# Self-Dual Abelian Codes in Some Nonprincipal Ideal Group Algebras.

1. Introduction

Information media, such as communication systems and storage devices of data, are not 100 percent reliable in practice because of noise or other forms of introduced interference. The art of error correcting codes is a branch of Mathematics that has been introduced to deal with this problem since the 1960s. Linear codes with additional algebraic structures and self-dual codes are important families of codes that have been extensively studied for both theoretical and practical reasons (see [1-10] and references therein). Some major results on Euclidean self-dual cyclic codes have been established in [11]. In [12], the complete characterization and enumeration of such self-dual codes have been given. These results on Euclidean self-dual cyclic codes have been generalized to abelian codes in group algebras [6] and the complete characterization and enumeration of Euclidean self-dual abelian codes in principal ideal group algebras (PIGAs) have been established. Extensively, the characterization and enumeration of Hermitian self-dual abelian codes in PIGAs have been studied in [9]. To the best of our knowledge, the characterization and enumeration of self-dual abelian codes in nonprincipal ideal group algebras (non-PIGAs) have not been well studied. It is therefore of natural interest to focus on this open problem.

In [6,9], it has been shown that there exists a Euclidean (resp., Hermitian) self-dual abelian code in [mathematical expression not reproducible] if and only if p = 2 and [absolute value of G] is even. In order to study self-dual abelian codes, it is therefore restricted to the group algebra [mathematical expression not reproducible], where A is an abelian group of odd order and B is a nontrivial abelian group of two power order. In this case, [mathematical expression not reproducible] is a PIGA if and only if [mathematical expression not reproducible] is a cyclic group (see [13]). Equivalently, [mathematical expression not reproducible] is a non-PIGA if and only if B is noncyclic. To avoid tedious computations, we focus on the simplest case where [mathematical expression not reproducible], where s is a positive integer. Precisely, the goal of this paper is to determine the algebraic structures and the numbers of Euclidean and Hermitian self-dual abelian codes in [mathematical expression not reproducible].

It turns out that every Euclidean (resp., Hermitian) self-dual abelian code in [mathematical expression not reproducible] is isomorphic to a suitable Cartesian product of cyclic codes, Euclidean self-dual cyclic codes, and Hermitian self-dual cyclic codes of length [2.sup.s] over some Galois extension of the ring [mathematical expression not reproducible], where [u.sup.2] = 0 (see Section 2). Hence, the number of self-dual abelian codes in [mathematical expression not reproducible] can be determined in terms of the cyclic codes mentioned earlier. Subsequently, useful properties of cyclic codes, Euclidean self-dual cyclic codes, and Hermitian self-dual cyclic codes of length [p.sup.s] over [mathematical expression not reproducible] are given for all primes p. Combining these results, the characterizations and enumerations of Euclidean and Hermitian self-dual abelian codes in [mathematical expression not reproducible] are rewarded.

The paper is organized as follows. In Section 2, some basic results on abelian codes are recalled together with a link between abelian codes in [mathematical expression not reproducible] and cyclic codes of length [2.sup.s] over Galois extensions of [mathematical expression not reproducible]. General results on the characterization and enumeration of cyclic codes of length [p.sup.s] over [mathematical expression not reproducible] are provided in Section 3. In Section 4, the characterizations and enumerations of Euclidean and Hermitian self-dual cyclic codes of length [p.sup.s] over [mathematical expression not reproducible] are established. Summary and remarks are given in Section 5.

2. Preliminaries

In this section, we recall some definitions and basic properties of rings and abelian codes. Subsequently, a link between an abelian code in nonprincipal ideal algebras and a product of cyclic codes over rings is given. This link plays an important role in determining the algebraic structures and the numbers of Euclidean and Hermitian self-dual abelian codes in nonprincipal ideal algebras.

2.1. Rings and Abelian Codes in Group Rings. For a prime p and a positive integer k, denote by [mathematical expression not reproducible] the finite field of order [p.sup.k]. Let [mathematical expression not reproducible] be a ring, where the addition and multiplication are defined as in the usual polynomial ring over [mathematical expression not reproducible] with indeterminate u together with the condition [u.sup.2] = 0. We note that [mathematical expression not reproducible] is isomorphic to [mathematical expression not reproducible] as rings. The Galois extension of [mathematical expression not reproducible] of degree m is defined to be the quotient ring [mathematical expression not reproducible], where f(x) is an irreducible polynomial of degree m over [mathematical expression not reproducible]. It is not difficult to see that the Galois extension of [mathematical expression not reproducible] of degree m is isomorphic to [mathematical expression not reproducible] as rings. In the case where k is even, the mapping [mathematical expression not reproducible] is a ring automorphism of order 2 on [mathematical expression not reproducible]. The readers may refer to [14,15] for properties of the ring [mathematical expression not reproducible].

For a commutative ring R with identity 1 and a finite abelian group G, written additively, let R[G] denote the group ring of G over R. The elements in R[G] will be written as [[summation].sub.g[member of]G][[alpha].sub.g][Y.sup.g], where [[alpha].sub.g] [member of] R. The addition and the multiplication in R[G] are given as in the usual polynomial ring over R with indeterminate Y, where the indices are computed additively in G. Note that [Y.sup.0] := 1 is the multiplicative identity of R[G] (resp., R), where 0 is the identity of G. We define a conjugation - on R[G] to be the map that fixes R and sends [Y.sup.g] to [Y.sup.-g] for all g [member of] G; that is, for u = [[summation].sub.g[member of]G][[alpha].sub.g][Y.sup.g] [member of] R[G], we set [bar.u] := [[summation].sub.g[member of]G][[alpha].sub.g][Y.sup.-g] = [[summation].sub.g[member of]G][[alpha].sub.g][Y.sup.g] [member of] R[G]. In the case where there exists a ring automorphism [rho] on R of order 2, we define [??] := [[summation].sub.g[member of]G][rho]([[alpha].sub.g])[Y.sup.-g] for all u = [[summation].sub.g[member of]G][[alpha].sub.g][Y.sup.g] [member of] R[G]. In the case where R is a finite field [mathematical expression not reproducible], then [mathematical expression not reproducible] can be viewed as an [mathematical expression not reproducible]-algebra and it is called a group algebra. The group algebra [mathematical expression not reproducible] is called a principal ideal group algebra (PIGA) if its ideals are generated by a single element.

An abelian code in R[G] is defined to be an ideal in R[G]. If G is cyclic, this code is called a cyclic code, a code which is invariant under the right cyclic shift. It is well known that cyclic codes of length n over R can be regarded as ideals in the quotient polynomial ring R[x]/{[x.sup.n] - 1} [congruent to] R[[Z.sub.n]].

The Euclidean inner product in R[G] is defined as follows. For

[mathematical expression not reproducible] (1)

in R[G], we set

[<u, v>.sub.E] := [summation over (g[member of]G)][[alpha].sub.g][[beta].sub.g]. (2)

In addition, if there exists a ring automorphism [rho] of order 2 on R, the [rho]-inner product of u and v is defined by

[<u, v>.sub.[rho]] := [summation over (g[member of]G)][[alpha].sub.g][rho]([[beta].sub.g]). (3)

If [mathematical expression not reproducible] and [rho](a) = [a.sup.q] (resp., [rho](a + ub) = [a.sup.q] + [ub.sup.q]) for all a [member of] [mathematical expression not reproducible], the [rho]-inner product is called the Hermitian inner product and denoted by [<u, v>.sub.H].

The Euclidean dual and Hermitian dual of C in R[G] are defined to be the sets

[mathematical expression not reproducible], (4)

[mathematical expression not reproducible], (5)

respectively.

An abelian code C is said to be Euclidean self-dual (resp., Hermitian self-dual) if [mathematical expression not reproducible].

For convenience, denote by N([p.sup.k], n), NE([p.sup.k], n), and NH([p.sup.k], n) the number of cyclic codes, the number of Euclidean self-dual cyclic codes, and the number of Hermitian self-dual cyclic codes of length n over [mathematical expression not reproducible], respectively.

2.2. Decomposition of Abelian Codes in [mathematical expression not reproducible]. In [6,9], it has been shown that there exists a Euclidean (resp., Hermitian) self-dual abelian code in [mathematical expression not reproducible] if and only if p = 2 and [absolute value of G] is even. To study self-dual abelian codes, it is sufficient to focus on [mathematical expression not reproducible], where A is an abelian group of odd order and B is a nontrivial abelian group of two power order. In this case, [mathematical expression not reproducible] is a PIGA if and only if [mathematical expression not reproducible] is a cyclic group for some positive integer s (see [13]). The complete characterization and enumeration of self-dual abelian codes in PIGAs have been given in [6,9]. Here, we focus on self-dual abelian codes in non-PIGAs, or equivalently, B is non-cyclic. To avoid tedious computations, we establish results for the simplest case where [mathematical expression not reproducible]. Useful decompositions of [mathematical expression not reproducible] are given in this section.

First, we consider the decomposition of [mathematical expression not reproducible]. In this case, R is semisimple [2] which can be decomposed using the Discrete Fourier Transform in [7] (see [6,9] for more details). For completeness, the decompositions used in this paper are summarized as follows.

For an odd positive integer i and a positive integer fc, let [ord.sub.i]([2.sup.k]) denote the multiplicative order of [2.sup.k] modulo i. For each a [member of] A, denote by ord(a) the additive order of a in A. A [2.sup.k]-cyclotomic class of A containing a [member of] A, denoted by [mathematical expression not reproducible], is defined to be the set

[mathematical expression not reproducible], (6)

where [mathematical expression not reproducible] a in A.

An idempotent in R is a nonzero element e such that [e.sup.2] = e. It is called primitive if, for every other idempotent f, either ef = e or ef = 0. The primitive idempotents in R are induced by the [2.sup.k]-cyclotomic classes of A (see [5, Proposition II.4]). Let {[a.sub.1], [a.sub.2],...,[a.sub.t]} be a complete set of representatives of [2.sup.k]-cyclotomic classes of A and let [e.sub.i] be the primitive idempotent induced by [mathematical expression not reproducible] for all 1 [less than or equal to] i [less than or equal to] t. From [7], R can be decomposed as

R = [t.direct sum over (i=1)] [Re.sub.i], (7)

and hence,

[mathematical expression not reproducible]. (8)

It is well known (see [6,9]) that [mathematical expression not reproducible], where [k.sub.i] is a multiple of k. Precisely, [mathematical expression not reproducible] provided that [e.sub.i] is induced by [mathematical expression not reproducible]. It follows that [mathematical expression not reproducible]. Under the ring isomorphism that fixes the elements in [mathematical expression not reproducible] and [mathematical expression not reproducible] is isomorphic to the ring [mathematical expression not reproducible], where [u.sup.2] = 0. We note that this ring plays an important role in coding theory and codes over rings in this family have extensively been studied in [14-17] and references therein.

From (8) and the ring isomorphism discussed above, we have

[mathematical expression not reproducible], (9)

where [mathematical expression not reproducible] for all 1 [less than or equal to] i [less than or equal to] t.

In order to study the algebraic structures of Euclidean and Hermitian self-dual abelian codes in [mathematical expression not reproducible], the two rearrangements of [R.sub.i]'s in the decomposition (9) are needed. The details are given in the following two subsections.

2.2.1. Euclidean Case. A [2.sup.k]-cyclotomic class [mathematical expression not reproducible] is said to be of type I if a = -a (in this case, [mathematical expression not reproducible], type II if [mathematical expression not reproducible] and a [not equal to] -a, or type III if [mathematical expression not reproducible]. The primitive idempotent e induced by [mathematical expression not reproducible] is said to be of type [lambda] [member of] {I, II, III} if [mathematical expression not reproducible] is a [2.sup.k]-cyclotomic class of type [lambda].

Without loss of generality, the representatives [a.sub.1], [a.sub.2],..., [a.sub.t] of [2.sup.k]-cyclotomic classes of A can be chosen such that [mathematical expression not reproducible] and [mathematical expression not reproducible] are sets of representatives of [2.sup.k]-cyclotomic classes of types I, II, and III, respectively, where t = [r.sub.I] + [r.sub.II] + [2r.sub.III].

Rearranging the terms in the decomposition of R in (7) based on the 3 types of primitive idempotents, we have

[mathematical expression not reproducible], (10)

where [mathematical expression not reproducible] for all i = 1,2,...,[r.sub.I], [mathematical expression not reproducible] for all j = 1,2,...,[r.sub.II], and [mathematical expression not reproducible] for all l = 1,2,...,[r.sub.III].

From (10), we have

[mathematical expression not reproducible]. (11)

It follows that an abelian code C in [mathematical expression not reproducible] can be viewed as

[mathematical expression not reproducible], (12)

where [B.sub.i], [C.sub.j], [D.sub.l], and [D'.sub.l] are cyclic codes in [mathematical expression not reproducible], [mathematical expression not reproducible], and [mathematical expression not reproducible], respectively, for all i = 1,2,...,[r.sub.I], j = 1,2,...,[r.sub.II] and l = 1,2,...,[r.sub.III].

Using the analysis similar to those in [6, Section II.D], the Euclidean dual of C in (12) is of the form

[mathematical expression not reproducible]. (13)

Similar to [6, Corollary 2.9], necessary and sufficient conditions for an abelian code in [mathematical expression not reproducible] to be Euclidean self-dual can be given using the notions of cyclic codes of length [2.sup.s] over [R.sub.i], [S.sub.j], and [T.sub.l] in the following corollary.

Corollary 1. An abelian code C in [mathematical expression not reproducible] is Euclidean self-dual if and only if in the decomposition (12)

(i) [B.sub.i] is a Euclidean self-dual cyclic code of length [2.sup.s] over [R.sub.i] for all i = 1,2,...,[r.sub.I],

(ii) [C.sub.j] is a Hermitian self-dual cyclic code of length [2.sup.s] over [S.sub.j] for all i = 1,2,...,[r.sub.II],

(iii) [mathematical expression not reproducible] is a cyclic code of length [2.sup.s] over [T.sub.l] for all l = 1,2,...,[r.sub.III].

Given a positive integer k and an odd positive integer j, the pair (j, [2.sup.k]) is said to be good if j divides [2.sup.ki] + 1 for some integer i [greater than or equal to] 1 and bad otherwise. These notions have been introduced in [6,12] for the enumeration of self-dual cyclic codes and self-dual abelian codes over finite fields.

Let [chi] be a function defined on the pair (j, [2.sup.k]), where j is an odd positive integer, as follows:

[mathematical expression not reproducible]. (14)

The number of Euclidean self-dual abelian codes in [mathematical expression not reproducible] can be determined as follows.

Theorem 2. Let k and s be positive integers and let A be a finite abelian group of odd order and exponent M. Then the number of Euclidean self-dual abelian codes in [mathematical expression not reproducible] is

[mathematical expression not reproducible], (15)

where [N.sub.A](d) denotes the number of elements in A of order d determined in [18].

Proof. From (12) and Corollary 1, it suffices to determine the numbers of cyclic codes [B.sub.i]'s, [C.sub.j]'s, and [D.sub.l]'s such that [B.sub.i] and [C.sub.j] are Euclidean and Hermitian self-dual, respectively.

From [9, Remark 2.5], the elements in A of the same order are partitioned into [2.sup.k]-cyclotomic classes of the same type. For each divisor d of M, a [2.sup.k]-cyclotomic class containing an element of order d has cardinality [ord.sub.d]([2.sup.k]) and the number of such [2.sup.k]-cyclotomic classes is [N.sub.A](d)/[ord.sub.d]([2.sup.k]). We consider the following 3 cases.

Case 1 ([chi](d, [2.sup.k]) = 0 and [ord.sub.d]([2.sup.k]) = 1). By [6, Remark 2.6], every [2.sup.k]-cyclotomic class of A containing an element of order d is of type I. Since there are [N.sub.A](d)/[ord.sub.d]([2.sup.k]) such [2.sup.k] -cyclotomic classes, the number of Euclidean self-dual cyclic codes [B.sub.i]'s of length [2.sup.s] corresponding to d is

[mathematical expression not reproducible]. (16)

Case 2 ([chi](d, [2.sup.k]) = 0 and [ord.sub.d]([2.sup.k]) [not equal to] 1). By [6, Remark 2.6], every [2.sup.k]-cyclotomic class of A containing an element of order d is of type II. Hence, the number of Hermitian self-dual cyclic codes [C.sub.j]'s of length 2s corresponding to d is

[mathematical expression not reproducible]. (17)

Case 3 ([chi](d, [2.sup.k]) = 1). By [6, Lemma 4.5], every [2.sup.k]-cyclotomic class of A containing an element of order d is of type III. Then the number of cyclic codes [D.sub.l]'s of length [2.sup.s] corresponding to d is

[mathematical expression not reproducible]. (18)

The desired result follows since d runs over all divisors of M.

This enumeration will be completed by counting the above numbers NE, NH, and N in Corollaries 22, 25, and 17, respectively.

2.2.2. Hermitian Case. We focus on the case where k is even. A [2.sup.k]-cyclotomic class [mathematical expression not reproducible] is said to be of type I' if [mathematical expression not reproducible] or type II' if [mathematical expression not reproducible]. The primitive idempotent e induced by [mathematical expression not reproducible] is said to be of type [lambda] [member of] {I', II'} if [mathematical expression not reproducible] is a [2.sup.k]-cyclotomic class of type [lambda].

Without loss of generality, the representatives [a.sub.1],[a.sub.2],..., [a.sub.t] of [2.sup.k]-cyclotomic classes can be chosen such that {[a.sub.i] | i = 1,2,...,[r.sub.I']} and [mathematical expression not reproducible] are sets of representatives of [2.sup.k]-cyclotomic classes of types I' and II', respectively, where t = [r.sub.I'] + [2r.sub.II'].

Rearranging the terms in the decomposition of R in (7) based on the above 2 types of primitive idempotents, we have

[mathematical expression not reproducible], (19)

where [mathematical expression not reproducible] for all j = 1,2,...,[r.sub.I'] and [mathematical expression not reproducible] for all l = 1,2,...,[r.sub.II'].

From (19), we have

[mathematical expression not reproducible]. (20)

Hence, an abelian code C in [mathematical expression not reproducible] can be viewed as

[mathematical expression not reproducible], (21)

where [C.sub.j], [D.sub.l], and [D'.sub.l] are cyclic codes in [mathematical expression not reproducible], and [mathematical expression not reproducible], respectively, for all j = 1,2,...,[r.sub.I'] and l = 1,2,...,[r.sub.II'].

Using the analysis similar to those in [9, Section II.D], the Hermitian dual of C in (21) is of the form

[mathematical expression not reproducible]. (22)

Similar to [9, Corollary 2.8], necessary and sufficient conditions for an abelian code in [mathematical expression not reproducible] to be Hermitian self-dual are now given using the notions of cyclic codes of length [2.sup.s] over [S.sub.j] and [T.sub.l] in the following corollary.

Corollary 3. An abelian code C in [mathematical expression not reproducible] is Hermitian self-dual if and only if in the decomposition (21)

(i) [C.sub.j] is a Hermitian self-dual cyclic code of length [2.sup.s] over [S.sub.j] for all j = 1,2,...,[r.sub.I'],

(ii) [mathematical expression not reproducible] is a cyclic code of length [2.sup.s] over [T.sub.l] for all l = 1,2,...,[r.sub.II'].

Given a positive integer fc and an odd positive integer j, the pair (j, [2.sup.k]) is said to be oddly good if j divides [2.sup.ki] + 1 for some odd integer i [greater than or equal to] 1 and evenly good if j divides [2.sup.ki] + 1 for some even integer i [greater than or equal to] 2. These notions have been introduced in [9] for characterizing the Hermitian self-dual abelian codes in PIGAs.

Let [lambda] be a function defined on the pair (j, [2.sup.k]), where j is an odd positive integer, as

[mathematical expression not reproducible]. (23)

The number of Hermitian self-dual abelian codes in [mathematical expression not reproducible] can be determined as follows.

Theorem 4. Let k be an even positive integer and let s be a positive integer. Let A be a finite abelian group of odd order and exponent M. Then the number of Hermitian self-dual abelian codes in [mathematical expression not reproducible] is

[mathematical expression not reproducible], (24)

where [N.sub.A](d) denotes the number of elements of order d in A determined in [18].

Proof. By Corollary 3 and (21), it is enough to determine the numbers of cyclic codes [C.sub.j]'s and [D.sub.l]'s of length 2s in (21) such that [C.sub.j] is Hermitian self-dual.

The desired result can be obtained using arguments similar to those in the proof of Theorem 2, where [9, Lemma 3.5] is applied instead of [6, Lemma 4.5].

This enumeration will be completed by counting the above numbers NH and N in Corollaries 25 and 17, respectively.

3. Cyclic Codes of Length [p.sup.s] over [mathematical expression not reproducible]

The enumeration of self-dual abelian codes in non-PIGAs in the previous section requires properties of cyclic codes of length [2.sup.s] over [mathematical expression not reproducible]. In this section, a more general situation is discussed. Precisely, properties cyclic of length [p.sup.s] over [mathematical expression not reproducible] are studied for all primes p. We note that algebraic structures of cyclic codes of length [p.sup.s] over [mathematical expression not reproducible] were studied in [14,15]. Here, based on [19], we give an alternative characterization of such codes which is useful in studying self-dual cyclic codes of length [p.sup.s] over [mathematical expression not reproducible].

First, we note that there exists a one-to-one correspondence between the cyclic codes of length [p.sup.s] over [mathematical expression not reproducible] and the ideals in the quotient ring [mathematical expression not reproducible]. Precisely, a cyclic code C of length [p.sup.s] can be represented by the ideal

[mathematical expression not reproducible] (25)

in [mathematical expression not reproducible].

From now on, a cyclic code C will be referred to as the above polynomial presentation. Note that the map [mathematical expression not reproducible] defined by

[mu](f(x)) = f(x)(mod u) (26)

is a surjective ring homomorphism. For each cyclic code C in [mathematical expression not reproducible] and i [member of] {0,1}, let

[mathematical expression not reproducible]. (27)

For each i [member of] {0,1}, [Tor.sub.i](C) is called the ith torsion code of C. The codes [Tor.sub.0](C) = [mu](C) and [Tor.sub.1](C) are sometimes called the residue and torsion codes of C, respectively.

It is not difficult to see that, for each i [member of] {0,1}, c(x) [member of] [Tor.sub.i](C) if and only if [u.sup.i](c(x) + uz(x)) [member of] C for some z(x) [member of] [mathematical expression not reproducible]. Consequently, we have that [Tor.sub.0](C) [subset or equal to] [Tor.sub.1](C) are ideals in [mathematical expression not reproducible] (cyclic codes of length [p.sup.s] over [mathematical expression not reproducible]). We note that every ideal C in [mathematical expression not reproducible] is of the form <[(x - 1).sup.i]> for some 0 [less than or equal to] i [less than or equal to] [p.sup.s] and the cardinality of C is [p.sup.s-i].

From the structures of cyclic codes of length [p.sup.s] over [mathematical expression not reproducible] discussed above and [14, Proposition 2.5], we have the following properties of the torsion and residue codes.

Proposition 5. Let C be an ideal in [mathematical expression not reproducible] and let i [member of] {0,1}. Then the following statements hold:

(i) [Tor.sub.i](C) is an ideal of [mathematical expression not reproducible] and [mathematical expression not reproducible] for some 0 [less than or equal to] [T.sub.i] [less than or equal to] [p.sup.s].

(ii) If [mathematical expression not reproducible], then [mathematical expression not reproducible].

(iii) [mathematical expression not reproducible].

With the notations given in Proposition 5, for each i [member of] {0,1}, [T.sub.i](C) := [T.sub.i] is called the ith-torsional degree of C.

Remark 6. From Proposition 5 and the definition above, we have the following facts:

(i) Since [Tor.sub.0](C) [subset or equal to] [Tor.sub.1](C), we have 0 [less than or equal to] [T.sub.1](C) [less than or equal to] [T.sub.0](C) [less than or equal to] [p.sup.s].

(ii) If u[(x - 1).sup.t] [member of] C, then t [greater than or equal to] [T.sub.1](C).

Next, we determine a generator set of an ideal in [mathematical expression not reproducible].

Theorem 7. Let C be an ideal of [mathematical expression not reproducible]. Then

C = <[s.sub.0](x), [us.sub.1](x)>, (28)

where, for each i [member of] {0,1},

(i) either [mathematical expression not reproducible] for some [mathematical expression not reproducible] and 0 [less than or equal to] [t.sub.j] < [p.sup.s],

(ii) [s.sub.j](x) [not equal to] 0 if and only if [Tor.sub.j](C) [not equal to] {0} and [Tor.sub.0](C) [not equal to] [Tor.sub.1](C),

(iii) if [s.sub.j](x) [not equal to] 0, then [mathematical expression not reproducible].

Proof. The statement can be obtained using a slight modification of the proof of [20, Theorem 6.5]. For completeness, the details are given as follows.

For each ideal C in [mathematical expression not reproducible], it can be represented as C = {0} or [mathematical expression not reproducible], where 0 [less than or equal to] [t.sub.0] < [p.sup.s]. If C = {0}, then we are done by choosing [s.sub.0](x) = 0. For 0 [less than or equal to] [t.sub.0] < [p.sup.s], let [mathematical expression not reproducible]. By abuse of notation, [mathematical expression not reproducible]. Hence, [s.sub.0](x) satisfies conditions (i), (ii), and (iii).

Since C is an ideal of the ring [mathematical expression not reproducible] and [mu] is a surjective ring homomorphism, [mu](C) is an ideal of [mathematical expression not reproducible] which implies that [mu](C) = <[s'.sub.0](x)>, where [s'.sub.0](x) satisfies conditions (i), (ii), and (iii). If [s'.sub.0](x) = 0, then take [s.sub.0](x) = 0. Assume that [s'.sub.0](x) = 0; then [mathematical expression not reproducible], where 0 [less than or equal to] [t.sub.0] < [p.sup.s]. Then there exists [z.sub.0](x) [member of] [mathematical expression not reproducible] such that [mathematical expression not reproducible] [member of] C and [mu]([s.sub.0](x)) = [s'.sub.0](x); that is, [s.sub.0](x) = [s'.sub.0](x) + [uz.sub.0](x). Since [Tor.sub.1](C) is an ideal of [mathematical expression not reproducible], it follows that [mathematical expression not reproducible] for some 0 [less than or equal to] [t.sub.1] [less than or equal to] [p.sup.s]. Let [mathematical expression not reproducible]. Claim that C = <[s.sub.0](x), [us.sub.1](x)>. Since C is an ideal of [mathematical expression not reproducible], we have [mathematical expression not reproducible]. Thus, <[s.sub.0](x), [us.sub.1](x)> [subset or equal to] C. To show that C [subset or equal to] <[s.sub.0](x), [us.sub.1](x), let c(x) [member of] C. Then [mu](c(x)) = w(x)[s'.sub.0](x) for some w(x) [member of] [mathematical expression not reproducible]. Thus, [mu](c(x)) = w(x)[s'.sub.0](x) which implies that c(x) = w(x)[s'.sub.0](x) + uw'(x) = w(x)[s.sub.0](x) - uw(x)[z.sub.0](x) + uw'(x) = w(x)[s.sub.0](x) + u(-[z.sub.0](x)w(x) + w'(x)) for some w'(x) [member of] [Tor.sub.1](C). Since w'(x) [member of] [Tor.sub.1] (C), it follows that c(x) [member of] <[s.sub.0](x), [us.sub.1](x)>. Therefore, C = <[s.sub.0](x), [us.sub.1](x)> as desired.

Note that [mathematical expression not reproducible] implies C = {0}. Assume that C [not equal to] {0}. Then [mathematical expression not reproducible], where 0 [less than or equal to] [t.sub.1] < [p.sup.s]. If [s.sub.0](x) = 0, then we are done. Assume that [s.sub.0](x) [not equal to] 0. Then [s.sub.0](x) = [(x - 1).sup.t], where 0 [less than or equal to] t < [p.sup.s]. Hence, [Tor.sub.0](C) = <[(x - 1).sup.t]>. Since [Tor.sub.0](C) [subset or equal to] [Tor.sub.1](C), we have [t.sub.1] [less than or equal to] t. If [t.sub.1] < t, then we are done. Assume that [t.sub.1] = t. Then [s.sub.0](x) = [(x - 1).sup.t] + [uz.sub.0](x) for some [mathematical expression not reproducible]. It follows that [mathematical expression not reproducible], a contradiction. Therefore, C = <[s.sub.0](x)? = <[s.sub.0](x), [us.sub.0](x)> = <[s.sub.0](x), [us.sub.1](x)>.

However, the generator set given in Theorem 7 does not need to be unique. The unique presentation is given in the following theorem.

Theorem 8. Let C be an ideal in [mathematical expression not reproducible], [T.sub.0] := [T.sub.0](C), and [T.sub.1] := [T.sub.1](C). Then

C = <[f.sub.0](x), [f.sub.1](x)>, (29)

where

[mathematical expression not reproducible], (30)

with [mathematical expression not reproducible] being either zero or a unit with t + deg(h(x)) < [T.sub.0].

Moreover, ([f.sub.0](x), [f.sub.1](x)) is unique in the sense that if there exists a pair ([g.sub.0](x), [g.sub.1](x)) of polynomials satisfying the conditions in the theorem, then [f.sub.0](x) = [g.sub.0](x) and [f.sub.1](x) = [g.sub.1](x).

Proof. If C = {0}, then [Tor.sub.1](C) = {0} and [Tor.sub.0](C) = {0} which imply that [T.sub.0] = [p.sup.s] and [T.sub.1] = [p.sup.s]. The polynomials [f.sub.0](x) = 0 and [f.sub.1](x) = 0 have the desired properties.

Next, assume that C [not equal to] {0}. Then there exists the smallest nonnegative integer r [member of] {0,1} such that [T.sub.r] < [p.sup.s]. From Theorem 7, it can be concluded that

C = <[s.sub.0](x), [us.sub.1](x)), (31)

where

[mathematical expression not reproducible], (32)

for some [mathematical expression not reproducible].

Case 1 (r = 0). Then [mathematical expression not reproducible] for some [mathematical expression not reproducible] and [mathematical expression not reproducible]. It follows that [mathematical expression not reproducible]. Let [f.sub.1](x) = [us.sub.1](x). Since [mathematical expression not reproducible], we have

[mathematical expression not reproducible], (33)

where [a.sub.j] [member of] [mathematical expression not reproducible] for all j = 0,1,...,[p.sup.k-1]. Since [mathematical expression not reproducible], we have [ua.sub.j][(x - 1).sup.j] [member of] C for all j = [T.sub.1], [T.sub.1] + 1,...,[p.sup.s] - 1. It follows that [mathematical expression not reproducible]. Let [mathematical expression not reproducible]. Then [mathematical expression not reproducible].

We show that C = <[f.sub.0](x), [f.sub.1](x)>. From the discussion above, we have <[f.sub.0](x), [f.sub.1](x)> [subset or equal to] C. Since [mathematical expression not reproducible], it follows that [ua.sub.j][(x - 1).sup.j] [member of] <[f.sub.0](x), [f.sub.1](x)> for all j = [T.sub.1],[T.sub.1+1],...,[p.sup.s] - 1. Hence, [mathematical expression not reproducible] which implies that

[mathematical expression not reproducible]. (34)

Therefore, C = <[s.sub.0](x), [us.sub.1](x)> [subset or equal to] <[f.sub.0](x), [f.sub.1](x)>. As desired, C = <[f.sub.0](x), [f.sub.1](x)>.

Case 2 (r = 1). Then [s.sub.0](x) = 0 and [mathematical expression not reproducible] which implies that [mathematical expression not reproducible] and [Tor.sub.0](C) = {0}. By choosing [f.sub.0](x) = 0 and [mathematical expression not reproducible], the result follows.

To prove the uniqueness, let C = <[g.sub.0], [g.sub.1](x))> be such that [g.sub.0](x) and [g.sub.1](x) satisfy the conditions in the theorem. Then [mathematical expression not reproducible].

Write [mathematical expression not reproducible], where [c.sub.j] [member of] [mathematical expression not reproducible].

Then

[mathematical expression not reproducible]. (35)

It can be seen that [f.sub.0](x) - [g.sub.0](x) = u[(x - 1).sup.l] h(x), where h(x) = 0 or h(x) is a unit with l [less than or equal to] [T.sub.1] -1 < [T.sub.1]. If is a unit, then u[(x - 1).sup.l] [member of] C which implies that l [greater than or equal to] [T.sub.1], a contradiction. Hence, h(x) = 0 which means that [f.sub.0](x) = [g.sub.0](x) as desired.

Definition 9. For each ideal C in [mathematical expression not reproducible], denote by C = <<[f.sub.0](x), [f.sub.1](x)>> the unique representation of the ideal C obtained in Theorem 8.

Illustrative examples of the representations in Theorems 7 and 8 are given as follows.

Example 10. Consider the ideal C = <[(x - 1).sup.2]> in ([F.sub.2] + [uF.sub.2]) [x]/<[x.sup.4] - 1>. Using Theorem 8, we obtain that C has the unique representation <<[(x - 1).sup.2], u[(x - 1).sup.2]>>. Based on Theorem 7, C can be represented as <[(x - 1).sup.2], 0), <[(x - 1).sup.2] + u[(x - 1).sup.2], 0), and <[(x - 1).sup.2] + u[(x - 1).sup.3], 0>.

The annihilator of an ideal C in [mathematical expression not reproducible] is key to determine properties C as well as the number of ideals in [mathematical expression not reproducible].

Definition 11. Let C be an ideal in [mathematical expression not reproducible]. The annihilator of C, denoted by Ann(C), is defined to be the set [mathematical expression not reproducible].

The following properties of the annihilator can be obtained using arguments similar to those in the case of Galois rings in [19].

Theorem 12. Let C be an ideal of [mathematical expression not reproducible]. Then the following statements hold:

(i) Ann(C) is an ideal of [mathematical expression not reproducible].

(ii) If [absolute value of C] = [([p.sup.k]).sup.d], then [mathematical expression not reproducible].

(iii) Ann(Ann(C)) = C.

Theorem 13. Let J denote the set of ideals of [mathematical expression not reproducible] and let A ={C [member of] J | [T.sub.0](C) + [T.sub.1](C) [less than or equal to] [p.sup.s]} and A' = {C [member of] J | [T.sub.0](C) + [T.sub.1](C) [greater than or equal to] [p.sup.s]}. Then the map [phi] : A [right arrow] A' defined by C [??] Ann(C) is a bijection.

The rest of this section is devoted to the determination of all ideals in [mathematical expression not reproducible]. In view of Theorem 13, it suffices to focus on the ideals in A.

For each C = <<[f.sub.0](x), [f.sub.1](x)>> in A, if [f.sub.0](x) = 0, then [T.sub.0](C) = [p.sup.s] and [T.sub.1](C) = 0. Hence, the only ideal in A with [f.sub.0](x) = 0 is of the form <<0, u>>. In the following two theorems, we assume that [f.sub.0](x) [not equal to] 0.

Theorem 14. Let [mathematical expression not reproducible] be the representation of an ideal in [mathematical expression not reproducible]. Then it is a representation of an ideal in A if and only if [i.sub.0], [i.sub.1], and t are integers and h(x) [member of] [mathematical expression not reproducible] such that 0 [less than or equal to] [i.sub.0] < [p.sup.s], 0 [less than or equal to] [i.sub.1] [less than or equal to] min{[i.sub.0], [p.sup.s] - [i.sub.0]}, t [greater than or equal to] 0, t + deg(h(x)) < [i.sub.1], and h(x) is either zero or a unit in [mathematical expression not reproducible].

Proof. Form Theorem 8, we have that [i.sub.0], [i.sub.1], and t are integers and h(x) [member of] [mathematical expression not reproducible] such that 0 [less than or equal to] [i.sub.0] < [p.sup.s], 0 [less than or equal to] [i.sub.1] [less than or equal to] [i.sub.0], t [greater than or equal to] 0, t + deg(h(x)) < [i.sub.1], and h(x) is either zero or a unit in [mathematical expression not reproducible].

Assume that [mathematical expression not reproducible] is a representation of an ideal in A. Then [i.sub.0] + [i.sub.1] [less than or equal to] [p.sup.s] which implies that [i.sub.1] [less than or equal to] [p.sup.s] - [i.sup.0]. Hence, we have [i.sub.1] [less than or equal to] min{[i.sub.0], [p.sup.s] - [i.sub.0]}.

Conversely, assume that [mathematical expression not reproducible]), where [i.sub.0], [i.sub.1], and t are integers and [mathematical expression not reproducible] such that 0 [less than or equal to] [i.sub.0] < [p.sup.s], 0 [less than or equal to] [i.sub.1] < min{[i.sub.0], [p.sup.s] - [i.sub.0]}, t [greater than or equal to] 0, t + deg(^(x)) < [i.sub.1], and h(x) is either zero or a unit in [mathematical expression not reproducible]. Clearly, [i.sub.0] + [i.sub.1] [less than or equal to] [p.sup.s]. To show that [mathematical expression not reproducible] in A, it remains to prove that [T.sub.0](C) = [i.sub.0] and [T.sub.1](C) = [i.sub.1].

Let [mathematical expression not reproducible]. It is not difficult to see that

[mathematical expression not reproducible]. (36)

Since

[mathematical expression not reproducible], (37)

we have D [subset or equal to] Ann(C). By Proposition 5, we obtain that [T.sub.j](C) [less than or equal to] [i.sub.j] and [mathematical expression not reproducible] for all j [member of] {0,1}. Hence, [mathematical expression not reproducible] and [mathematical expression not reproducible]. Since [mathematical expression not reproducible] we have Ann(C) = D. Therefore, [T.sub.0](C) = [i.sub.0] and [T.sub.1](C) = [i.sub.1] as desired.

Since every polynomial [[summation].sup.m.sub.i=0] [a.sub.i][(x - 1).sup.i] in [mathematical expression not reproducible] is either 0 or [(x - 1).sup.t] h(x), where h(x) is a unit in [mathematical expression not reproducible] and 0 [less than or equal to] t [less than or equal to] m - deg(h(x)), Theorem 14 is rewritten as follows.

Theorem 15. The expression [mathematical expression not reproducible] represents an ideal in A if and only if [i.sub.0] and [i.sub.1] are integers such that 0 [less than or equal to] [i.sub.0] < [p.sup.s], 0 [less than or equal to] [i.sub.1] [less than or equal to] min{[i.sub.0], [p.sup.s] - [i.sub.0]}, [i.sub.0] + [i.sub.1] [less than or equal to] [p.sup.s], and [h.sub.j] [member of] [mathematical expression not reproducible] for all 0 [less than or equal to] j < [i.sub.1].

The number of distinct ideals of [mathematical expression not reproducible] of a fixed d = [T.sub.0] + [T.sub.1] is given in the following proposition.

Proposition 16. Let 0 [less than or equal to] d [less than or equal to] [p.sup.s]. Then the number of distinct ideals in [mathematical expression not reproducible] with [T.sub.0] + [T.sub.1] = dis

[p.sup.k(K+1)] - 1/[p.sup.k] - 1, (38)

where K = min{|d/2], [p.sup.s] - [d/2]}.

Proof. Let [T.sub.1] = [i.sub.1] and [i.sub.0] := [T.sub.0] = d - [T.sub.1] be fixed.

Case 1 (d < [p.sup.s]). Then [i.sub.0] [less than or equal to] [i.sub.0] + [i.sub.1] = [T.sub.0] + [T.sub.1] = d < [p.sup.s]. By Theorem 15, it follows that [mathematical expression not reproducible]. Then the choice for [mathematical expression not reproducible]. By Theorem 15 again, we also have [T.sub.1] [less than or equal to] min{[T.sub.0], [p.sup.s] - [T.sub.0]}. Since [T.sub.0] + [T.sub.1] = d, we obtain that [T.sub.1] [less than or equal to] [d/2] [less than or equal to] [T.sub.0], and hence, [T.sub.1] [less than or equal to] min{[d/2], [p.sup.s] - [T.sub.0]} [less than or equal to] min{[d/2], [p.sup.s] -[d/2]}. Now, vary [T.sub.1] from 0 to K; we obtain that there are 1 + [p.sup.k] + ... + [([p.sup.k]).sup.K] = ([p.sup.k(K+1)] - 1)/([p.sup.k] - 1) ideals with [T.sub.0] + [T.sub.1] = d.

Case 2 (d = [p.sup.s]). If [i.sub.0] = [p.sup.s], then the only ideal with [T.sub.0] + [T.sub.1] = [p.sup.s] is the ideal represented by <<0, u)). If [i.sub.0] < [p.sup.s], then we have [p.sup.k] + [([p.sup.k]).sup.2] + ... + [([p.sup.k]).sup.K] ideals by arguments similar to those in Case 1.

For a cyclic code C in A, we have C [not equal to] Ann(C) whenever [T.sub.0](C) + [T.sub.1](C) < [p.sup.s]. In the case where [T.sub.0](C) + [T.sub.1](C) = [p.sup.s], by the proof of Theorem 14, the annihilator of the cyclic code [mathematical expression not reproducible] is of the form [mathematical expression not reproducible]. If p is odd, then C = Ann(C) occurs only in the case h(x) = 0. In the case where p = 2, C = Ann(C) is always true. By Proposition 16 and the bijection given in Theorem 13, the number of cyclic codes of length [p.sup.s] over [mathematical expression not reproducible] can be summarized as follows.

Corollary 17. The number of cyclic codes of length [p.sup.s] over [mathematical expression not reproducible] is

[mathematical expression not reproducible]. (39)

Proof. From Theorem 13, the number of cyclic codes of length [p.sup.s] over [mathematical expression not reproducible] is [absolute value of A [union]A'] = [absolute value of A] + [absolute value of A'] - [absolute value of A [union] A']. The desired results follow immediately form the discussion above. ?

4. Self-Dual Cyclic Codes of Length [p.sup.s] over [mathematical expression not reproducible]

In this section, characterization and enumeration self-dual cyclic codes of length [p.sup.s] over [mathematical expression not reproducible] are given under the Euclidean and Hermitian inner products.

4.1. Euclidean Self-Dual Cyclic Codes of Length [p.sup.s] over [mathematical expression not reproducible]. Characterization and enumeration of Euclidean self-dual cyclic codes of length [p.sup.s] over [mathematical expression not reproducible] are given in this subsection.

For each subset A of [mathematical expression not reproducible], denote by [bar.A] the set of polynomials [bar.f(x)] for all f(x) in A, where - is viewed as the conjugation on the group ring [mathematical expression not reproducible] defined in Section 2. From the definition of the annihilator, the next theorem can be derived similar to [21, Proposition 2.12].

Theorem 18. Let C be an ideal in [mathematical expression not reproducible]. Then [mathematical expression not reproducible].

Using the unique generators of an ideal C in [mathematical expression not reproducible] determined in Theorem 15, the Euclidean dual of C can be given in the following theorem.

Theorem 19. Let [mathematical expression not reproducible] be an ideal in A, where [h.sub.j] [member of] [mathematical expression not reproducible] for all 0 [less than or equal to] j [less than or equal to] [i.sub.i] - 1. Then [mathematical expression not reproducible] is in the form of

[mathematical expression not reproducible]. (40)

Proof. From the proof of Theorem 14, we have

[mathematical expression not reproducible]. (41)

By Theorem 18, it follows that [mathematical expression not reproducible]. Hence, [mathematical expression not reproducible] contains the elements [mathematical expression not reproducible] and

[mathematical expression not reproducible]. (42)

By writing x = (x - 1) + 1 and using the Binomial Theorem, it follows that [mathematical expression not reproducible] contains the element

[mathematical expression not reproducible]. (43)

Hence,

[mathematical expression not reproducible]. (44)

By counting the number of elements, the two sets are equal as desired. Updating the indices, it can be concluded that

[mathematical expression not reproducible]. (45)

Assume that an ideal [mathematical expression not reproducible] in [mathematical expression not reproducible] is Euclidean self-dual; that is, [mathematical expression not reproducible]. By Theorem 18, it is equivalent to [p.sup.s] = [i.sub.0] + [i.sub.1] and

[mathematical expression not reproducible] (46)

in [mathematical expression not reproducible] for all 0 [less than or equal to] t [less than or equal to] [i.sub.i] - 1.

We note that if [i.sub.1] = 0, then it is not difficult to see that only the ideal generated by u is Euclidean self-dual.

For the case [i.sub.1] [greater than or equal to] 1, the situation is more complicated. First, we recall an [i.sub.1] x [i.sub.1] matrix M([p.sup.s], [i.sub.1]) over [mathematical expression not reproducible] defined in [22] as

[mathematical expression not reproducible]. (47)

It is not difficult to see that [i.sub.1] equations from (46) are equivalent to the matrix equation

M([p.sup.s], [i.sup.1]) h = 0, (48)

where [mathematical expression not reproducible] and 0 = [(0,0,...,0).sup.T].

Moreover, it can be concluded that the ideal C is Euclidean self-dual if and only if [p.sup.s] = [i.sub.0] + [i.sub.1] and [mathematical expression not reproducible] satisfy (48). Since [mathematical expression not reproducible] is a solution of (48), the corresponding idea [mathematical expression not reproducible] is Euclidean self-dual in [mathematical expression not reproducible]. Hence, for a fixed first torsion degree 1 [less than or equal to] [i.sub.1] [less than or equal to] [p.sup.s], a Euclidean self-dual ideal in [mathematical expression not reproducible] always exists. By solving (48), all Euclidean self-dual ideals in [mathematical expression not reproducible] can be constructed. Therefore, for a fixed first torsion degree 1 [less than or equal to] [i.sub.1] [less than or equal to] [p.sup.s], the number of Euclidean self-dual ideals in [mathematical expression not reproducible] equals the number of solutions of (48) which is [p.sup.k[kappa]], where [kappa] is the nullity of M([p.sup.s], [i.sup.1]) determined in [22].

Proposition 20 (see [22, Proposition 3.3]). Let [kappa] be the nullity of M([p.sup.s], [i.sub.1]). Then

[mathematical expression not reproducible]. (49)

The number of Euclidean self-dual cyclic codes in [mathematical expression not reproducible] with first torsional degree [i.sub.1] is given in terms of the nullity of M([p.sup.s], [i.sub.1]) as follows.

Proposition 21. Let [i.sub.1] > 0 and let k be the nullity of M([p.sup.s], [i.sub.1]) over [mathematical expression not reproducible]. Then the number of Euclidean self-dual cyclic codes of length [p.sup.s] over [mathematical expression not reproducible] with first torsional degree [i.sub.1] is [([p.sup.k]).sup.[kappa]].

From Theorem 14, we have 0 [less than or equal to] [i.sub.1] [less than or equal to] [[p.sup.s]/2] since [i.sub.0] + [i.sub.1] = [p.sup.s]. Hence, the number of Euclidean self-dual cyclic codes of length [p.sup.s] over [mathematical expression not reproducible] is given by the following corollary.

Corollary 22. Let p be a prime and let s and k be positive integers. Then the following statements hold:

(i) If p is odd, then the number of Euclidean self-dual cyclic codes of length [p.sup.s] over [mathematical expression not reproducible] is

[mathematical expression not reproducible]. (50)

(ii) If p = 2, then the number of Euclidean self-dual cyclic codes of length [2.sup.s] over [mathematical expression not reproducible] is

[mathematical expression not reproducible]. (51)

Proof. From Propositions 20 and 21, the number of Euclidean self-dual cyclic codes of length [2.sup.s] over [mathematical expression not reproducible] is [mathematical expression not reproducible]. Apply a suitable geometric sum; the results follow.

4.2. Hermitian Self-Dual Cyclic Codes of Length [p.sup.s] over [mathematical expression not reproducible]. Under the assumption that k is even, characterization and enumeration of Hermitian self-dual cyclic codes of length [p.sub.s] over [mathematical expression not reproducible] are given in this section.

For a subset A of [mathematical expression not reproducible], let

[mathematical expression not reproducible], (52)

where [mathematical expression not reproducible].

Based on the structural characterization of C given in Theorem 15, the Hermitian dual of C is determined as follows.

Theorem 23. Let C be an ideal in A and

[mathematical expression not reproducible], (53)

where [h.sub.j] [member of] [mathematical expression not reproducible]. Then [mathematical expression not reproducible] has the representation

[mathematical expression not reproducible]. (54)

Proof. From Theorem 19 and the fact that [mathematical expression not reproducible], the result follows.

Assume that C is Hermitian self-dual. Then [mathematical expression not reproducible] which implies that [mathematical expression not reproducible] and [i.sub.0] + [i.sub.1] = [p.sup.s].

If [i.sub.1] = 0, then it is not difficult to see that the ideal generated by u is only Hermitian self-dual cyclic code of length [p.sup.s] over [mathematical expression not reproducible].

Assume that [i.sub.1] [greater than or equal to] 1. Then

[mathematical expression not reproducible] (55)

for all 0 [less than or equal to] t [less than or equal to] [i.sub.1] - 1.

From [i.sub.i] equations above and the definition of M([p.sup.s], [i.sub.1]), we have

[mathematical expression not reproducible], (56)

where [mathematical expression not reproducible], and 0 = (0,0,...,0).

From Theorem 14, we have 0 [less than or equal to] [i.sub.1] [less than or equal to] [[p.sup.s]/2] since [i.sub.0] + [i.sub.1] = [p.sup.s].

Proposition 24. Let k be an even positive integer and let [i.sub.1] be a positive integer such that [i.sub.1] [less than or equal to] [[p.sup.k]/2]. Then the number of solutions of (56) in [mathematical expression not reproducible] is [mathematical expression not reproducible].

Proof. Let [mathematical expression not reproducible] defined by [mathematical expression not reproducible] for all [alpha] [member of] [mathematical expression not reproducible]. Using the fact that [PSI](1) = 0 = [PSI](0) and arguments similar to those in [23, Proposition 3.3], the result follows.

For a prime number p, a positive integer s, and an even positive integer k, the number of Hermitian self-dual cyclic codes of length [p.sup.s] over [mathematical expression not reproducible] can be determined in the following corollary.

Corollary 25. Let p be a prime and let s and k be positive integers such that fc is even. Then the number of Hermitian self-dual cyclic codes of length [p.sup.s] over [mathematical expression not reproducible] is

[mathematical expression not reproducible]. (57)

5. Conclusions and Remarks

Euclidean and Hermitian self-dual abelian codes in non-PIGAs [mathematical expression not reproducible] are studied. The complete characterization and enumeration of such abelian codes are given and summarized as follows.

In Corollaries 1 and 3, self-dual abelian codes in [mathematical expression not reproducible] are shown to be a suitable Cartesian product of cyclic codes, Euclidean self-dual cyclic codes, and Hermitian self-dual cyclic codes of length [2.sup.s] over some Galois extension of the ring [mathematical expression not reproducible]. Subsequently, the characterizations and enumerations of cyclic and self-dual cyclic codes of length [p.sup.s] over [mathematical expression not reproducible] are studied for all primes p. Combining these results, the following enumerations of Euclidean and Hermitian self-dual abelian codes in [mathematical expression not reproducible] are rewarded.

For each abelian group A of odd order and positive integers s and k, the number of Euclidean self-dual abelian codes in [mathematical expression not reproducible] is given in Theorem 2 in terms of the numbers N, and NH of cyclic codes, Euclidean self-dual cyclic codes, and Hermitian self-dual cyclic codes of length [2.sup.s] over a Galois extension of [mathematical expression not reproducible], respectively.

In addition, if fc is even, the number of Hermitian self-dual abelian codes in [mathematical expression not reproducible] is given in Theorem 4 in terms of the numbers N and NH of cyclic codes and Hermitian self-dual cyclic codes of length [2.sup.s] over a Galois extension of [mathematical expression not reproducible], respectively.

We note that all numbers N, and NH are determined in Corollaries 17, 22, and 25, respectively. Therefore, the complete enumerations of Euclidean and Hermitian self-dual abelian codes in [mathematical expression not reproducible] are established.

One of the interesting problems concerning the enumeration of self-dual abelian codes in [F.sup.k.sub.2][A x B], where A is an abelian group of odd order, is the case where B is a 2-group of other types.

http://dx.doi.org/10.1155/2016/9020173

Competing Interests

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

Acknowledgments

This work is supported by the Thailand Research Fund under Research Grant TRG5780065 and the DPST Research Grant 005/2557.

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Parinyawat Choosuwan, (1) Somphong Jitman, (2) and Patanee Udomkavanich (1)

(1) Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand

(2) Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000, Thailand

Correspondence should be addressed to Somphong Jitman; sjitman@gmail.com

Received 25 June 2016; Accepted 27 September 2016

Academic Editor: Kishin Sadarangani
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