Self-Dual Abelian Codes in some Non-Principal Ideal Group Algebras

The main focus of this paper is the complete enumeration of self-dual abelian codes in non-principal ideal group algebras $\mathbb{F}_{2^k}[A\times \mathbb{Z}_2\times \mathbb{Z}_{2^s}]$ with respect to both the Euclidean and Hermitian inner products, where $k$ and $s$ are positive integers and $A$ is an abelian group of odd order. Based on the well-know characterization of Euclidean and Hermitian self-dual abelian codes, we show that such enumeration can be obtained in terms of a suitable product of the number of cyclic codes, the number of Euclidean self-dual cyclic codes, and the number of Hermitian self-dual cyclic codes of length $2^s$ over some Galois extensions of the ring $\mathbb{F}_{2^k}+u\mathbb{F}_{2^k}$, where $u^2=0$. Subsequently, general results on the characterization and enumeration of cyclic codes and self-dual codes of length $p^s$ over $\mathbb{F}_{p^k}+u\mathbb{F}_{p^k}$ are given. Combining these results, the complete enumeration of self-dual abelian codes in $\mathbb{F}_{2^k}[A\times \mathbb{Z}_2\times \mathbb{Z}_{2^s}]$ is therefore obtained.


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 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 [2][3][4][5][6]14,16,[21][22][23] and references therein). Some major results on Euclidean self-dual cyclic codes have been established in [18]. In [13], 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 [14] 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 [16]. To the best of our knowledge, the characterization and enumeration of self-dual abelian codes in non-principal ideal group algebras (non-PIGAs) have not been well studied. It is therefore of natural interest to focus on this open problem.
In [14] and [16], it has been shown that there exists a Euclidean (resp., Hermitian) self-dual abelian code in F p k [G] if and only if p = 2 and |G| is even. In order to study self-dual abelian codes, it is therefore restricted to the group algebra F 2 k [A×B], where A is an abelian group of odd order and B is a non-trivial abelian group of two power order. In this case, F 2 k [A × B] is a PIGA if and only if B = Z 2 s is a cyclic group (see [12]). Equivalently, F 2 k [A × B] is a non-PIGA if and only if B is non-cyclic. To avoid tedious computations, we focus on the simplest case where B = Z 2 × Z 2 s , 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 F 2 k [A × Z 2 × Z 2 s ].
It turns out that every Euclidean (resp., Hermitian) self-dual abelian code in F 2 k [A× Z 2 × Z 2 s ] is a suitable Cartesian product of cyclic codes, Euclidean self-dual cyclic codes, and Hermitian self-dual cyclic codes of length 2 s over some Galois extension of the ring F 2 k + uF 2 k , where u 2 = 0 (see Section 2). Hence, the number of self-dual abelian codes in F 2 k [A × Z 2 × Z 2 s ] 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 s over F p k + uF p k are given for all primes p. Combining these results, the characterizations and enumerations of Euclidean and Hermitian self-dual abelian codes in F 2 k [A × Z 2 × Z 2 s ] 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 F 2 k [A × Z 2 × Z 2 s ] and cyclic codes of length 2 s over Galois extensions of F 2 k + uF 2 k . General results on the characterization and enumeration of cyclic codes of length p s over F p k + uF p k are provided in Section 3. In Section 4, the characterizations and enumerations of Euclidean and Hermitian self-dual cyclic codes of length p s over F p k + uF p k are established. Summary and remarks are given in Section 5.

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

Rings and Abelian Codes in Group Rings
For a prime p and a positive integer k, denote by F p k the finite field of order p k . Let where the addition and multiplication are defined as in the usual polynomial ring over F p k with indeterminate u together with the condition u 2 = 0. We note that F p k + uF p k is isomorphic to F p k [u]/ u 2 as rings. The Galois extension of F p k + uF p k of degree m is defined to be the quotient ring is an irreducible polynomial of degree m over F p k . It is not difficult to see that the Galois extension of F p k + uF p k of degree m is isomorphic to F p km + uF p km as rings. In the case where k is even, the mapping a + ub → a p k/2 + ub p k/2 is a ring automorphism of order 2 on F p k + uF p k . The readers may refer to [7,8] for properties of the ring F p k + uF p k .
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 g∈G α g Y g , where α g ∈ 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 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 g to Y −g for all g ∈ G, i.e., for u = In the case where, there exists a ring automorphism ρ on R of order 2, we define u := can be viewed as an F p k -algebra and it is called a group algebra. The group algebra F p k [G] 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 The Euclidean inner product in R[G] is defined as follows. For In addition, if there exists a ring automorphism ρ of order 2 on R, the ρ-inner product of u and v is defined by If R = F q 2 (resp., R = F q 2 + uF q 2 ) and ρ(a) = a q (resp., ρ(a + ub) = a q + ub q ) for all a ∈ F q 2 (resp., a + ub ∈ F q 2 + uF q 2 ), the ρ-inner product is called the Hermitian inner product and denoted by u, v H .
The Euclidean dual and Hermitian dual of C in R[G] are defined to be the sets respectively.
An abelian code C is said to be Euclidean self-dual (resp., Hermitian self-dual) if C = C ⊥ E (resp., C = C ⊥ H ).
For convenience, denote by N(p k , n), NE(p k , n), and NH(p k , n) the number of cyclic codes, the number of Euclidean self-dual cyclic codes, and the number of Hermitian selfdual cyclic codes of length n over F p k + uF p k , respectively.

Decomposition of Abelian Codes in
In [14] and [16], it has been shown that there exists a Euclidean (resp., Hermitian) selfdual abelian code in F p k [G] if and only if p = 2 and |G| is even. To study self-dual abelian codes, it is sufficient to focus on F 2 k [A × B], where A is an abelian group of odd order and B is a non-trivial abelian group of two power order. In this case, is a PIGA if and only if B = Z 2 s is a cyclic group for some positive integer s (see [12]). The complete characterization and enumeration of self-dual abelian codes in PIGAs have been given in [14] and [16]. 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 B = Z 2 × Z 2 s . Useful decompositions of F 2 k [A × Z 2 × Z 2 s ] are given in this section.
First, we consider the decomposition of R := F 2 k [A]. In this case, R is semi-simple [2] which can be decomposed using the Discrete Fourier Transform in [22] (see [16] and [14] 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 k, let ord i (2 k ) denote the multiplicative order of 2 k modulo i. For each a ∈ A, denote by ord(a) the additive order of a in A. A 2 k -cyclotomic class of A containing a ∈ A, denoted by S 2 k (a), is defined to be the set An idempotent in R is a non-zero element e such that e 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 k -cyclotomic classes of A (see [6,Proposition II.4]). Let {a 1 , a 2 , . . . , a t } be a complete set of representatives of 2 k -cyclotomic classes of A and let e i be the primitive idempotent induced by S 2 k (a i ) for all 1 ≤ i ≤ t. From [22], R can be decomposed as and hence, It is well known (see [14,16]) that Under the ring isomorphism that fixes the elements in F 2 k i and ] is isomorphic to the ring F 2 k i + uF 2 k i , where u 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 [7,8,10,17] and references therein. From (2.2) and the ring isomorphism discussed above, we have In order to study the algebraic structures of Euclidean and Hermitian self-dual abelian codes in F 2 k [A × Z 2 × Z 2 s ], the two rearrangements of R i 's in the decomposition (2.3) are needed. The details are given in the following two subsections.

Euclidean Case
A 2 k -cyclotomic class S 2 k (a) is said to be of type I if a = −a (in this case, S 2 k (a) = S 2 k (−a)), type II if S 2 k (a) = S 2 k (−a) and a = −a, or type III if S 2 k (−a) = S 2 k (a). The primitive idempotent e induced by S 2 k (a) is said to be of type λ ∈ {I, II, III} if S 2 k (a) is a 2 k -cyclotomic class of type λ.
Without loss of generality, the representatives a 1 , a 2 , . . . , a t of 2 k -cyclotomic classes of A can be chosen such that {a i | i = 1, 2, . . . , r I }, {a r I +j | j = 1, 2, . . . , r II } and {a r I +r I I +l , a r I +r I I +r I I I +l = −a r I +r I I +l | l = 1, 2, . . . , r III } are sets of representatives of 2 kcyclotomic classes of types I, II, and III, respectively, where t = r I + r II + 2r III .
Using the analysis similar to those in [14, Section II.D], the Euclidean dual of C in (2.6) is of the form Similar to [14, Corollary 2.9], necessary and sufficient conditions for an abelian code in to be Euclidean self-dual can be given using the notions of cyclic codes of length 2 s over R i , S j , and T l in the following corollary.
ii) C j is a Hermitian self-dual cyclic code of length 2 s over S j for all j = 1, 2, . . . , r II , and is a cyclic code of length 2 s over T l for all l = 1, 2, . . . , r III .
Given a positive integer k and an odd positive integer j, the pair (j, 2 k ) is said to be good if j divides 2 kt + 1 for some integer t ≥ 1, and bad otherwise. This notions have been introduced in [13,14] for the enumeration of self-dual cyclic codes and self-dual abelian codes over finite fields.
Let χ be a function defined on the pair (j, 2 k ), where j is an odd positive integer, as follows.
The number of Euclidean self-dual abelian codes in F 2 k [A × Z 2 × Z 2 s ] can be determined as follows.
Theorem 2.2. 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 where N A (d) denotes the number of elements in A of order d determined in [1].
Proof. From (2.6) and Corollary 2.1, it suffices to determine the numbers of cyclic codes B i 's, C j 's, and D l 's such that B i and C j are Euclidean and Hermitian self-dual, respectively.
From [16,Remark 2.5], the elements in A of the same order are partitioned into 2 k -cyclotomic classes of the same type. For each divisor d of M, a 2 k -cyclotomic class containing an element of order d has cardinality ord d (2 k ) and the number of such 2 kcyclotomic classes is N A (d) ord d (2 k ) . We consider the following 3 cases. Case 1: χ(d, 2 k ) = 0 and ord d (2 k ) = 1. By [14, Remark 2.6], every 2 k -cyclotomic class of A containing an element of order d is of type I. Since there are N A (d) ord d (2 k ) such 2 k -cyclotomic classes, the number of Euclidean self-dual cyclic codes B i 's of length 2 s corresponding to d is Case 2: χ(d, 2 k ) = 0 and ord d (2 k ) = 1. By [14, Remark 2.6], every 2 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 j 's of length 2 s corresponding to d is Case 3: χ(d, 2 k ) = 1. By [14,Lemma 4.5], every 2 k -cyclotomic class of A containing an element of order d is of type III. Then the number of cyclic codes D l 's of length 2 s corresponding to d is 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 4.5, 4.8, and 3.13, respectively.

Hermitian Case
We focus on the case where k is even. A 2 k -cyclotomic class S 2 k (a) is said to be of type Without loss of generality, the representatives a 1 , a 2 , . . . , a t of 2 k -cyclotomic classes can be chosen such that {a i | i = 1, 2, . . . , r I ′ } and {a r I ′ +j , a r I ′ +r I I ′ +j = −2 k 2 a r I ′ +j | j = 1, 2, . . . , r II ′ } are sets of representatives of 2 k -cyclotomic classes of types I ′ and II ′ , respectively, where t = r I ′ + 2r II ′ .
Rearranging the terms in the decomposition of R in (2.1) based on the above 2 types primitive idempotents, we have where S j := F 2 k j + uF 2 k j for all j = 1, 2, . . . , r I ′ and T l := F 2 k r I ′ +l + uF 2 k r I ′ +l for all l = 1, 2, . . . , r II ′ .
Using the analysis similar to those in [16, Section II.D], the Hermitian dual of C in (2.10) is of the form Similar to [16, Corollary 2.8], necessary and sufficient conditions for an abelian code in F 2 k [A × Z 2 × Z 2 s ] to be Hermitian self-dual are now given using the notions of cyclic codes of length 2 s over S j and T l in the following corollary.
is Hermitian self-dual if and only if in the decomposition (2.10), i) C j is a Hermitian self-dual cyclic code of length 2 s over S j for all j = 1, 2, . . . , r I ′ , and is a cyclic code of length 2 s over T l for all l = 1, 2, . . . , r II ′ .
Given a positive integer k and an odd positive integer j, the pair (j, 2 k ) is said to be oddly good if j divides 2 kt + 1 for some odd integer t ≥ 1, and evenly good if j divides 2 kt + 1 for some even integer t ≥ 2. These notions have been introduced in [16] for characterizing the Hermitian self-dual abelian codes in PIGAs.
Let λ be a function defined on the pair (j, 2 k ), where j is an odd positive integer, as The number of Hermitian self-dual abelian codes in F 2 k [A × Z 2 × Z 2 s ] can be determined as follows.
Theorem 2.4. 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 where N A (d) denotes the number of elements of order d in A determined in [1].
Proof. By Corollary 2.3 and (2.10), it is enough to determine the numbers cyclic codes C j 's and D l 's of length 2 s in (2.10) such that C j is Hermitian self-dual. The desired result can be obtained using arguments similar to those in the proof of Theorem 2.2, where [16,Lemma 3.5] is applied instead of [14,Lemma 4.5].
This enumeration will be completed by counting the above numbers NH and N in Corollaries 4.8 and 3.13, respectively.
3 Cyclic Codes of Length p s over F p k + uF p k The enumeration of self-dual abelian codes in non-PIGAs in the previous section requires properties of cyclic codes of length 2 s over F 2 k + uF 2 k . In this section, a more general situation is discussed. Precisely, properties cyclic of length p s over F p k +uF p k are studied for all primes p. We note that algebraic structures of cyclic codes of length p s over F p k + uF p k was studied in [7] and [8]. Here, based on [20], we give an alternative characterization of such codes which is useful in studying self-dual cyclic codes of length p s over F p k + uF p k .
First, we note that there exists a one-to-one correspondence between the cyclic codes of length p s over F p k + uF p k and the ideals in the quotient ring (F p k + uF p k )[x]/ x p s − 1 . Precisely, a cyclic code C of length p s can be represented by the ideal Form now on, a cyclic code C will be referred to as the above polynomial presentation. Note that the map µ : is a surjective ring homomorphism. For each cyclic code C in (F p k + uF p k )[x]/ x p s − 1 and i ∈ {0, 1}, let For each i ∈ {0, 1}, Tor i (C) is called the ith torsion code of C. The codes Tor 0 (C) = µ(C) and Tor 1 (C) are some time called the residue and torsion codes of C, respectively.
It is not difficult to see that for each i ∈ {0, 1}, c(x) ∈ Tor i (C) if and only if From the structures of cyclic codes of length p s over F p k discussed above and [8, Proposition 2.5], we have the following properties of the torsion and residue codes.
Then the following statements hold.
Next, we determine a generator set of an ideal in (F p k + uF p k )[x]/ x p s − 1 .
where, for each i ∈ {0, 1}, Proof. The statement can be obtained using a slight modification of the proof of [11,Theorem 6.5]. For completeness, the details are given as follows.
Since C is an ideal of the ring (F p k + uF p k )[x]/ x p s − 1 and µ is a surjective ring homomorphism, µ(C) is an ideal of F p k [x]/ x p s − 1 which implies that µ(C) = s ′ 0 (x) where s ′ 0 (x) satisfies the conditions (i), (ii) and (iii). If s ′ 0 (x) = 0, then take s 0 (x) = 0. us 1 (x) . Therefore, C = s 0 (x), us 1 (x) as desired.
However, the generator set given in Theorem 3.3 does not need to be unique. The unique presentation is given in the following theorem.
Moreover, (f 0 (x), f 1 (x)) is unique in the sense that if there exists a pair (g 0 (x), g 1 (x)) of polynomials satisfying the conditions in the theorem, then f 0 (x) = g 0 (x) and f 1 (x) = g 1 (x).
To prove the uniqueness, let C = g 0 (x), g 1 (x) be such that g 0 (x) and g 1 (x) satisfying the conditions in the theorem. Then

It can be seen that
is a unit, then u(x − 1) l ∈ C which implies that l ≥ T 1 , a contradiction. Hence, h(x) = 0 which means that f 0 (x) = g 0 (x) as desired.
The annihilator of an ideal C in (F p k +uF p k )[x]/ x p s −1 is key to determine properties C as well as the number of ideals in (F p k + uF p k )[x]/ x p s − 1 .
The following properties of the annihilator can be obtained using arguments similar to those in the case of Galois rings in [20].
Theorem 3.8. Let C be an ideal of (F p k + uF p k )[x]/ x p s − 1 . Then the following statements hold.
(iii) Ann(Ann(C)) = C Theorem 3.9. Let I denote the set of ideals of (F p k + uF p k )[x]/ x p s − 1 and let A = {C ∈ I | T 0 (C) + T 1 (C) ≤ p s } and A ′ = {C ∈ I | T 0 (C) + T 1 (C) ≥ p s }. Then the map φ : A → A ′ defined by C → Ann(C) is a bijection.
The rest of this section is devoted to the determination of all ideals in (F p k + uF p k )[x]/ x p s − 1 . In view of Theorem 3.9, it suffices to focus on the ideals in A.
For each C = f 0 (x), f 1 (x) in A, if f 0 (x) = 0, then T 0 (C) = p s and T 1 (C) = 0. Hence, the only ideal in A with f 0 (x) = 0 is of the form 0, u . In the following two theorems, we assume that f 0 (x) = 0.
Assume that (x − 1) i 0 + u(x − 1) t h(x), u(x − 1) i 1 is a representation of an ideal in A. Then i 0 +i 1 ≤ p s which implies that i 1 ≤ p s −i 0 . Hence, we have i 1 ≤ min{i 0 , p s −i 0 }.
Proof. Let T 1 = i 1 and i 0 := T 0 = d − T 1 be fixed. Case 1 : d < p s . Then i 0 ≤ i 0 + i 1 = T 0 + T 1 = d < p s . By Theorem 3.11, it follows that Then the choices for . Now, vary T 1 from 0 to K, we obtain that there are 1 + p k + . . . + (p k ) K = p k(K+1) −1 Case 2 : d = p s . If i 0 = p s , then the only ideal with T 0 + T 1 = p s is the ideal represented by 0, u . If i 0 < p s , then we have p k + (p k ) 2 . . . + (p k ) K ideals by arguments similar to those in Case 1.
For a cyclic code C in A, we have C = Ann(C) whenever T 0 (C) + T 1 (C) < p s . In the case where T 0 (C) + T 1 (C) = p s , by the proof of Theorem 3.10, the annihilator of the cyclic code C = ( If p is odd, then C = Ann(C) occurs only the case h(x) = 0. In the case where p = 2, C = Ann(C) is always true. By Proposition 3.12 and the bijection given in Theorem 3.9, the number of cyclic codes of length p s over F p k + uF p k can be summarized as follows.
Corollary 3.13. The number of cyclic codes of length p s over F p k + uF p k is Proof. From Theorem 3.9, the number of cyclic codes of length p s over The desired results follow immediately form the discussion above.
4 Self-Dual Cyclic Codes of Length p s over F p k + uF p k In this section, characterization and enumeration self-dual cyclic codes of length p s over F p k + uF p k are given under the Euclidean and Hermitian inner products.

Euclidean Self-Dual Cyclic
Codes of Length p s over F p k + uF p k Characterization and enumeration of Euclidean self-dual cyclic codes of length p s over F p k + uF p k are given in this subsection. For each subset A of (F p k +uF p k )[x]/ x p s −1 , denote by A the set of polynomials f (x) for all f (x) in A, where¯is viewed as the conjugation on the group ring (F p k +uF p k )[Z p s ] defined in Section 2. From the definition of the annihilator, the next theorem can be derived similar to [9,Proposition 2.12].
Using the unique generators of an ideal C in (F p k + uF p k )[x]/ x p s − 1 determined in Theorem 3.11, the Euclidean dual of C can be given in the following theorem.
Proof. From the proof of Theorem 3.10, we have By Theorem 4.1, it follows that C ⊥ E = Ann(C). Hence, C ⊥ E contains the elements By writing x = (x−1) + 1 and using the Binomial Theorem, it follows that C ⊥ E contains the element Hence, By counting the number of elements, the two sets are equal as desired. Updating the indices, it can be concluded that We note that, if i 1 = 0, then it is not difficult to see that only the ideal generated by u is Euclidean self-dual.
For the case i 1 ≥ 1, the situation is more complicated. First, we recall an i 1 × i 1 matrix M(p s , i 1 ) over F p k defined in [19] as Hence, for a fixed first torsion degree 1 ≤ i 1 ≤ p s , a Euclidean self-dual ideal in (F p k + uF p k )[x]/ x p s − 1 always exists. By solving (4.4), all Euclidean self-dual ideals in (F p k + uF p k )[x]/ x p s − 1 can be constructed. Therefore, for a fixed first torsion degree 1 ≤ i 1 ≤ p s , the number of Euclidean self-dual ideals in (F p k + uF p k )[x]/ x p s − 1 equals the number of solutions of (4.4) which is p kκ , where κ is the nullity of M(p s , i 1 ) determined in [19].
The number of Euclidean self-dual cyclic codes in (F p k + uF p k )[x]/ x p s − 1 with first torsional degree i 1 is given in terms of the nullity of M(p s , i 1 ) as follows.
Proposition 4.4. Let i 1 > 0 and let κ be the nullity of M(p s , i 1 ) over F p k . Then the number of Euclidean self-dual cyclic codes of length p s over F p k with first torsional degree i 1 is (p k ) κ .
From Theorem 3.10, we have 0 ≤ i 1 ≤ ⌊ p s 2 ⌋ since i 0 + i 1 = p s . Hence, the number of Euclidean self-dual cyclic codes of length p s over F p k + uF p k is given by the following corollary. (i) If p is odd, then the number of Euclidean self-dual cyclic codes of length p s over if p s ≡ 1 mod 4.
(ii) If p = 2, then the number of Euclidean self-dual cyclic codes of length 2 s over Proof. From Propositions 4.3 and 4.4, the number of Euclidean self-dual cyclic codes of Apply a suitable geometric sum, the results follow.
4.2 Hermitian Self-Dual Cyclic Codes of Length p s over F p k + uF p k Under the assumption that k is even, characterization and enumeration Hermitian selfdual cyclic codes of length p s over F p k + uF p k are given in this section. For a subset A of (F p k + uF p k )[x]/ x p s − 1 , let where ρ(a + ub) = a p k 2 + ub p k 2 . Based on the structural characterization of C given in Theorem 3.11, the Hermitian dual of C is determined as follows.
Theorem 4.6. Let C be an ideal in A and where h j ∈ F p k . Then C ⊥ H has the representation Proof. From Theorem 4.2 and the fact that C ⊥ H = ρ(C ⊥ E ), the result follows.. Assume that C is Hermitian self-dual. Then C = C ⊥ H which implies that |C| = (p k ) p s and i 0 + i 1 = p s .
If i 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 s over F p k + uF p k .
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 s over F p k + uF p k can be determined in the following corollary.
Corollary 4.8. Let p be a prime and let s and k be positive integers such that k is even. Then the number of Hermitian self-dual cyclic codes of length p s over F p k + uF p k is NH(p k , p s ) = ⌊ p s 2 ⌋ i 1 =0 p ki 1 /2 = (p k/2 ) ⌊ p s 2 ⌋+1 − 1 p k/2 − 1 .

Conclusions and Remarks
Euclidean and Hermitian self-dual abelian codes in non-PIGAs F 2 k [A × Z 2 × Z 2 s ] are studied. The complete characterization and enumeration of such abelian codes are given and summarized as follows.
In Corollaries 2.1 and 2.3, self-dual abelian code in F 2 k [A × Z 2 × Z 2 s ] 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 s over some Galois extension of the ring F 2 k + uF 2 k . Subsequently, the characterizations and enumerations of cyclic and self-dual cyclic codes of length p s over F p k + uF p k are studied for all primes p. Combining these results, the following enumerations of Euclidean and Hermitian self-dual abelian codes in F 2 k [A × Z 2 × Z 2 s ] 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 F 2 k [A × Z 2 × Z 2 s ] is given in Theorem 2.2 in terms of the numbers N, NE, and NH of cyclic codes, Euclidean self-dual cyclic codes, and Hermitian self-dual cyclic codes of length 2 s over a Galois extension of F 2 k + uF 2 k , respectively.
In addition, if k is even, the number of Hermitian self-dual abelian codes in F 2 k [A × Z 2 × Z 2 s ] is given in Theorem 2.4 in terms of the numbers N and NH of cyclic codes and Hermitian self-dual cyclic codes of length 2 s over a Galois extension of F 2 k + uF 2 k , respectively. We note that all numbers N, NE, and NH are determined in Corollaries 3.13, 4.5, and 4.8, respectively. Therefore, the complete enumerations of Euclidean and Hermitian self-dual abelian codes in F 2 k [A × Z 2 × Z 2 s ] are established.
One of the interesting problems concerning the enumeration of self-dual abelian codes in F k 2 [A × B], where A is an abelian group of odd order, is the case where B is a 2-group of other types.