ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.784982 J. Algebra Comb. Discrete Appl. 7(3) • 209–227 Received: 7 September 2019 Accepted: 6 May 2020 Journal of Algebra Combinatorics Discrete Structures and Applications Self-dual codes over Fq + uFq + u2Fq and applications∗ Research Article Parinyawat Choosuwan, Somphong Jitman Abstract: Self-dual codes over finite fields and over some finite rings have been of interest and extensively studied due to their nice algebraic structures and wide applications. Recently, characterization and enumeration of Euclidean self-dual linear codes over the ring Fq + uFq + u2Fq with u3 = 0 have been established. In this paper, Hermitian self-dual linear codes over Fq + uFq + u2Fq are studied for all square prime powers q. Complete characterization and enumeration of such codes are given. Subsequently, algebraic characterization of H-quasi-abelian codes in Fq[G] is studied, where H ≤ G are finite abelian groups and Fq[H] is a principal ideal group algebra. General characterization and enumeration of H-quasi-abelian codes and self-dual H-quasi-abelian codes in Fq[G] are given. For the special case where the field characteristic is 3, an explicit formula for the number of self-dual A× Z3-quasi-abelian codes in F3m[A× Z3 ×B] is determined for all finite abelian groups A and B such that 3 - |A| as well as their construction. Precisely, such codes can be represented in terms of linear codes and self-dual linear codes over F3m + uF3m + u2F3m. Some illustrative examples are provided as well. 2010 MSC: 94B15, 94B05, 94B60 Keywords: Hermitian self-dual linear codes, Quasi-abelian codes, Finite chain rings, Group algebras 1. Introduction Self-dual linear codes over finite fields form an interesting class of linear codes that have been extensively studied due to their nice algebraic structures and wide applications (see [8], [11], [12], [22] and references therein). Codes over finite rings have been of interest after it was shown that some binary ∗ P. Choosuwan was partially supported by the Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT). S. Jitman was supported by the Thailand Research Fund and Silpakorn Uni- versity under Research Grant RSA6280042. Parinyawat Choosuwan; Department of Mathematics and Computer Science, Faculty of Science and Tech- nology, Rajamangala University of Technology Thanyaburi (RMUTT), Pathum Thani 12110, Thailand (email: parinyawat_c@rmutt.ac.th). Somphong Jitman (Corresponding Author); Department of Mathematics, Faculty of Science, Silpakorn Univer- sity, Nakhon Pathom 73000, Thailand (email: sjitman@gmail.com). 209 https://orcid.org/0000-0003-0817-282X https://orcid.org/0000-0003-1076-0866 P. Choosuwan, S. Jitman / J. Algebra Comb. Discrete Appl. 7(3) (2020) 209–227 non-linear codes such as the Kerdock, Preparata and Goethal codes are the Gray images of linear codes over Z4 in [7]. In general, families of linear codes and self-dual linear codes over finite chain rings are now become of interest. In [16], the mass formula for Euclidean self-dual linear codes over Zp3 has been studied. Characterization and enumeration of Euclidean self-dual linear codes over the ring Fq + uFq + u2Fq with u3 = 0 have been given in [3]. Algebraically structured codes over finite fields such as cyclic codes, abelian codes and quasi-abelian codes are another important family of linear codes that have been extensively studied for both theoretical and practical reasons (see [2], [8], [10], [11], [12] and references therein). In [10], H-quasi-abelian codes and self-dual H-quasi-abelian codes in Fq[G] have been studied in the case where Fq[H] is semisimple To the best of our knowledge, Hermitian self-dual linear codes over Fq + uFq + u2Fq and non- semisimple H-quasi-abelian codes in Fq[G] have not been well studied. The goals of this paper are to investigate the following families of linear codes and their links. 1) Hermitian self-dual linear codes over Fq +uFq +u2Fq where q is a square prime power. 2) H-quasi-abelian codes and self-dual H-quasi-abelian codes in group algebras Fq[G], where H ≤ G are finite abelian groups and Fq[H] is a principal ideal group algebra. The paper is organized as follows. In Section 2, some results on linear codes and Euclidean self- dual linear codes over Fq + uFq + u2Fq are recalled. In Section 3, characterization and enumeration Hermitian self-dual linear codes of length n over Fq + uFq + u2Fq are established for all square prime powers q together with an algorithm to determine all Hermitian self-dual codes and illustrative examples. In Section 4, the study of H-quasi-abelian codes in Fq[G] is given, where Fq[H] is a principal ideal group algebra. In the special case where the field characteristic is 3, the characterization and enumeration of A×Z3-quasi-abelian codes and self-dual A×Z3-quasi-abelian codes in F3m [A×Z3 ×B] are completely determined in terms of linear and self-dual linear codes over F3m + uF3m + u2F3m obtained in Section 3 for all finite abelian groups A and B such that 3 - |A|. Summary and remarks are given in Section 5. 2. Preliminaries In this section, basic results on linear codes and Euclidean self-dual linear codes over rings are recalled. 2.1. Linear codes over Fq + uFq + · · ·+ ue−1Fq For a prime power q, denote by Fq the finite field of order q. Let Fq + uFq + · · · + ue−1Fq := {a0 +ua1 +· · ·+ue−1ae−1 | ai ∈ Fq for all 0 ≤ i < e} be a ring, where the addition and multiplication are defined as in the usual polynomial ring over Fq with indeterminate u together with the condition ue = 0. It is easily seen that Fq +uFq +· · ·+ue−1Fq is isomorphic to Fq[u]/〈ue〉 as rings. The Galois extension of Fq +uFq +· · ·+ue−1Fq of degree m is defined to be the quotient ring (Fq +uFq +· · ·+ue−1Fq)[x]/〈f(x)〉, where f(x) is an irreducible polynomial of degree m over Fq. It is not difficult to see that the Galois extension of Fq + uFq + · · ·+ ue−1Fq of degree m is isomorphic to Fqm + uFqm + · · ·+ ue−1Fqm. The ring Fq + uFq + · · ·+ ue−1Fq is a finite chain ring with maximal ideal 〈u〉, nilpotency index e and residue field Fq. In addition, if q is a square, the mapping ¯ : Fq → Fq defined by a 7→ a √ q is a field automorphism on Fq of order 2. Extend ¯ to be a ring automorphism of order 2 on Fq + uFq + · · · + ue−1Fq of the form a0 + ua1 + · · · + ue−1ae−1 = a0 + ua1 + · · · + ue−1ae−1. Let n be a positive integer and let R be a finite ring. The Euclidean inner product of u = (u0,u1, . . . ,un−1) and v = (v0,v1, . . . ,vn−1) in Rn is defined to be 〈u,v〉E := n−1∑ i=0 uivi. In the case where q is a square and R ∈{Fq,Fq + uFq + · · ·+ ue−1Fq}, the Hermitian inner product of u 210 P. Choosuwan, S. Jitman / J. Algebra Comb. Discrete Appl. 7(3) (2020) 209–227 and v in Rn is defined to be 〈u,v〉H := n−1∑ i=0 uivi. A linear code C of length n over the ring R is defined to be an R-submodule of the R-module Rn. A linear code over R is said to be free if it is a free R-module. Denote by wt(v) the Hamming weight of an element v ∈ Rn. Precisely, wt(v) is the number of non-zero components in v. For a linear code C over R, let wt(C) = min{wt(c) | c ∈C} be the minimum Hamming weight of C. If R = Fq, a linear code C of length n and dimension k over R with wt(C) = d is referred as an [n,k,d]q code. The parameters of a linear code C of length n over R satisfies the Singleton bond [14], i.e., wt(C) ≤ n−log|R|(|C|) + 1. A linear code C is called a Maximum Distance Separable (MDS) code if the equality in the Singleton bound holds. A matrix G over R is called a generator matrix for C if the rows of G generate all the elements of C and none of the rows can be written as a linear combination of the others. Linear codes C1 and C2 over R are said to be equivalent if there exists a monomial matrix P such that C2 = C1P := {cP | c ∈ C1}. Denote by C⊥E = {v ∈ Rn | 〈u,v〉E = 0} and C⊥H = {v ∈ Rn | 〈u,v〉H = 0} the Euclidean and Hermitian duals of C, respectively. A linear code C is said to be Euclidean (resp., Hermitian) self-orthogonal if C ⊆ C⊥E (resp., C ⊆C⊥H). It is called Euclidean (resp., Hermitian) self-dual if C = C⊥E (resp., C = C⊥H). In Section 3 and the remaining parts of this section, we focus on linear and self-dual linear codes over Fq + uFq + u2Fq. In [19], it has been shown that every linear code of length n over Fq + uFq + u2Fq is permutation equivalent to a code C with generator matrix G =  Ik A2 A3 A40 uIl uB3 uB4 0 0 u2Im u 2C4   =   A′uB′ u2C   , (1) where Ir is the identity matrix of order r, A3 = A30 + uA31,B4 = B40 + uB41,A4 = A40 + uA41 + u2A42, and A2,B3,C4, Aij and Bij are matrices of appropriate sizes over Fq. In this case, the code C is said to be of type {k,l,m} and it contains q3k+2l+m codewords. For each linear code C of length n over Fq + uFq + u2Fq and i ∈ {0, 1, 2}, the ith torsion code of C is a linear code of length n over Fq defined to be Tori(C) = { v(modu) | v ∈ ( Fq + uFq + u2Fq )n and uiv ∈C } . The code Tor0(C) is sometime called the residue code of C and denoted it by Res(C). From the definitions, it is obvious that Res(C) = Tor0(C) ⊆ Tor1(C) ⊆ Tor2(C). For a linear code C of length n over Fq +uFq +u2Fq with generator matrix G given in (1), the residue code Res(C) has dimension k and generator matrix G = [ Ik A2 A30 A40 ] , (2) the first torsion code Tor1(C) has dimension k + l and generator matrix[ A B ] = [ Ik A2 A30 A40 0 Il B3 B40 ] , (3) and the second torsion code Tor2(C) has dimension k + l + m and generator matrix AB C   =  Ik A2 A30 A400 Il B3 B40 0 0 Im C4   . (4) For 0 ≤ k ≤ n, the Gaussian coefficient is defined to be[ n k ] q = (qn − 1)(qn −q) · · ·(qn −qk−1) (qk − 1)(qk −q) · · ·(qk −qk−1) . 211 P. Choosuwan, S. Jitman / J. Algebra Comb. Discrete Appl. 7(3) (2020) 209–227 Let Ne(q,n) denote the number of distinct linear codes of length n over Fq + uFq + · · · + ue−1Fq. The number Ne(q,n) has been studied and summarized in [4]. For e = 3, we have the following result. Proposition 2.1 ([4, Lemma 2.2]). Let q be a prime power and let n be a positive integer. Then N3(q,n) = 1 + 3∑ t=1 ∑ n≥h1≥h2≥···≥ht>ht+1=0 t∏ j=1 [ n−hj+1 hj −hj+1 ] q qhj+1(n−hj). 2.2. Euclidean self-dual linear codes over Fq + uFq + u2Fq Let C be a linear code of length n and type {k,l,m} over Fq +uFq +u2Fq and let h = n−(k +l +m). In [3], it has been shown that the Euclidean dual C⊥E of C is of type {h,m,l} and it contains q3h+2m+l codewords. Therefore, k = h and l = m whenever C is Euclidean self-dual. Consequently, every Euclidean self-dual code of type {k,l,m} over Fq + uFq + u2Fq has even length n = 2(k + l). Characterization of Euclidean self-dual linear codes of even length n over Fq + uFq + u2Fq has been established in [3]. Proposition 2.2 ([3, Proposition 1]). Let q be a prime power and let C be a linear code of length n and type {k,l,m} over Fq + uFq + u2Fq with generator matrix G in the form of (1). Then C is Euclidean self-dual if and only if k = h,l = m and the following conditions hold: A′A′ T ≡ 0 (mod u3), (5) A′B′ T ≡ 0 (mod u2), (6) B′B′ T ≡ 0 (mod u), (7) A′CT ≡ 0 (mod u). (8) For a positive integer n and a prime power q, let σE(q,n) denote the number of Euclidean self-dual linear codes of length n over Fq. Further, if q is a square prime power, let σH(q,n) denote the number of Hermitian self-dual linear codes of length n over Fq. The following results in [21] and [22] are useful in the enumeration of self-dual linear codes over Fq + uFq + u2Fq. Lemma 2.3 ([21] and [22]). Let q be a prime power and let n be a positive integer. Then σE(q, l) =   n 2 −1∏ i=1 (qi + 1) if q and n are even, 2 n 2 −1∏ i=1 (qi + 1) if q ≡ 1 (mod 4) and 2 | n, 2 n 2 −1∏ i=1 (qi + 1) if q ≡ 3 (mod 4) and 4 | n, 0 otherwise. (9) If q is square, then σH(q,n) =   n 2 −1∏ i=0 (qi+ 1 2 + 1) if n is even, 0 otherwise. (10) 212 P. Choosuwan, S. Jitman / J. Algebra Comb. Discrete Appl. 7(3) (2020) 209–227 The empty product is regarded as 1. Let NEe(q,n) denote the number of distinct Euclidean self-dual linear codes of length n over Fq + uFq + · · · + ue−1Fq. The value of NE3(q,n) has been completely determined in [3]. Theorem 2.4 ([3, Theorem 1]). Let q be a prime power and let n be a positive integer. Then NE3(q,n) =   σE(q,n) n/2∑ k=0 [n 2 k ] q qkn/2 if q is even and n is even, σE(q,n) n/2∑ k=0 [n 2 k ] q qk(n/2−1) if q is odd and n is even, 0 otherwise. 3. Hermitian self-dual linear codes over Fq + uFq + u2Fq In this section, we focus on characterization and enumeration of Hermitian self-dual linear codes of length n over Fq + uFq + u2Fq. Throughout this section, we assume that q is a square prime power. For each positive integer n, let NHe(q,n) denote the number of distinct Hermitian self-dual linear codes of length n over Fq + uFq + · · ·+ ue−1Fq. By extending techniques introduced in [3], the characterization and the number NH3(q,n) of Hermitian self-dual linear codes of length n over Fq + uFq + u2Fq are established. Let C be a linear code of length n over Fq + uFq + u2Fq of type {k,l,m} and let h = n−(k + l + m). Using argument similar to those in Section 2 of [3], it can be deduced that the Hermitian dual C⊥H of C is of type {h,m,l} and it contains q3h+2m+l codewords. It follows that k = h and l = m if C is Hermitian self-dual. Hence, every Hermitian self-dual code of type {k,l,m} over Fq + uFq + u2Fq has even length n = 2(k + l). For a matrix A = [aij]s×t over Fq + uFq + u2Fq, let A := [aij]s×t and A † := A T . Characterization of Hermitian self-dual linear codes of even length n over Fq +uFq +u2Fq is given in the following proposition. Proposition 3.1. Let q be a square prime power and let C be a linear code of even length n and type {k,l,m} over Fq +uFq +u2Fq with generator matrix G in the form of (1). Then C is Hermitian self-dual if and only if k = h, l = m and the following hold: A′A′ † ≡ 0 (mod u3), (11) A′B′ † ≡ 0 (mod u2), (12) B′B′ † ≡ 0 (mod u), (13) A′C† ≡ 0 (mod u). (14) Proof. Assume that C is Hermitian self-dual. By the above discussion, we have k = h, l = m and  A′uB′ u2C     A′uB′ u2C  † ≡ [0] (mod u3) which are equivalent to the conditions (11)–(14). 213 P. Choosuwan, S. Jitman / J. Algebra Comb. Discrete Appl. 7(3) (2020) 209–227 Conversely, assume that C is a linear code such that k = h,l = m and the conditions (11)–(14) hold. From (11)–(14), it is not difficult to see that C is Hermitian self-orthogonal. Equivalently, C ⊆C⊥H. Since k = h and l = m, we have |C| = |C⊥H| which implies that C = C⊥H. Therefore, C is Hermitian self-dual as desired. Corollary 3.2. Let C be a Hermitian self-dual linear code of length n over Fq + uFq + u2Fq. Then the following statements holds. 1) Tor1(C) is a Hermitian self-dual code of length n over Fq. 2) Tor2(C) = Res(C)⊥H. Proof. Assume that C is of type {k,l,m} over Fq + uFq + u2Fq. Then From (11)–(13), it follows that Tor1(C) is Hermitian self-orthogonal. Since dim(Tor1(C)) = k + l = n2 = dim((Tor1(C)) ⊥H ), Tor1(C) is Hermitian self-dual. From (11)–(14), we have Tor2(C) ⊆ Res(C)⊥H. Since dim(Tor2(C)) = k + 2l = n−k = dim((Res(C))⊥H ), we have Tor2(C) = Res(C)⊥H. Since Res(C) = Tor0(C) ⊆ Tor1(C) ⊆ Tor2(C), it can be concluded further that Res(C) is Hermitian self-orthogonal for all Hermitian self-dual linear codes C over Fq + uFq + u2Fq. From Corollary 3.2, a Hermitian self-dual code C of length n over Fq + uFq + u2Fq can be induced by Hermitian self-dual linear codes of length n over Fq. For a given Hermitian self-dual code C1 of length n over Fq, we first aim to determine the number of Hermitian self-dual linear codes C of length n over Fq + uFq + u2Fq such that Tor1(C) = C1. Proposition 3.3. Let q be a square prime power and let n be an even positive integer. Let C1 be a Hermitian self-dual linear code of length n over Fq. Then, for each 0 ≤ k ≤ n2 , there are q kn 2 Hermitian self-dual linear codes of length n over Fq + uFq + u2Fq corresponding to each subspace of C1 of dimension k. Proof. Let C1 be a Hermitian self-dual linear code of length n over Fq with dimension n2 = k + l and generator matrix [ A B ] = [ Ik A2 A30 A40 0 Il B3 B40 ] , (15) where the columns are grouped into blocks of sizes k,l, l and k. Since C1 is Hermitian self-dual, we have Ik + A2A † 2 + A30A † 30 + A40A † 40 = 0, (16) A2 + A30B † 3 + A40B † 40 = 0, (17) Il + B3B † 3 + B40B † 40 = 0. (18) Let H = [ A30 A40 B3 B40 ] . From (16)–(18), it can be deduced that H(−H†) = −HH† = −HH T = [ −A30AT30 −A40AT40 −A30BT3 −A40BT40( −A30BT3 −A40BT40 )T −B3BT3 −B40BT40 ] = [ Ik + A2A T 2 A2 AT2 Il ] . 214 P. Choosuwan, S. Jitman / J. Algebra Comb. Discrete Appl. 7(3) (2020) 209–227 Let J = [ Ik −A2 −AT2 Il + AT2 A2 ] . Then H(−H†)J = [ Ik 0 0 Il ] which implies that H is invertible. Let C0 be a k-dimensional Fq-subspace of C1 with generator matrix A. Since C1 is Hermitian self- dual, it follows that C0 is Hermitian self-orthogonal. Up to permutation of the last (k + l) columns (if necessary), its follows that C⊥H0 has a generator matrix of the form Ik A2 A30 A400 Il B3 B40 0 0 Il C4   . (19) Then A30 = −A40C † 4 which implies that H = [ −A40CT4 A40 B3 B40 ] . Since H is invertible, it follows that A40 is invertible. Next, we determined the matrices over Fq + uFq + u2Fq satisfying conditions (11)–(14) which are equivalent to Ik + A2A † 2 + A3A † 3 + A4A † 4 ≡ 0 (mod u 3) (20) A2 + A3B † 3 + A4B † 4 ≡ 0 (mod u 2) (21) Il + B3B † 3 + B4B † 4 ≡ 0 (mod u) (22) A3 + A4C † 4 ≡ 0 (mod u). (23) The matrices A2,B3 and C4 are considered modulo u, i.e. all the entries in A2,B3 and C4 are in Fq. The matrices A3 and B4 are considered modulo u2 while A4 is considered modulo u3. From these fact, let A3 = A30 + uA31,B4 = B40 + uB41 and A4 = A40 + uA41 + u2A42, where A31,B41,A41 and A42 are matrices of appropriate sizes over Fq. Therefore, we can write (20) as( Ik + A2A † 2 + A30A † 30 + A40A † 40 ) + u ( Ã30A † 31 + Ã40A † 41 ) +u2 ( A31A † 31 + A41A † 41 + Ã40A † 42 ) ≡ 0 (mod u3), where X̃ := X + X†. We can also rewrite (21) as( A2 + A30B † 3 + A40B † 40 ) + u ( A31B † 3 + A40B † 41 + A41B † 40 ) ≡ 0 (mod u2) By substituting (18) into (21), we obtain that B † 41 = −A −1 40 ( A31B † 3 + A41B † 40 ) , From (23), C4 is uniquely determined as C4 = ( −A−140 A30 )† . It is sufficient to focus on (20) because (18) is the same as (22). From (16), we have to determined the matrices satisfying the following: Ã30A † 31 + Ã40A † 41 = 0, (24) A31A † 31 + A41A † 41 + Ã40A † 42 = 0. (25) Hence, the code C with generator matrix of the form (1) is a Hermitian self-dual linear code if and only if conditions (24) and (25) are satisfied. 215 P. Choosuwan, S. Jitman / J. Algebra Comb. Discrete Appl. 7(3) (2020) 209–227 Therefore, the number of Hermitian self-dual linear codes of length n over Fq + uFq + u2Fq whose the 1st torsion is C1 is equal to the number of solutions of the system of matrix equations (24) and (25). We take an arbitrary matrix A31 ∈ Mk×l(Fq) and put [gij] = Ã30A † 31 and [xij] = A40A † 41. Then condition (24) is equivalent to gij + xij + xji = 0. Then −gii = xii + xii = Tr(xii) ∈ F√q for each 1 ≤ i ≤ k, where Tr : Fq → F√q is the trace map defined by α 7→ α + α for all α ∈ Fq. Note that |Tr−1(a)| = √ q for all a ∈ F√q. Then we have xii ∈ Tr−1(−gii) for all 1 ≤ i ≤ k, xji ∈ Fq and xij = −gij −xji for each 1 ≤ i < j ≤ k. Therefore, A41 = (A −1 40 [xij]) †. Thus we have qkl possible choices for A31 and q k(k−1) 2 + k 2 = q k2 2 for A41. For fixed matrices A31 and A41, let [hij] = A31A † 31 + A41A † 41 and [yij] = A40A † 42. Then (25) is equivalent to hij + yij + yji = 0. Using a similar argument as above, we have q k2 2 possible choices for A42 = (A −1 40 [yij]) †. Therefore, we have qkl ×q k2 2 ×q k2 2 = qk 2+kl = qk(k+l) = q kn 2 possible choices for the matrices A31,A41 and A42 over Fq. Therefore, the desired result follows immedi- ately. The number of distinct Hermitian self-dual linear codes of even length n over Fq + uFq + u2Fq can be summarized in the following theorem. Theorem 3.4. Let q be a square prime power and let n be a positive integer. Then the number of distinct Hermitian self-dual linear codes of length n over Fq + uFq + u2Fq is NH3(q,n) =   σH(q,n) n/2∑ k=0 [n 2 k ] q qkn/2 if n is even, 0 otherwise. From the proof of Theorem 3.3, we obtain not only the number of Hermitian self-dual linear codes of length n over Fq + uFq + u2Fq but also a construction of such Hermitian self-dual linear codes. The construction of Hermitian self-dual linear codes of length n over Fq + uFq + u2Fq induced by a Hermitian self-dual linear code of length n over Fq in the proof of Theorem 3.3 is summarized in Algorithm 1. Based on Algorithm 1, an illustrative example of a Hermitian self-dual linear code of length 6 over F9 +uF9 +u2F9 constructed from a Hermitian self-dual linear code of length 6 over F9 is given as follows. Example 3.5. Let F9 = F3[α] be the finite field of order 9, where α is a root of x2 + x + 2 over F3. Let C0 and C1 be linear codes of length 6 over F9 with generator matrices [ 1 0 1 α2 α5 α2 0 1 0 α 1 α ] and   1 0 1 α2 α5 α20 1 0 α 1 α 0 0 1 α5 α 1   , 216 P. Choosuwan, S. Jitman / J. Algebra Comb. Discrete Appl. 7(3) (2020) 209–227 Algorithm 1. Construction of Hermitian self-dual linear codes of length n over Fq + uFq + u2Fq For a given Hermitian self-dual linear code C1 of length n over Fq and its linear subcode C0 of dimension 0 ≤ k ≤ n 2 , do the following steps. 1. Define l = n 2 −k. 2. Construct a generator matrix A = [ Ik A2 A30 A40 ] for C0, where the columns are grouped into blocks of sizes k,l, l and k. 3. Extend A to be a generator matrix [ Ik A2 A30 A40 0 Il B3 B40 ] for C1. 4. Set C4 = − ( A−140 A30 )† . 5. Set A31 to be a k × l matrix over Fq. 6. Define [gij] = Ã30A † 31 and set [xij] to be a k×k matrix over Fq such that the strictly lower triangular elements are arbitrary in Fq, xii ∈ Tr−1(−gii), and xij = −gij −xji for all i < j. (If k = n2 , set [gij] to be the k ×k zero matrix over Fq.) 7. Set A41 = (A−140 [xij]) †. 8. Set B41 = − ( A−140 ( A31B † 3 + A41B † 40 ))† . 9. Define [hij] = A31A † 31 + A41A † 41 and set [yij] to be a k × k matrix over Fq such that the strictly lower triangular elements are arbitrary in Fq, yii ∈ Tr−1(−hii), and yij = −gij −xji for all i < j. (If k = n 2 , set [hij] = A41A † 41.) 10. Set A42 = (A−140 [yij]) †. 11. Define C to be a linear code of length n over Fq + uFq + u2Fq with generator matrix Ik A2 A30 + uA31 A40 + uA41 + u2A420 uIl uB3 uB40 + u2B41 0 0 u2Il u 2C4   . The C is Hermitian self-dual by Theorem 3.3. 12. Repeat 5.−11. with different choices of A31, A41, and A42. The Hermitian self-dual linear codes of length n over Fq + uFq + u2Fq determined by C0 ⊆C1 are obtained. respectively. Then C1 is Hermitian self-dual and C0 ⊆ C1. Based on Algorithm 1, we have k = 2, l = 1, A2 = [ 1 0 ] , A30 = [ α2 α ] , A40 = [ α5 α2 1 α ] , B3 = [ α5 ] , and B40 = [ α 1 ] . Then we have C4 = − ( A−140 A30 )† = −  [α5 α2 1 α ]−1 [ α2 α ]† = [0 2]. By choosing A31 = [ 1 1 ] , we have [gij] = Ã30A † 31 = ˜[ α2 α ][ 1 1 ] = [ 0 α α3 2 ] . We choose x11 = 0 ∈ {0,α2,α6} = Tr−1(0) = Tr−1(−g11), x22 = 2 ∈ {2,α5,α7} = Tr−1(1) = Tr−1(−g22), x21 = 1, and x12 = −g12 − x21 = −α − 1 = α3. It follows that [xij] = [ 0 α3 1 2 ] and A41 = (A −1 40 [xij]) † = [ 2 α α 1 ] . Consequently, B41 = − ( A−140 ( A31B † 3 + A41B † 40 ))† = [ α 0 ] . 217 P. Choosuwan, S. Jitman / J. Algebra Comb. Discrete Appl. 7(3) (2020) 209–227 Let [hij] = A31A † 31 + A41A † 41 = [ 1 1 ][ 1 1 ] + [ α4 α α 1 ][ α4 α3 α3 1 ] = [ 1 α3 α 1 ] . We choose y11 = 1 ∈ {1,α,α3} = Tr−1(2) = Tr−1(−h11), y22 = 1 ∈ {1,α,α3} = Tr−1(2) = Tr−1(−h22), y21 = 1, and y12 = −h12 −y21 = −α3 − 1 = α. Then [yij] = [ 1 a 1 1 ] and A42 = (A −1 40 [yij]) † = [ α3 α3 0 α5 ] . From Algorithm 1, the matrix  Ik A2 A30 + uA31 A40 + uA41 + u 2A42 0 uIl uB3 uB40 + u 2B41 0 0 u2Il u 2C4   =   1 0 1 α2 + u α5 + 2u + α3u2 α2 + αu + α3u2 0 1 0 α + u 1 + αu α + u + α5u2 0 0 u α5u αu + αu2 u 0 0 u2 0 2u2   is a generator matrix for a Hermitian self-dual code of length 6 over F9 + uF9 + u2F9 with type {2, 1, 1}. When k = n 2 in Theorem 3.3 (equivalently, in Algorithm 1), we have the following extension on the parameters of Hermitian self-dual linear codes over Fq + uFq + u2Fq. Let n be an even positive integer and let C1 = C0 be a Hermitian self-dual code of length n over Fq with parameters [n,k = n2 ,d]q and generator matrix A = [ In 2 A40 ] , (26) where A40 is a k×k invertible matrix over Fq. Based on Algorithm 1, the linear code C of length n and type {n 2 , 0, 0} over Fq + uFq + u2Fq with generator matrix G = [ Ik A40 + uA41 + u 2A42 ] (27) is Hermitian self-dual. Since C is a free code, wt(C) = wt(C1) = d by [18, Corollary 4.3]. Hence, the following two theorems can be derived directly. Theorem 3.6. Let q be a prime power and let n be an even positive integer. If there exists an [n, n 2 , n 2 +1]q MDS Hermitian self-dual code over Fq, then an MDS Hermitian self-dual code of length n over Fq +uFq + u2Fq of type {n2 , 0, 0} can be constructed with minimum Hamming weight n 2 + 1. Proof. Assume that there exists an [n, n 2 , n 2 + 1]q MDS Hermitian self-dual code C1 over Fq with generator matrix of the form (26). By Algorithm 1, a linear code C of length n and type {n 2 , 0, 0} over Fq + uFq + u2Fq with generator matrix of the form (27) is Hermitian self-dual. Since C is free, we have wt(C) = wt(C1) = n2 + 1 and logq3 (|C|) = n 2 = dim(C1) by the discussion above. Hence, wt(C) = n 2 + 1 = n− logq3 (|C|) + 1 which implies that C is MDS. Using the above theorem, numerous MDS Hermitian self-dual codes over Fq + uFq + u2Fq can be construct based on known MDS Hermitian self-dual codes over Fq (see, for example, [13], [17], [24]). 4. Self-dual quasi-abelian codes over principal ideal group alge- bras In this section, the study of quasi-abelian codes over principal ideal group algebras is given. In the special case where the field characteristic is 3 and the Sylow 3-subgroup of the underlying finite abelian group has order 3, complete characterization and enumeration of quasi-abelian codes and self-dual quasi- abelian codes are presented in terms linear codes and self-dual linear codes over F3m + uF3m + u2F3m obtained in [3], [4] and Section 3. 218 P. Choosuwan, S. Jitman / J. Algebra Comb. Discrete Appl. 7(3) (2020) 209–227 4.1. Group rings and quasi-abelian codes Let R be a finite commutative ring with nonzero identity and let G be a finite abelian group. Then R[G] =  ∑ g∈G αgY g | αg ∈ R,g ∈ G   is a commutative ring under the addition and multiplication given for the usual polynomial ring over R with indeterminate Y , where the indices are computed additively in G. The ring R[G] is called a group ring of G over R. In the case where R is the finite field Fpm, the group ring Fpm [G] is called a group algebra of G over Fpm and it is called a Principal Ideal Group Algebra (PIGA) if every ideal in Fpm [G] is principal. The readers may refer to [15] for more details on group rings. A linear code of length |G| over R can be viewed as an embedded R-submodule of the R-module in R[G] by indexing the |G|-tuples by the elements in G. Given a subgroup H of G with index n = [G : H], a linear code C of length |G| viewed as an R-submodule of R[G] is called an H-quasi-abelian code (specifically, an H-quasi-abelian code of index n) in R[G] if C is an R[H]-module, i.e., C is closed under the multiplication by the elements in R[H]. Such a code will be called a quasi-abelian code if H is not specified or where it is clear in the context. Let {g1,g2, . . . ,gn} be a fixed set of representatives of the cosets of H in G. Let R := Fq[H]. Define Φ : Fq[G] →Rn by Φ (∑ h∈H n∑ i=1 αh+giY h+gi ) = (α1(Y ),α2(Y ), . . . ,αn(Y )), where αi(Y ) = ∑ h∈H αh+giY h ∈ R for all i = 1, 2, . . . ,n. It is well known that Φ is an R-module isomorphism interpreted as follows. Lemma 4.1 ([10, Lemma 2.1]). The map Φ induces a one-to-one correspondence between H-quasi-abelian codes in Fq[G] and linear codes of length n over R. We note that a group algebra Fpm [H] is semisimple if and only if the Sylow p-subgroup of H is trivial (see [20, Chapter 2: Theorem 4.2]), and it is a PIGA if and only if he Sylow p-subgroup of H is cyclic (see [6]). In [10], complete characterization and enumeration of H-quasi-abelian codes in Fpm [G] have been established in the case where Fpm [H] is semisimple. Here, we focus on a more general case where Fpm [H] is a PIGA, or equivalently, the Sylow p-subgroup of H is cyclic. Precisely, H ∼= A × Zpmi and G ∼= A×Zps ×B, where s is a non-negative integer, A and B are finite abelian groups such that p - |A|. General characterization is given in Subsection 4.2. In the special case where p = 3 and s = 1, complete characterization and enumeration of A×Z3-quasi-abelian codes and self-dual A×Z3-quasi-abelian codes in F3m [A×Z3 ×B] are given in Subsection 4.3. 4.2. A×Zps-Quasi-Abelian Codes in Fpm[A×Zps ×B] We focus on H-quasi-abelian codes in Fpm [G], where Fpm [H] is a PIGA. Equivalently, H ∼= A×Zps and G ∼= A× Zps ×B, where s is a positive integer, A and B are finite abelian groups such that p - |A| (see [6] and [12]). Note that the group algebra Fpm [A] is semisimple [2] and it can be decomposed using the Discrete Fourier Transform in [23] (see [12] and [11] for more details). For completeness, the decomposition used in this paper are summarized as follows. For co-prime positive integers i and j, denote by ordj(i) the multiplicative order of i modulo j. For each a ∈ A, denote by ord(a) the additive order of a in A and the pm-cyclotomic class of A containing 219 P. Choosuwan, S. Jitman / J. Algebra Comb. Discrete Appl. 7(3) (2020) 209–227 a ∈ A is defined to be the set Spm (a) := {pmi ·a | i = 0, 1, . . .} = {pmi ·a | 0 ≤ i < ordord(a)(pm)}, where pki · a := pmi∑ j=1 a in A. A subset {a1,a2, . . . ,at} of A is called a complete set of representatives of pm-cyclotomic classes of A if Spm (a1),Spm (a2), . . . ,Spm (at) are distinct and t⋃ i=1 Spm (ai) = A. An idempotent in Fpm [A] is a nonzero element e such that e2 = e. It is called primitive if for every other idempotent f, either ef = e or ef = 0. The existence of primitive idempotent elements in Fpm [A] is proved in [5]. They are induced by the pm-cyclotomic classes of A (see [5, Proposition II.4]). Consequently, Fpm [A] can be viewed as a direct sum of principal ideals generated by these primitive idempotent elements. Proposition 4.2 ([5, Corollary III.6]). Let {a1,a2, . . . ,at} be a complete set of representatives of pm- cyclotomic classes of a finite abelian group A where p - |A| and let ei be the primitive idempotent induced by Spm (ai) for all 1 ≤ i ≤ t. Then Fpm [A] = t⊕ i=1 Fpm [A]ei ∼= t∏ i=1 Fpmi , where mi = m · ordord(ai)(p m). A PIGA Fpm [A×Zps ] can be decomposed in the following theorem. Theorem 4.3. Let s be a positive integer. Let {a1,a2, . . . ,at} be a complete set of representatives of pm-cyclotomic classes of a finite abelian group A where p - |A|. Then Fpm [A×Zps ] ∼= t∏ i=1 ( Fpmi + uFpmi + · · · + up s−1Fpmi ) where mi = m · ordord(ai)(p m) for all 1 ≤ i ≤ t. Proof. For each 1 ≤ i ≤ t, let ei be the primitive idempotent induced by Spm (ai). From Proposition 4.2, we have Fpm [A] ∼= Fpm [A×Zps ]ei ∼= t∏ i=1 Fpmi , (28) and hence, Fpm [A×Zps ] ∼= (Fpm [A])[Zps ] ∼= t∏ i=1 Fpmi [Zps ]. (29) Under the ring isomorphism that fixes the elements in Fpmi and Y 1 7→ u + 1, it is not difficult to see that Fpmi [Zps ] ∼= Fpmi + uFpmi + · · · + up s−1Fpmi (30) as rings. Therefore, Fpm [A×Zps ] ∼= t∏ i=1 ( Fpmi + uFpmi + · · · + up s−1Fpmi ) (31) as desired. 220 P. Choosuwan, S. Jitman / J. Algebra Comb. Discrete Appl. 7(3) (2020) 209–227 For each finite abelian group B of order n, every A × Zps-quasi-abelian code in Fpm [A × Zps × B] can be viewed as a linear code of length n over Fpm [A× Zps ] by Lemma 4.1. The next corollary follows directly from Theorem 4.3. Corollary 4.4. Let s and m be positive integers. Let A and B be finite abelian groups such that |B| = n and p - |A|. Then every A×Zps-quasi-abelian code C in Fpm [A×Zps ×B] can be viewed as C ∼= t∏ i=1 Ci, where Ci is a linear code of length n over Fpmi + uFpmi + · · · + up s−1Fpmi for all i = 1, 2, . . . , t. The enumeration of A×Zps-quasi-abelian code in Fpm [A×Zps ×B] is given as follows. Theorem 4.5. Let s and m be positive integers. Let A and B be finite abelian groups such that |B| = n and the exponent of A is M and p - M. Then the number of A×Zps-quasi-abelian codes in Fpm [A×Zps×B] is ∏ d|M ( Nps (p m·ordd(pm),n) ) NA(d) ordd(p m) , where NA(d) is the number of elements of order d in A determined in [1] and Nps (pm·ordd(p m),n) is the number of linear codes of length n over Fpm·ordd(pm) + uFpm·ordd(pm) + · · · + u ps−1Fpm·ordd(pm) determined in [4, Lemma 2.2]. Proof. From Theorem 4.3, it suffices to determine the number of linear codes of length n over the ring Fpmi + uFpmi + · · · + up s−1Fpmi for all i = 1, 2, . . . , t. For each divisor d of M, each pm-cyclotomic class containing an element of order d has ordd(pm) elements and the number of such pm-cyclotomic classes is NA(d) ordd(pm) . By Theorem 4.3, it follows that the number of linear codes of length n over Fpm·ordd(pm) +uFpm·ordd(pm) +· · ·+u ps−1Fpm·ordd(pm) corresponding to d is ( Nps (p m·ordd(pm),n) ) NA(d) ordd(p m) . By taking the summation over all the divisors d of M, the desired result follows. Example 4.6. Let A ≤ H ≤ G be finite abelian groups such that A ∼= Z2 × Z4, H ∼= A × Z3, and G ∼= H × Z4. Then the 3-cyclotomic classes of A are S3((0, 0)) = {(0, 0)}, S3((0, 2)) = {(0, 2)}, S3((1, 0)) = {(1, 0)}, S3((1, 2)) = {(1, 2)}, S3((0, 1)) = {(0, 1), (0, 3)}, and S3((1, 1)) = {(1, 1), (1, 3)}. It follows that ordord((0,0))(3) = ordord((0,2))(3) = ordord((1,0))(3) = ordord((1,2))(3) = 1 and ordord((0,1))(3) = ordord((1,1))(3) = 2. By Proposition 4.2, F3[A] has 4 primitive idempotents ei such that F3[A]ei ∼= F3 and 2 primitive idempotents ej such that F3[A]ej ∼= F9. Such primitive idempotents are induced by the above 6 cyclotomic classes while their explicit forms can be determined using [5, Proposition II.4]. Consequently, F3[A] ∼= F3 ×F3 ×F3 ×F3 ×F9 ×F9 and F3[H] = F3[A×Z3] ∼= F3[Z3] ×F3[Z3] ×F3[Z3] ×F3[Z3] ×F9[Z3] ×F9[Z3]. By Proposition 4.3, we have F3[Z3] ∼= F3 + uF3 + u2F3 and F9[Z3] ∼= F9 + uF9 + u2F9, where u3 = 0. Hence, F3[H] ∼= 4∏ i=1 (F3 + uF3 + u2F3) × 2∏ j=1 (F9 + uF9 + u2F9). (32) 221 P. Choosuwan, S. Jitman / J. Algebra Comb. Discrete Appl. 7(3) (2020) 209–227 Using Corollary 4.4, every H-quasi-abelian code in F3[G] is isomorphic to a code of the form C1 ×C2 ×C3 ×C4 ×C5 ×C6, where C1, C2, C3, and C4 are linear codes of length |Z4| = 4 over F3 + uF3 + u2F3, and C5 and C6 are linear codes of length 4 over F9 + uF9 + u2F9. In the next subsections, we focus on self-dual A×Zps-quasi-abelian codes in Fpm [A×Zps ×B] with respect to both the Euclidean and Hermitian inner products. 4.3. Euclidean self-dual A×Zps-quasi-abelian codes in Fpm[A×Zps ×B] Euclidean self-dual A×Zps-quasi-abelian codes in Fpm [A×Zps×B] is studied in terms of the following types of pm-cyclotomic classes. A pm-cyclotomic class Spm (a) is said to be of type I if a = −a (in this case, Spm (a) = Spm (−a)), type II if Spm (a) = Spm (−a) and a 6= −a, or type III if Spm (−a) 6= Spm (a). The primitive idempotent e induced by Spm (a) is said to be of type λ ∈{I,II,III} if Spm (a) is a pm-cyclotomic class of type λ. Rearrange the terms in the decomposition in Theorem 4.3 based on the pm-cyclotomic classes of types I, II and III, we have the next theorem. Theorem 4.7. Let m and s be positive integers and let A be a finite abelian group such that p - |A|. Then Fpm [A×Zps ] ∼= ( rI∏ i=1 Ri ) ×   rII∏ j=1 Sj  ×  (rIII)/2∏ l=1 (Tl ×Tl)   , where rI,rII and rIII are the numbers of elements in a complete set of representatives of pm-cyclotomic classes of A of types I,II, and III, respectively, Ri = Fpm + uFpm + · · · + up s−1Fpm for all i = 1, 2, . . . ,rI, Sj = FpmrI+j + uFpmrI+j + · · ·+ u ps−1FpmrI+j for all j = 1, 2, . . . ,rII, and Tl = FpmrI+rII+l + uFpmrI+rII+l + . . .up s−1FpmrI+rII+l for all l = 1, 2, . . . , (rIII)/2. Using Theorem 4.7 and the analysis similar to those in [11, Section II.D], a A × Zps-quasi-abelian code C in Fpm [A×Zps ×B] and its Euclidean dual are given. Proposition 4.8. Let s and m be positive integers. Let A and B be finite abelian groups such that |B| = n and p - |A|. Then an A×Zps-quasi-abelian code in Fpm [A×Zps ×B] can be viewed as C ∼= ( rI∏ i=1 Bi ) ×   rII∏ j=1 Cj  ×  (rIII)/2∏ l=1 (Dl ×D′l)   , (33) where Bi, Cj, Dl and D′l are linear codes of length n over Ri, Sj, Tl and Tl, respectively, for all i = 1, 2, . . . ,rI, j = 1, 2, . . . ,rII and l = 1, 2, . . . , (rIII)/2. Furthermore, the Euclidean dual of C in (33) is of the form C⊥E ∼= ( rI∏ i=1 B⊥Ei ) ×   rII∏ j=1 C⊥Hj  ×  (rIII)/2∏ l=1 ( (D′l) ⊥E ×D⊥El ) . The characterization of Euclidean self-dual A × Zps-quasi-abelian codes in Fpm [A × Zps × B] is established in terms of a product of linear codes, Euclidean self-dual linear codes, and Hermitian self- dual linear codes over Galois extensions of the ring Fpm + uFpm + · · · + up s−1Fpm. 222 P. Choosuwan, S. Jitman / J. Algebra Comb. Discrete Appl. 7(3) (2020) 209–227 Corollary 4.9. Let s and m be positive integers. Let A and B be finite abelian groups such that |B| = n and p - |A|. Then a A×Zps-quasi-abelian code C in Fpm [A×Zps ×B] is Euclidean self-dual if and only if in the decomposition (33), i) Bi is a Euclidean self-dual linear code of length n over Ri for all i = 1, 2, . . . ,rI, ii) Cj is a Hermitian self-dual linear code of length n over Sj for all j = 1, 2, . . . ,rII, and iii) D′l = D ⊥E l is a linear code of length n over Tl for all l = 1, 2, . . . , (rIII)/2. From Corollary 4.9, the Euclidean self-duality of A×Zps-quasi-abelian code C in Fpm [A×Zps ×B] depends only on the structure of A×Zps and the index n = |B| but not the structure of B itself. Given positive integers m and j, the pair (j,pm) is said to be good if j divides pmt + 1 for some positive integer t, and bad otherwise. This notion have been introduced in [8] and [11] for the enumeration of self-dual cyclic codes and self-dual abelian codes over finite fields and it is completely determined in [9]. Let χ be a function defined on pairs (j,pm) as follows. χ(j,pm) = { 0 if (j,pm) is good, 1 otherwise. (34) The number of Euclidean self-dual A×Zps-quasi-abelian code C in Fpm [A×Zps×B] can be determined as follows. Theorem 4.10. Let s and m be positive integers. Let A and B be finite abelian groups such that |B| = n is even and the exponent of A is M and p - M. Then the number of Euclidean self-dual A× Zps-quasi- abelian codes in Fpm [A×Zps ×B] is (NEps (p m,n)) ∑ d|M,ordd(p m)=1 (1−χ(d,pm))NA(d) × ∏ d|M ordd(p m) 6=1 ( NHps (p m·ordd(pm),n) )(1−χ(d,pm)) NA(d) ordd(p m) × ∏ d|M ( Nps (p m·ordd(pm),n) )χ(d,pm) NA(d) 2ordd(p m) , where NA(d) denotes the number of elements in A of order d determined in [1]. Proof. From Corollary 4.9, it suffices to determine the numbers of linear codes Bi’s, Cj’s, and Dl’s such that Bi and Cj are Euclidean and Hermitian self-dual, respectively. From [12, Remark 2.5], the elements in A of the same order are partitioned into pm-cyclotomic classes of the same type. For each divisor d of M, a pm-cyclotomic class containing an element of order d has cardinality ordd(pm) and the number of such pm-cyclotomic classes is NA(d) ordd(pm) . We consider the following 3 cases. Case 1: χ(d,pm) = 0 and ordd(3k) = 1. By [11, Remark 2.6], every 3k-cyclotomic class of A containing an element of order d is of type I. Since there are NA(d) ordd(pm) such pm-cyclotomic classes, the number of Euclidean self-dual linear codes Bi’s of length n corresponding to d is (NEps (p m,n)) NA(d) ordd(p m) = (NEps (p m,n)) (1−χ(d,pm))NA(d) . Case 2: χ(d,pm) = 0 and ordd(pm) 6= 1. By [11, Remark 2.6], every pm-cyclotomic class of A containing an element of order d is of type II and of even cardinality ordd(pm). Hence, the number of Hermitian self-dual linear codes Cj’s of length n corresponding to d is( NHps (p m·ordd(pm),n) ) NA(d) ordd(p m) = ( NHps (p m·ordd(pm),n) )(1−χ(d,pm)) NA(d) ordd(p m) . 223 P. Choosuwan, S. Jitman / J. Algebra Comb. Discrete Appl. 7(3) (2020) 209–227 Case 3: χ(d,pm) = 1. By [11, Lemma 4.5], every pm-cyclotomic class of A containing an element of order d is of type III. Then the number of linear codes Dl’s of length n corresponding to d is ( Nps (p m·ordd(pm),n) ) NA(d) 2ordd(p m) = ( Nps (p m·ordd(pm),n) )χ(d,pm) NA(d) 2ordd(p m) . The formula for the number of Euclidean self-dual A×Zps-quasi-abelian codes in Fpm [A×Zps×B] follows since d runs over all divisors of M. Remark 4.11. In general, the numbers NEps (pm,n) and NHps (pm,n) in Theorem 4.10 have not been well studied. In the case where the field characteristic is 3, we have the following conclusions. 1. The numbers N3(3m,n), NE3(3m,n) and NH3(3m,n) have been determined in Proposition 2.1, [3, Theorem 1] and Theorem 3.4. By Theorem 4.10, the enumeration for Euclidean self-dual A× Z3- quasi-abelian codes in F3m [A×Z3 ×B] is completed. . 2. The construction/characterization of linear, Euclidean self-dual and Hermitian self-dual codes of length n over F3m + uF3m + u2F3m have been given in [3], [4] and in the proof of Proposition 3.3. Hence, the construction/characterization of Euclidean self-dual A × Z3-quasi-abelian code in F3m [A×Z3 ×B] can be obtained from Corollary 4.9. 3. Note that, if n is odd, there are no Hermitian self-dual linear codes of length n over F3m + uF3m + u2F3m by Theorem 3.4. Hence, there are no Euclidean self-dual A × Z3-quasi-abelian codes in F3m [A×Z3 ×B] for all abelian groups B of odd order. Example 4.12. Let A ≤ H ≤ G be finite abelian groups such that A ∼= Z2 × Z4, H ∼= A × Z3, and G ∼= H × Z4. Form Example 4.6, it is easily seen that the 3-cyclotomic classes S3((0, 0)) = {(0, 0)}, S3((0, 2)) = {(0, 2)}, S3((1, 0)) = {(1, 0)}, S3((1, 2)) = {(1, 2)} of A ∼= Z2 × Z4 are of type I, the 3- cyclotomic classes S3((0, 1)) = {(0, 1), (0, 3)} and S3((1, 1)) = {(1, 1), (1, 3)} are of type II, and there are no 3-cyclotomic classes of type III. Then rI = 4, rII = 2, and rIII = 0. In view of Theorem 4.7, (32) is recalled as F3[H] ∼= 4∏ i=1 (F3 + uF3 + u2F3) × 2∏ j=1 (F9 + uF9 + u2F9). Hence, by Corollary 4.9, each Euclidean self-dual H-quasi-abelian code in F3[G] is isomorphic to a code of the form C1 ×C2 ×C3 ×C4 ×C5 ×C6, where C1, C2, C3, and C4 are Euclidean self-dual linear codes of length 4 over F3 + uF3 + u2F3, and C5 and C6 are Hermitian self-dual linear codes of length 4 over F9 + uF9 + u2F9. 4.4. Hermitian self-dual A×Zps-quasi-abelian codes in Fpm[A×Zps ×B] In this subsection, we focus on the case where m is even and study Hermitian self-dual A × Zps- quasi-abelian codes in Fpm [A×Zps ×B]. The characterization and enumeration of Hermitian self-dual A×Zps-quasi-abelian codes in Fpm [A× Zps ×B] are given based on the decomposition of a group algebra Fpm [A×Zps ] in terms of the following types of pm-cyclotomic classes of A. A pm-cyclotomic class Spm (a) is said to be of type I ′ if Spm (a) = Spm (−p m 2 a) or type II′ if Spm (a) 6= Spm (−p m 2 a). The primitive idempotent e induced by Spm (a) is said to be of type λ ∈{I′,II′} if Spm (a) is a pm-cyclotomic class of type λ. Rearrange the terms in the decomposition in Theorem 4.3 based on the pm-cyclotomic classes of types I′ and II′, the next theorem follows. 224 P. Choosuwan, S. Jitman / J. Algebra Comb. Discrete Appl. 7(3) (2020) 209–227 Theorem 4.13. Let m be an even positive integer and let A be a finite abelian group such that p - |A|. Fpm [A×Zps ] ∼=   rI′∏ j=1 S  ×  (rII′)/2∏ l=1 (Tl ×Tl)   , where r′I and rII′ are the numbers of elements in a complete set of representatives of p m-cyclotomic classes of A of types I′ and II′, respectively, Sj = Fpmj + uFpmj + · · · + up s−1Fpmj for all j = 1, 2, . . . ,rI′ and Tl = FpmrI′+l + uFpkrI′+l + · · · + u ps−1F p mrI′+l for all l = 1, 2, . . . , (rII′)/2. Using Theorem 4.13 and the analysis similar to those in [12, Section II.D], the A×Zps-quasi-abelian code C in Fpm [A×Zps ×B] and its Hermitian dual are given. Proposition 4.14. Let s and m be positive integers such that m is even. Let A and B be finite abelian groups such that |B| = n and p - |A|. Then an A× Zps-quasi-abelian code in Fpm [A× Zps ×B] can be viewed as C ∼=   rI′∏ j=1 Cj  ×  (rII′)/2∏ l=1 (Dl ×D′l)   , (35) where Cj, Dl and D′l are linear codes of length n over Sj, Tl and Tl, respectively, for all j = 1, 2, . . . ,rI′ and l = 1, 2, . . . , (rII′)/2. Furthermore, the Hermitian dual of C in (35) is of the form C⊥H ∼=   rI′∏ j=1 C⊥Hj  ×  (rII′)/2∏ l=1 ( (D′l) ⊥E ×D⊥El ) . The characterization of Hermitian self-dual A × Zps-quasi-abelian codes in Fpm [A × Zps × B] in term of a product of linear codes, and Hermitian self-dual linear codes over Galois extensions of the ring Fpm + uFpm + · · · + up s−1Fpm is established. Corollary 4.15. Let s and m be positive integers such that m is even. Let A and B be finite abelian groups such that |B| = n and p - |A|. Then an A × Zps-quasi-abelian code in Fpm [A × Zps × B] is Hermitian self-dual if and only if in the decomposition (35), i) Cj is a Hermitian self-dual linear code of length n over Sj for all j = 1, 2, . . . ,rI′, and ii) D′l = D ⊥E l is a linear code of length n over Tl for all l = 1, 2, . . . , (rII′)/2. From Corollary 4.15, it follows that the Hermitian self-duality of A × Zps-quasi-abelian codes in Fpm [A×Zps ×B] depends only on the structure of A×Zps and the index n = |B| but not the structure of B itself. Given a positive integer m and a positive integer j, the pair (j,pm) is said to be oddly good if j divides pmt + 1 for some odd positive integer t. This notion has been introduced in [12] for characterizing the Hermitian self-dual abelian codes in principal ideal group algebra and completely determined in [9]. Let λ be a function defined on the pair (j,pm) as λ(j,pm) = { 0 if (j,pm) is oddly good, 1 otherwise. (36) The number of Hermitian self-dual A×Zps-quasi-abelian codes in Fpm [A×Zps×B] can be determined as follows. 225 P. Choosuwan, S. Jitman / J. Algebra Comb. Discrete Appl. 7(3) (2020) 209–227 Theorem 4.16. Let s and m be positive integers such that m is even. Let A and B be finite abelian groups such that |B| = n is even and the exponent of A is M and p - M. Then the number of Euclidean self-dual A×Zps-quasi-abelian codes in Fpm [A×Zps ×B] is ∏ d|M ( NHps (p m·ordd(pm),ps) )(1−λ(d,p m2 )) NA(d) ordd(p m) × ∏ d|M ( Nps (p m·ordd(pm),ps) )λ(d,p m2 ) NA(d) 2ordd(p m) , where NA(d) denotes the number of elements of order d in A determined in [1]. Proof. By Corollary 4.15, it is enough to determine the numbers linear codes Cj’s and Dl’s of length n in (35) such that Cj is Hermitian self-dual. The result can be deduced using arguments similar to those in the proof of Theorem 4.10, where [12, Lemma 3.5] is applied instead of [11, Lemma 4.5]. Remark 4.17. In general, the number NHps (pm,n) of Hermitian self-dual linear codes of length n over Fpm + uFpm + · · · + up s−1Fpm in Theorem 4.16 has not been well studied. In the case where the field characteristic is 3, we have the following results. 1. The numbers N3(pm,n) and NH3(3m,n) have been determined in Proposition 2.1 and Theorem 3.4. Hence, the complete enumeration of Hermitian self-dual A×Z3-quasi-abelian codes in F3m [A×Z3× B] follows. 2. The construction/characterization of linear and Hermitian self-dual dual linear codes of length n over F3m + uF3m + u2F3m have been given in [4] and in the proof of Proposition 3.3. Hence, the construction/characterization of Hermitian self-dual A×Z3-quasi-abelian code in F3m [A×Z3 ×B] can be obtained from Corollary 4.15. 3. Note that, if n is odd, there are no Hermitian self-dual linear codes of length n over F3m + uF3m + u2F3m by Theorem 3.4. Hence, there are no Hermitian self-dual A × Z3-quasi-abelian codes in F3m [A×Z3 ×B] for all abelian groups B of odd order. 5. Conclusion and remarks By extending the technique used in the study of Euclidean self-dual linear codes over Fq +uFq +u2Fq in [3], complete characterization and enumeration of Hermitian self-dual linear codes over Fq +uFq +u2Fq have been established for all square prime powers q. An algorithm for constructions of such self-dual codes has veen provided as well as an illustrative example. Subsequently, algebraic characterization of H-quasi-abelian codes in Fpm [G] has been studied, where H ≤ G are finite abelian groups and the Sylow p-subgroup of H is cyclic, or equivalently, Fpm [H] is a principal ideal group algebra. In the special case where H ∼= A × Z3 with 3 - |A|, characterization and enumeration of H-quasi-abelian codes and self-dual H-quasi-abelian codes in F3m [H × B] have been completely determined for all finite abelian group B. 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