprime polynomials, then, after renumbering,. A regular polynomial is primary only is primary in. Equivalently,, where is irreducible in. Combining this with Propositions 2.3, 2.5, 2.6, we have the following. 2.7 Proposition: If is a monic polynomial over is square free, then factors uniquely as a product of monic basic irreducible pairwise coprime polynomial. The following version of the Euclidean algorithm holds true for polynomials over finite commutative local rings. 2.8 Proposition: (cf. [16, Exercise XIII.6]) Let be nonzero polynomials in. If is regular, then there exist polynomials. Let. is called an associate of there is an invertible element. A polynomial is called irreducible whenever, where, then either or is an invertible element in. A polynomial is called reducible it is not irreducible. Note that the irreducibility of a polynomial depends on the ring, for example, is irreducible over, but it is reducible over. The following

3 1730 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004 so-called Eisenstein criterion makes it easier to check the irreducibility of certain class of polynomials in over (cf. [12, Theorem ]). 2.9 Theorem: (Eisenstein criterion). Let 2.11 Proposition: Let be a finite ring of order. The number of codewords in any linear code of length over is, for some integer. Moreover, the dual code has codewords, where Proposition: Let be a finite commutative ring If there is a prime number then is irreducible in. Note that the Eisenstein criterion does not hold for.for example, consider, then satisfies the conditions of Eisenstein criterion for,but is reducible in,as.in Section VI, we will extend this criterion to check irreducibility of polynomials in (Lemma 6.2). Classical coding theory took place in the setting of vector spaces over finite fields. We refer to the textbooks [3], [13], [15], [21] for background on algebraic codes over finite fields. Natural modications lead us to codes over finite rings. For a finite ring, consider the set of -tuples of elements from as a module over in the usual way. A subset is called a linear code of length over is an -submodule of. is called cyclic, for every codeword, its cyclic sht is also in. An -tuple is identied with the polynomial in, which is called the polynomial representation of. It is well known that a code of length over is cyclic only the set of polynomial representations of its codewords is an ideal of. Given, their scalar product (or dot product) is (evaluated in ). Two words are called orthogonal. For a linear code over, its dual code is the set of words over that are orthogonal to all codewords of, i.e., A code is called self-dual. For a finite chain ring with maximal ideal the nilpotency of is even, the code is self-dual is called the trivial self-dual code. In Section III, we will discuss in more detail nontrivial cyclic self-dual codes over finite chain rings. The following result is well known (cf. [8]) Proposition: The number of codewords in any linear code of length over is, for some integer. Moreover, the dual code has codewords, where. The proof of Proposition 2.10 can be used to prove a more general result. Then in only is orthogonal to all its cyclic shts. Proof: Let denote the cyclic sht for codewords of length, i.e., for each Thus,, are all cyclic shts of. Let Then for Therefore, only for only for only is orthogonal to all its cyclic shts, as desired. III. STRUCTURE OF CYCLIC CODES OVER FINITE CHAIN RINGS Let be a finite chain ring with the maximal ideal, be the nilpotency of. By Proposition 2.2, there exist a prime an integer,, the characteristic of are powers of. In this section, we assume to be a positive integer which is not divisible by ; that implies is not divisible by the characteristic of the residue field, so that is square free in. Therefore, has a unique decomposition as a product of basic irreducible pairwise coprime polynomials in (by Proposition 2.7). 3.1 Lemma: Let be a finite chain ring with the maximal ideal, be the nilpotency of. If is a regular basic irreducible polynomial of the ring, then is also a chain ring with precisely the following ideals: Proof: First we show that for distinct values of,. Suppose for

4 DINH AND LÓPEZ-PERMOUTH: CYCLIC AND NEGACYCLIC CODES OVER FINITE CHAIN RINGS 1731 Then by Proposition 2.8, there exists means.as with. That in is a sum of ideals of the form, where. Proof: By the Chinese Reminder theorem, we have it follows that. Multiplying by gives, which is a contradiction to our hypothesis that has nilpotency. Let be a nonzero ideal of a nonzero element of. By assumption, is a basic irreducible polynomial in, hence, is irreducible in. Therefore,,or. If, i.e., are coprime in, then are coprime in. So there exist. That implies whence is invertible in. Therefore, For the case, for all, which means divides, hence, there exist. Whence for all implying. Let be the greatest integer such that. Then, as, there is a (nonzero) element. Since, there exist such that. Now, or. Suppose, then divides so there exist. Hence, It follows that, a contradiction. Thus,. The same arguement as above yields that is invertible in, which means that there exists Therefore, Thus, any ideal of is of the form, where is an ideal of. By Lemma 3.1, for, or, for some. Then correspond to in. Consequently, is a sum of ideals of the form. 3.3 Corollary: Let be a finite chain ring with maximal ideal, be the nilpotency of. The numbers of cyclic codes over of length is, where is the number of factors in the unique factorization of into a product of monic basic irreducible pairwise coprime polynomials. From now on, in order to simply notation, we will just write for the corresponding coset in. 3.4 Theorem: Let be a cyclic code of length over a finite chain ring ( has maximal ideal is the nilpotency of ). Then there exists a unique family of pairwise coprime monic polynomials in. Moreover Proof: Let be the unique factorization of into a product of monic basic irreducible pairwise coprime polynomials. By Theorem 3.2, is a direct sum of ideals of the form, where. After reordering necessary, we can assume that where. Let, be a nonnegative integer.for, define Consequently,. Customarily, for a polynomial of degree, its reciprocal polynomial will be denoted by. Thus, for example,, then Then by our construction, it is clear that coprime,, To prove the uniqueness, assume coprime monic polynomials in. Then are pairwise are pairwise Moreover, is a factor of, we denote. 3.2 Theorem: Assume is a finite chain ring with maximal ideal, that is the nilpotency of. Let be a representation of as a product of basic irreducible pairwise-coprime polynomials in. Then any ideal thus,. Now there exist nonnegative integers with, a permutation of such that, for

5 1732 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004 Hence,,wehave. Thus, for any integer Now for, it follows that,, furthermore, is a permutation of. Therefore,, for. To calculate the order, note that Multiplying the left-h side of this equation out, we get that there exist polynomials for In particular, when, we have Multiplying both sides of the above equation by yields Hence, By hypothesis which implies 3.5 Theorem: Let be a cyclic code of length over a finite chain ring, which has maximal ideal is the nilpotency of. Then there exist polynomials in. Proof: By Theorem 3.4, there exists a family of pairwise coprime monic polynomials in. Define Then clearly,wehave.. Moreover, for Therefore,. On the other h,. Since are coprime polynomials in, there exist polynomials. It follows that hence, Therefore, that Consequently,.. Continuing this process, we obtain 3.7 Corollary: is a principal ideal ring. Proof: Let be any ideal of. Then is a cyclic code of length, so by Theorem 3.6, is a principal ideal, proving that is a principal ideal ring. 3.8 Theorem: Let be a cyclic code over with where, as in Theorem 3.4. Then whence. Continuing this process, we obtain for, which implies Consequently,. 3.6 Theorem: Let be a cyclic code of length with notation as in Theorem 3.4,. Then is a generating polynomial of, i.e.,. Proof: For any distinct, we have, so in. Moreover, for any with, are coprime, hence, there exist Proof: By Proposition 2.2,,, hence, By Proposition 2.11,, where

6 DINH AND LÓPEZ-PERMOUTH: CYCLIC AND NEGACYCLIC CODES OVER FINITE CHAIN RINGS 1733 Therefore, That means Denote Let For,, i.e.,, then ;, i.e.,, then. Hence, in. Therefore,. Note that, for, we have Then in Hence, Consequently,, as desired. 3.9 Lemma: Let as in Theorem 3.8. Then in a), for any distinct ; b) are coprime for any. Proof: a) For any distinct, in, hence, there exists with. Whence which implies that Therefore, are coprime Theorem: With the notations as in Theorem 3.8, let. Then is a generating polynomial of, i.e.,. Proof: Making use of Lemma 3.9 a similar arguement as in the proof of Theorem 3.6 gives the result. IV. CYCLIC DUAL CODES Corollary 4.3 in [14] provided a characterization for a cyclic code over to be self-dual; we now generalize that result to cyclic codes over a finite chain ring. 4.1 Proposition: Let (as in Theorem 3.4), then is self-dual only is an associate of for all. Proof: By Theorem 3.8,, hence, is an associate of for all such that, then i.e., is self-dual. On the other h, assume, let denote the constants of,. Since, it follows that. Therefore, s are invertible elements of s are leading coefficents of s. For all, denote, where s are suitable invertible elements in s are monic polynomials. Note that,.now Therefore, Thus, in. b) Since are coprime, there exist. It follows that Also, i.e., From the uniqueness in Theorem 3.4, complete. the proof is

7 1734 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST Lemma: a) If, then b). Proof: Straightforward calculations. 4.3 Theorem: Assume that is an even integer, then nontrivial self-dual cyclic codes exist only there exists a basic irreducible factor of are not associate. Proof: Assume there exists a basic irreducible factor of are not associate. Then the constant of is nonzero, it follows that, is also a factor of,, furthermore, is a factor of. Write consider Since Lemma 4.2,wehave, so. Now by Theorem 3.8, Conversely, assume that there is a nontrivial self-dual cyclic code, in view of Theorem 3.4, can be written as, where are monic polynomial. Suppose, to the contrary, that every basic irreducible factor of is an associate of. Then are associate for.as is self-dual, Proposition 4.1 implies that for all, is an associate of, hence, of.as has no repeated roots, it follows that for with, either or, whence, for all,. Therefore,, which is the trivial self-dual code, a contradiction. That means there must exist a basic irreducible factor of are not associate. Consider, since, has an inverse modulo, thus there exist positive integers. Let be the smallest such integer, the set, where each is reduced modulo, is called the cyclotomic coset modulo containing (cf., [15], [21]). It is straightforward to very that for, only for some integer. Using the above observation Proposition 4.3, we arrive at the following result which gives a necessary sufficent condition for the existence of nontrivial self-dual cyclic codes of length over when is even. This result is a generalization of Theorem 3 in [23] ( ), Theorem 4.4 in [14]. 4.4 Theorem: Assume that is a finite chain ring with the maximal ideal,, where is the nilpotency of. If is even, then nontrivial self-dual cyclic codes of length over exist only for all positive integers. Proof: Let denote the cyclotomic coset modulo containing be a primitive th root of unity. If is a monic basic irreducible factor of then there exists a cyclotomic coset so where is an invertible element in. Hence, by Proposition 4.3, nontrivial self-dual cyclic codes of length exist only there is a basic irreducible factor of are not associate, only for all cyclotomic cosets, only for all. When, the integers, where for all positive integers, was completely characterized by Moree in [23, Appendix B] more details in [17]. 4.5 Theorem: (cf. [23, Theorem 4]) Let be a finite chain ring with the maximal ideal where, where is the nilpotency of. If is even, is odd, then nontrivial self-dual cyclic codes of length over exist only is divisible by either of the following: a prime,or a prime, where the order of is odd, or dferent odd primes the order of is the order of is, where is odd, is even,. When is an odd prime, a characterization of integers, where for all positive integers, is still unknown. However, there are cases where for some integer, which leads to the nonexistence of nontrivial self-dual cyclic codes for certain values of. Recall that for relatively prime integers, is called a quadratic residue or quadratic nonresidue of according to whether the congruence has a solution or not. We discuss important properties of quadratic residues related concepts in the Appendix. We return now to our consideration of a finite chain ring with the maximal ideal,, where is the nilpotency of, is even. Theorem 4.4 the properties of quadratic residues in the Appendix give the following results about the nonexistence of nontrivial self-dual cyclic codes. 4.6 Corollary: If is a prime, then nontrivial self-dual cyclic codes of length do not exist in the following cases:, ;, ;, ;, ;,.

8 DINH AND LÓPEZ-PERMOUTH: CYCLIC AND NEGACYCLIC CODES OVER FINITE CHAIN RINGS 1735 Proof: Follows from Theorem 4.4, Remark A.4, Proposition A Corollary: Assume that is an odd prime dferent than, is a quadratic nonresidue of, where. Then nontrivial self-dual cyclic codes of length do not exist. Proof: Follows from Theorem 4.4, Remark A.4, Proposition A Corollary: If is an odd prime, then nontrivial self-dual cyclic codes of length do not exist in the following cases:, there exists a positive integer such that is a quadratic nonresidue of ;, there exist positive integers, is a quadratic nonresidue of. Furthermore, let be the prime factorization of. Assume that, is a quadratic nonresidue of, but then there exists an integer nontrivial self-dual cyclic codes of length do not exist. Proof: Follows from Theorem 4.4, Remark A.4, Theorem A.12. V. NEGACYCLIC CODES OVER FINITE CHAIN RINGS A negacyclic code of length over is defined to be a subset of, where is the negasht of (i.e.,, for all ). Similar to cyclic codes, it is known that a subset of is a negacyclic code of length only its polynomial representation is an ideal of the factor ring. In 1968, Berkelamp started the study of negacyclic codes over finite fields [2], [3]. Recently, Wolfmann [29] gave various interesting results about negacyclic codes of odd length over, proposed questions about such codes when the length is even. We will start by studying negacyclic codes of odd length in a more general setting, namely, over finite chain rings. Then, in Section VI, we will focus on negacyclic codes of length over. In this section, we discuss the structure of negacyclic codes of length over a finite chain ring, with the same sting hypotheses as in the previous sections, i.e., is a finite chain ring with the maximal ideal, is the nilpotency of,,, the characteristic of are powers of ; is assumed to be a positive integer which is not divisible by, so that is not divisible by the characteristic of the residue field, that means is square-free in, therefore, has a unique decomposition into basic irreducible pairwise coprime polynomials in. Assume furthermore that is an odd integer. 5.1 Proposition: Let be a map given by Then is a ring isomorphism. Proof: For polynomial only there exist a polynomial only only That means, for, only, whence is well-defined one-to-one. It is obvious that is onto it is easy to very that is a ring homomorphism. Therefore, is a ring isomorphism. The following is straightforward from Proposition Corollary: Let be. Then is an ideal of only is an ideal of. Equivalently, is a cyclic code of length over only is a negacyclic code of length over. The condition that the length is odd is critical here, as the one-to-one correspondence between cyclic negacyclic codes of the same length does not hold when the length is even. This can be seen easily from the fact that may have dferent factorizations in. For instance, while is always reducible, we will show in Proposition 6.3 that is irreducible in. In fact, Blackford has showed in [5] that over, there are 1183 cyclic codes of length,but only 125 negacyclic codes of length. For a factor of, denote. Using the isomorphism in Proposition 5.1 ( Corollary 5.2), results about cyclic codes of length over obtained in Sections III IV, can be carried over respectively to negacyclic codes of length over, with the sting hypotheses given at the beginning of the section. We list them here for the sake of completeness. 5.3 Theorem: (cf. Theorem 3.2) Assume is a finite chain ring with maximal ideal, that is the nilpotency of. Let be a representation of as a product of basic irreducible pairwise-coprime polynomials in. Then any ideal in is a sum of ideals of the form, where. 5.4 Corollary: (cf. Corollary 3.3) Let be a finite chain ring with maximal ideal, be the nilpotency of. The numbers of negacyclic codes over of length is, where is the number of factors in the unique factorization of into

9 1736 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004 a product of monic basic irreducible pairwise coprime polynomials. 5.5 Theorem: (cf. Theorem 3.4) Let be a negacyclic code of length over a finite chain ring ( has maximal ideal, is the nilpotency of ). Then there exists a unique family of pairwise coprime monic polynomials in. Moreover orthogonal to as desired. all its negacyclic shts, With the help of Proposition 5.9 the above results, we can now carry results in Sections III IV about cyclic dual codes to negacyclic dual codes correspondingly Theorem: (cf. Theorem 3.8) Let be a negacyclic code over with 5.6 Theorem: (cf. Theorem 3.5) Let be a negacyclic code of length over a finite chain ring, which has maximal ideal, is the nilpotency of. Then there exist polynomials in. 5.7 Theorem: (cf. Theorem 3.6) Let be a negacyclic code of length with notation as in Theorem 5.5, Then is a generating polynomial of, i.e.,. 5.8 Corollary: (cf. Corollary 3.7) is a principal ideal ring. To study negacyclic dual codes, we need the following result, which is an analogous version of Proposition 2.12, but for negacyclic codes. 5.9 Proposition: Let be a finite commutative ring Then in only is orthogonal to all its negacyclic shts. Proof: Let denote the negacyclic sht for codewords of length, i.e., for each Thus,,, are all negacyclic shts of, for Then for. Let Therefore, only for only for only is where, as in Theorem 5.5. Then 5.11 Lemma: (cf. Lemma 3.9) Let as in Theorem Then in a) for any distinct ; b) are coprime for any Theorem: (cf. Theorem 3.10) With the notations as in Theorem 5.9, let. Then is a generating polynomial of, i.e., Proposition: (cf. Proposition 4.1) Let (as in Theorem 5.5), then is self-dual only is an associate of for all Theorem: (cf. Theorem 4.3) Assume that is an even integer, then nontrivial self-dual negacyclic codes exist only there exists a basic irreducible factor of are not associate Theorem: (cf. Theorem 4.4) Assume that is a finite chain ring with the maximal ideal,, where is the nilpotency of. If is even, then nontrivial self-dual negacyclic codes of length over exist only for all positive integers Theorem: (cf. Theorem 4.5) Let be a finite chain ring with the maximal ideal where, where is the nilpotency of. If is even, is odd, then nontrivial self-dual negacyclic codes of length over exist only is divisible by either of the following: a prime,or a prime, where the order of is odd, or dferent odd primes the order of is the order of is, where is odd, is even,.

10 DINH AND LÓPEZ-PERMOUTH: CYCLIC AND NEGACYCLIC CODES OVER FINITE CHAIN RINGS Corollary: (cf. Corollary 4.6) If is a prime, then nontrivial self-dual negacyclic codes of length do not exist in the following cases:,,,,,,,,, Corollary: (cf. Corollary 4.7) Assume that is an odd prime dferent than, is a quadratic nonresidue of, where. Then nontrivial self-dual negacyclic codes of length do not exist Corollary: (cf. Corollary 4.8) If is an odd prime, then nontrivial self-dual negacyclic codes of length do not exist in the following cases:, there exists a positive integer such that is a quadratic nonresidue of,, there exist positive integers, is a quadratic nonresidue of. Furthermore, let be the prime factorization of. Assume that, is a quadratic nonresidue of, but then there exists an integer nontrivial self-dual negacyclic codes of length do not exist. VI. NEGACYCLIC CODES OF LENGTH OVER Section V discussed negacyclic codes of odd length over a finite chain ring. Results about negacyclic of odd length in the specic case were obtained by Wolfmann in [29], where he also proposed various questions about negacyclic codes of even length over. Afterwards, a transform approach was used recently by Blackford in [5] to classy negacyclic codes of even length over. That classication is aimed at determining which binary cyclic codes are images of negacyclic codes under the Gray map. In this section, we will use an approach dferent from the one in [5] to obtain the structure of negacyclic codes of length over for any integer. We start our discussion with a couple of observations. 6.1 Lemma: For any positive integer, there exists a polynomial, in. Furthermore, is invertible in. Proof: We proceed by induction on. For,,, in, clearly, is invertible in. Now assume the statement is true for all positive integer less than. Then where. Since is invertible in, is also invertible in. Therefore, as is nilpotent in, we can write as, where is a nilpotent element in. Now let be an odd integer such that,wehave That means, is invertible in, hence, is invertible in. Recall that a polynomial is irreducible is not a unit whenever then either or is a unit. is called a divisor of, a proper divisor. It is well known easy to very that for a finite commutative local ring, is a proper divisor of only is a divisor of is a proper divisor of in (cf. [16, Ch. XIII]). We now produce a criterion that allows us to check irreducibility of some polynomials over the ring,. Recall that a polynomial is irreducible is not a unit whenever then either or is a unit. is called a divisor of, a proper divisor. It is well known easy to very that for a finite commutative local ring, is a proper divisor of only is a divisor of is a proper divisor of in (cf. [16, Ch. XIII]). We now produce a criterion that allows us to check irreducibility of some polynomials over the ring,. 6.2 Lemma: Let be a prime, be positive integers with, If in,, then is irreducible in. Proof: Suppose, to the contrary, that is reducible, i.e., where are both noninvertible polynomials in. Since are proper divisors of in, the reductions modulo of must be proper divisors of the reduction modulo of over (cf. [16, Example XIII.8]). It follows that for every. On the other h, By hypothesis,, it follows that either or. Without loss of generality, assume that. Since, wehave. Hence,

11 1738 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004 TABLE I REPRESENTATION OF f (x) 2 AS f (x) = f (x) + 2f (x), WHERE f (x) = a + a (x + 1);f (x) = a + a (x + 1) AND a ;a ;a ;a 2 there exists, necessarily less than,. Let be the first coefficient in.as in. In other words, there exist we have. Thus,, a contradiction. Therefore, is irreducible in. 6.3 Proposition: For any nonnegative integer, is irreducible in. Proof: Let For, each has a unique binary representation where. Hence, can be written as where,. Furthermore, can be represented as Denote. Consequently, can be written (uniquely) as Then, for. Furthermore, for, are even integers, thus,. Hence, by Lemma 6.2, is irreducible in. Therefore, is also irreducible in. Next, we explore alternative representations of the elements of. As is customary, every element can be viewed as a polynomial with degree less than or equal to where,. 6.4 Example: For all polynomials Table I lists the representation of, where Lemma: For integers, is nilpotent in, with nilpotency. as

12 DINH AND LÓPEZ-PERMOUTH: CYCLIC AND NEGACYCLIC CODES OVER FINITE CHAIN RINGS 1739 Proof: By Lemma 6.1, there exists an invertible polynomial, it follows that in. Hence, i.e., is nilpotent in. As the nilpotency of in is, it follows that the nilpotency of in is. 6.6 Proposition: Let be written as in TABLE II NEGACYCLIC CODES OF LENGTH 4 OVER AND THEIR ORDERS where,. Then is invertible only. Proof: If, then. Since are nilpotent in, it follows that are nilpotent in. Therefore, is nilpotent in, i.e., is not invertible. On the other h,, then. As above is nilpotent. Thus, there exists an odd positive integer.now Hence, is invertible. 6.7 Lemma: For any positive integer Now Theorem 6.8 provides us a complete list of negacyclic codes of length over Theorem: The negacyclic codes of length over (i.e., the ideals of ) are precisely where is the nilpotency of in Proposition: For, there are codewords in the negacyclic code. Proof: The chain of ideals of is a strictly chain Proof: It is clear from the fact that for all 6.8 Theorem: For any integers,, is a local ring with the maximal ideal. Proof: We have, so by Lemma 6.4,. Thus, in,. Let, write as where,. Then, by Proposition 6.6, either ( ), or is invertible ( ). Consequently, is a local ring with the maximal ideal. 6.9 Remark: In view of Proposition 2.1, is a principal ideal ring (as well as a chain ring). By Proposition 2.10, each of these ideals has order for some integer. Clearly, there are precisely such orders furthermore Therefore, for, Example: Consider negacyclic codes of length over. Here is nilpotent in, with the nilpotency. Hence, the negacyclic codes of length over are,. Table II gives those negacyclic codes (i.e., ideals of ), their orders Lemma: Let be a finite commutative ring a negacyclic code of length over. Then the dual code is also a negacyclic code. Proof: Let be a codeword of.for any codeword, since is negacyclic,. Therefore, That means, whence,. Thus,

13 1740 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004 TABLE III NEGACYCLIC CODES OF LENGTH 8 OVER AND THEIR DUAL CODES Hence, is negacyclic., consequently, 6.14 Theorem: Let be a negacyclic code of length over. Then for some is calculated modulo, the followings hold: a),. b),. Proof: a) Follows from Theorem 6.10 Proposition For b), by Proposition 2.10,. Moreover, by Lemma 6.13, is also a negacyclic code, so Proposition 6.11 implies that Corollary: is the only self-dual negacyclic code of length over. Proof: Let be a self-dual negacyclic code of length over. Then, by Theorem 6.14, for some. Self-duality of forces, i.e.,, as desired Corollary: If is even, then in. Proof: Since is even, is a self-dual negacyclic code over (the trivial self-dual code). On the other h, by Corollary 6.15, is the only self-dual negacyclic code over. It forces in Example: Consider the negacyclic codes of length over. We list in Table III all such negacyclic codes, their dual codes, orders Example: Consider the negacyclic codes of length over. Table IV lists all such negacyclic codes, their dual codes, orders. APPENDIX QUADRATIC RESIDUES A.1 Definition: Let be relatively prime integers, i.e.,. If the quadratic congruence has a solution, then is said to be a quadratic residue of, otherwise, is called a quadratic nonresidue of. The problem of determining whether an integer is a quadratic residue or quadratic nonresidue of has played an important role in Number Theory. It interested several of the greatest number theorists of the 18th century, including Leonhard Euler ( ), Joseph Louis Lagrange ( ), Adrien Marie Legendre ( ), Carl Friedrich Gauss ( ), who made the greatest contribution to the subject (cf. [6], [26]). Euler gave a simple criterion for deciding whether an integer is a quadratic residue of a given odd prime. A.2 Theorem (Euler s Criterion): Let be an odd prime. Then a is a quadratic residue or quadratic nonresidue of according to whether or. For the sake of simplication, we will make use of the following so-called Legendre symbol, named after the French mathematician Adrien Marie Legendre ( ), who first introduced this symbol in 1798.

14 DINH AND LÓPEZ-PERMOUTH: CYCLIC AND NEGACYCLIC CODES OVER FINITE CHAIN RINGS 1741 TABLE IV NEGACYCLIC CODES OF LENGTH 8 OVER AND THEIR DUAL CODES A.3 Definition: Let be an odd prime. The Legendre symbol is defined by is a quadratic residue of is a quadratic nonresidue of. are called the numerator denominator of the symbol, respectively. A.4 Remark: By Euler s Criterion A.2, Legendre symbol s definition A.3, for an odd integer an integer with,wehave only only. We gather some elementary properties of the Legendre symbol in the following proposition.

15 1742 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004 A.5 Proposition: Let be an odd prime be integers which are relatively prime to. Then i), then ; ii) ; iii) ; iv) ; v). The following powerful result, proved by Gauss named after him, gives the quadratic character of an integer, which allows us to obtain a variety of interesting results. A.6 Theorem (Gauss Lemma): Let be an odd prime.if denotes the number of integers in the set whose remainders by dividing exceed, then. For distinct odd primes, both of the Legendre symbols are well defined. It is natural to wonder there is a connection between the values of these two symbols. Fortunately, the answer is yes the result has been referred as the Quadratic Reciprocity Law. The relationship was first conjectured by Euler in 1783 imperfectly proved by Legendre in 1785 ( another of his proof in 1798 also contained a gap). The first complete proof was obtained by Gauss in 1795 when he was 18 years old. Later, Gauss contributed seven dferent proofs, since then, a total of more than a hundred dferent proofs have been published by numerous authors, including some of the greatest number theorists of the 19th century such as: Cauchy ( ), Karl Gustav Jacobi ( ), Peter Gustav Dirichlet ( ), Ferdin Eisenstein ( ), Leopold Kronecker ( ), Richard Dedekind ( ). A.7 Theorem (Quadratic Reciprocity Law): If are distinct odd primes, then The Quadratic Reciprocity Law A.7 has the following immediate consequences. A.8 Corollary: If are distinct odd primes, then i) or. ii) or. Gauss s Lemma A.6 the Quadratic Reciprocity Law A.7 enable us to compute for specic values of. The following proposition lists some of them. A.9 Proposition: Let be an odd prime, then: i) ii) ; iii) or or ; iv) or or ; v) ; vi) ; vii). viii) See the equation at the bottom of the page. If in addition, the odd prime is then ix) or or ; x). Proof: i) vii), ix), v) can be found in [6], [26], [24]. We will just give the proof of viii) as an illustration of applications of Proposition A.5, Theorem A.7, Corollary A.8. Since, Corollary A.8 implies that. Also by Proposition A.5 i) vii). Therefore, we get the expression at the bottom of the following page proving viii). A.10 Example: Using Propositions A.5 A.9, we compute for nonzero integers odd primes. Table V lists those values..

16 DINH AND LÓPEZ-PERMOUTH: CYCLIC AND NEGACYCLIC CODES OVER FINITE CHAIN RINGS 1743 TABLE V VALUES OF (ajq) FOR NONZERO INTEGERS a = 03; 02;...; 10; 11 AND ODD PRIMES q<50. A.11 Proposition: Let be an odd prime, be a positive integer,. Then is a quadratic residue of only. Theoretically, the following theorem settles completely the problem of determining whether is a quadratic residue of. A.12 Theorem: Let be the prime factorization of,. Then is a quadratic residue of only i) for, ; ii) but. The Legendre symbol is defined when is an integer relatively prime to the prime. There are other generalizations of this definition defined based on the Legendre symbols. One of which is the Jacobi symbol, named after Karl Gustav Jacobi ( ), who first introduced it. The Jacobi symbol is also denoted by but is only required to be an odd integer. Legendre Jacobi symbols share a lot of the same properties. It is also worth noting that there are concepts of higher degree residues, namely, is called an th degree residue of the congruence is solvable. We refer to [24] for detailed discussion about Jacobi symbols th degree residues. ACKNOWLEDGMENT The authors would like to sincerely thank the referees for a very meticulous reading of this manuscript. Their suggestions were valuable to create an improved final version. REFERENCES [1] T. Abualrub R. Oehmke, On the generators of cyclic codes of length 2, IEEE Trans. Inform. Theory, vol. 49, pp , Sept [2] E. R. Berlekamp, Negacyclic codes for the Lee metric, in Proc. Conf. Combinatorial Mathematics its Applications, Chapel Hill, NC, 1968, pp [3], Algebraic Coding Theory, revised 1984 ed. Laguna Hills, CA: Aegean Park, [4] T. Blackford, Negacyclic codes over of even length, IEEE Trans. Inform. Theory, vol. 49, pp , June [5], Cyclic codes over of oddly even length, Appl. Discr. Math., vol. 128, pp , [6] D. M. Burton, Elementary Number Theory, 4th ed. New York: Mc- Graw-Hill, 1998, Series in Pure Applied Mathematics. [7] A. R. Calderbank, A. R. Hammons, Jr., P. V. Kumar, N. J. A. Sloane, P. Solé, A linear construction for certain Kerdock Preparata codes, Bull. Amer. Math. Soc., vol. 29, pp , [8] A. R. Calderbank N. J. A. Sloane, Modular p-adic codes, Des., Codes Cryptogr., vol. 6, pp , [9] G. Castagnoli, J. L. Massey, P. A. Schoeller, N. von Seemann, On repeated-root cyclic codes, IEEE Trans. Inform. Theory, vol. 37, pp , Mar [10] J. H. Conway N. J. A. Sloane, Self-dual codes over the integers modulo 4, J. Combin. Theory Ser. A, vol. 62, pp , [11] A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, P. Solé, The -linearity of Kerdock, Preparata, Goethals, related codes, IEEE Trans. Inform. Theory, vol. 40, pp , Mar [12] I. N. Herstein, Topics in Algebra, 2nd ed. New York: Wiley, [13] W. C. Huffman V. Pless, Fundamentals of Error-Correcting Codes. Cambridge, U.K.: Cambridge Univ. Press, [14] P. Kanwar S. R. López-Permouth, Cyclic codes over the integers modulo p, Finite Fields Appl., vol. 3, pp , [15] F. J. MacWilliams N. J. A. Sloane, The Theory of Error-Eorrecting Eodes, 10th ed. Amsterdam, The Netherls: North-Holl, [16] B. R. McDonald, Finite rings with identity, in Pure Applied Mathematics. New York: Marcel Dekker, 1974, vol. 28. [17] P. Moree, On the divisors of a + b, Acta Arithm., vol. 80, pp , [18] A. A. Nechaev, Trace functions in Galois ring noise stable codes (in Russian), in V All-Union Symp. Theory of Rings, Algebras Modules, Novosibirsk, U.S.S.R., 1982, p. 97.

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