Hermitian self-dual quasi-abelian codes

Quasi-abelian codes constitute an important class of linear codes containing theoretically and practically interesting codes such as quasi-cyclic codes, abelian codes, and cyclic codes. In particular, the sub-class consisting of 1-generator quasi-abelian codes contains large families of good codes. Based on the well-known decomposition of quasi-abelian codes, the characterization and enumeration of Hermitian self-dual quasi-abelian codes are given. In the case of 1-generator quasi-abelian codes, we offer necessary and sufficient conditions for such codes to be Hermitian self-dual and give a formula for the number of these codes. In the case where the underlying groups are some p-groups, the actual number of resulting Hermitian self-dual quasi-abelian codes are determined. 2010 MSC: 94B15, 94B60, 16A26


Introduction
Quasi-cyclic codes form an important class of linear codes due to their rich algebraic structures, large number of codes with good parameters, and various applications (see [9], [10], [11], [12], [14], [17], and references therein).Let F q denote a finite field of order q.It is known that quasi-cyclic codes of length ml and index l over F q can be regarded as F q [Z m ]-submodules of the F q [Z m ]-module (F q [Z m ]) l , where Z m denotes the cyclic group of order m and F q [Z m ] is the group algebra of Z m over F q (see [10]).
In a more general setting, quasi-abelian codes are defined by replacing Z m with a finite abelian group.Particularly, if G is a finite abelian group and H ≤ G, then an H-quasi-abelian code is defined to be an F q [H]-submodule of the F q [H]-module F q [G].This class of codes was first introduced in [18] and further studies of their properties have been made in [4, Section 7] and [1].More recently in [6], via the Herbert S. Palines; Institute of Mathematical Sciences and Physics, University of the Philippines Los Baños, College, Laguna 4031, Philippines, and Institute of Mathematics, College of Science, University of the Philippines Diliman, Quezon City 1101, Philippines (email: herbertpalines@gmail.com).
Discrete Fourier Transform, the structural characterization of quasi-abelian codes have been established together with the existence of asymptotically good quasi-abelian codes.Quasi-abelian codes serve as the general case for quasi-cyclic codes (if H = G is cyclic), abelian codes (if H = G), and cyclic codes (if H = G is cyclic).Since the theory of quasi-abelian codes generalizes that of quasi-cyclic codes, a link can be established between 1-generator quasi-abelian codes and irreducible or minimal cyclic codes which plays a central role in the theory of cyclic codes [2].
Self-dual codes form another fascinating family of codes and are known to be closely related with other objects such as lattices and possess variety of practical applications (see [13]).Moreover, both Euclidean and Hermitian self-dual codes have close connection with quantum stabilizer codes [8].In [6], the authors presented necessary and sufficient conditions for quasi-abelian codes to be Euclidean self-dual and gave enumeration of those codes based on the classification of q-cyclotomic classes of the underlying group.Moreover, they have shown that some class of binary Euclidean self-dual strictly-quasi-abelian codes are asymptotically good.
To the best of our knowledge, no study has been done yet on Hermitian self-dual quasi-abelian codes.It is therefore of natural interest to investigate such family of codes and compare the result of this study with that of [6].In this work, considering finite abelian groups H ≤ G, we offer sufficient and necessary conditions for an H-quasi-abelian code in F q [G] to be Hermitian self-dual using similar decomposition given in [6, Section 3] (see Proposition 2.3).Consequently, enumeration of Hermitian self-dual H-quasi-abelian codes is presented (see Corollary 3.1).In similar fashion, the sufficient and necessary conditions for a 1-generator quasi-abelian code to be Hermitian self-dual are obtained (see Corollary 4.3).Enumeration of Hermitian self-dual 1-generator quasi-abelian codes is also given.In the case H ∼ = (Z p k ) s is a p-group, p is a prime, k > 0 and s > 0, we classify completely the q-cyclotomic classes of H (see Propositions 3.6 and 3.10) which lead to the actual number of the resulting Hermitian self-dual H-quasi-abelian codes.The asymptotic goodness of Hermitian self-dual strictly-quasi-abelian codes over F 2 2s is guaranteed by [6,Section 7] since every code over F 2 2s with generator matrix containing only elements from F 2 is Hermitian self-dual if and only if such a matrix generates a Euclidean self-dual code over F 2 .
The paper is organized as follows.In Section 2, we recall notations and definitions which are essential to this work as well as the well-known decomposition of semi-simple group algebras.Enumeration of Hermitian self-dual quasi-abelian codes, where the underlying groups are some p-groups, is established in Section 3. Finally in Section 4, we focus on the characterization and enumeration of Hermitian self-dual 1-generator quasi-abelian codes.

Preliminaries
For a prime power q and positive integer n, let F q denote a finite field of order q and let G be a finite abelian group of order n, written additively.Denote by F q [G] the group algebra of G over F q .The elements in F q [G] will be written as g∈G α g Y g , where α g ∈ F q .The addition and the multiplication in ] are given as in the usual polynomial rings over F q with the indeterminate Y , where the indices are computed additively in G.As convention, Y 0 is treated as the multiplicative identity of F q [G], where 0 is the identity of G.
Let R be a finite commutative ring with unity.A linear code of length n over R is defined to be an R-submodule of R n .A (linear) code C in F q [G] refers to an F q -subspace of F q [G].This can be viewed as a linear code of length n over F q by indexing the n-tuples by the elements of G.For more details, the reader is referred to [6].
]-module, i.e., C is closed under addition and multiplication by the elements in F q [H].If H is a non-cyclic subgroup of G, then we say that C is a strictly-quasi-abelian code.If it is clear in the context or if H is not specified, such a code will be called simply a quasi-abelian code.An H-quasi-abelian code C is said to be of 1-generator if C is a cyclic F q [H]-module.
Let {g 1 , g 2 , . . ., g l } be a fixed set of representatives of the cosets of where α i (Y ) = h∈H α h+gi Y h ∈ R, for all i = 1, 2, . . ., l.It is well known that Φ is an R-module isomorphism interpreted as follows.
Lemma 2.1.The map Φ induces a one-to-one correspondence between H-quasi-abelian codes in F q [G] and linear codes of length l over R.
From this point, we assume q = q 2 0 , where q 0 is a prime power.Consequently, the Hermitian inner product of u and v is defined as u, v H := n i=1 u i v i , where ¯is the automorphism on F q defined by α → α q0 for all α ∈ F q .For a code C of length n over F q , let C ⊥E and C ⊥H denote its Euclidean dual and Hermitian dual, respectively.The code C is said to be Euclidean (resp., Similarly, the code C in F q [G] is said to be Hermitian self-dual if C ⊥H = C.Note that without confusion, we use the symbol ⊥ H to indicate both the Hermitian dual of a code over F q and the Hermitian dual of a code in F q [G].All throughout, the self-duality of quasi-abelian codes is studied with respect to the given Hermitian inner product in F q [G].

Decomposition and Hermitian dual codes
The main tool of this work appears in this subsection.The idea is to have a convenient decomposition of quasi-abelian codes using the well-known decomposition of semi-simple group algebras introduced in [16].Then, combining this technique with the results of [7, Proposition 2.7] and [6, Proposition 4.1], we obtain characterization of Hermitian self-dual quasi-abelian codes (see Proposition 2.3).This will lead to enumeration of such class of codes.
For completeness, we discuss the concepts of q-cyclotomic classes and primitive idempotents as appeared in [7, Section II.C].Given coprime positive integers i and j, the multiplicative order of j modulo i, denoted by ord i (j), is defined to be the smallest positive integer s such that i divides j s − 1.For each a ∈ H, denote by ord(a) the additive order of a in H.
From this point, we assume that gcd(|H|, q) = 1.A q-cyclotomic class of H containing a ∈ H, denoted by S q (a), is defined to be the set For a positive integer r and a ∈ H, denote by −r • a the element r • (−a) ∈ H.A q-cyclotomic class S q (a) is said to be of type I if S q (a) = S q (−q 0 • a) and it is of type II if S q (−q 0 • a) = S q (a).Clearly, S q (0) is a q-cyclotomic class of type I.
An idempotent in a ring is a non-zero element e such that e 2 = e, and it is called primitive idempotent if, for every other idempotent f , either ef = e or ef = 0.The primitive idempotents in R := F q [H] are induced by the q-cyclotomic classes of H (see [5,Proposition II.4]).
Assume that H contains t q-cyclotomic classes.Without loss of generality, let {a 1 = 0, a 2 , . . ., a t } be a set of representatives of the q-cyclotomic classes of H such that {a i | i = 1, 2, . . ., r I } and {a r I +j , a r I +r II +j = −q 0 • a r I +j | j = 1, 2, . . ., r II } are sets of representatives of q-cyclotomic classes of types I and II, respectively, where t = r I + 2r II .Let {e 1 , e 2 , . . ., e t } be the set of primitive idempotents of R induced by {S q (a i ) | i = 1, 2, . . ., t}, respectively.It is well known that Re i is isomorphic to an extension field of F q of degree |S q (a i )| for each i = 1, 2, . . ., t.
In [16], R := F q [H] is decomposed in terms of e i 's.Later, the components in the decomposition of R are rearranged in [7] and obtain the following. where , and K j ∼ = Re r I +r II +j are finite extension fields of F q for all i = 1, 2, . . ., r I and j = 1, 2, . . ., r II .
From (1), we have where the isomorphisms are R-module isomorphisms.They can be viewed as F q -linear isomorphisms as well.Consequently, every quasi-abelian code C in F q [G] can be viewed as where C i , D j and D j are linear codes of length l over E i , K j , and K j , respectively, for all i = 1, 2, . . ., r I and j = 1, 2, . . ., r II .
Using arguments similar to the proofs of [7, Proposition 2.7] and [6, Proposition 4.1], it can be concluded that the Hermitian dual of C is of the form From ( 3) and (4), we have the following necessary and sufficient conditions for quasi-abelian codes to be Hermitian self-dual.
] is Hermitian self-dual if and only if, in the decomposition (3), i) C i is Hermitian self-dual for all i = 1, 2, . . ., r I , and ii) D j = D ⊥E j for all j = 1, 2, . . ., r II .

Enumeration of Hermitian self-dual quasi-abelian codes
In this section, we enumerate Hermitian self-dual quasi-abelian codes by using the decomposition in (3), Proposition 2.3 and the following formulas.Let N (q, l) (resp., N H (q, l)) denote the number of linear codes (resp., Hermitian self-dual codes) of length l over F q .It is well known (see [15] and [13]) that where the empty product is set to be 1.
In general, to count the number of Hermitian self-dual quasi-abelian codes in F q [G], in (3), we count the number of Hermitian self-dual codes C i of length l over F q s i for all i = 1, 2, . . ., r I and multiply it with the number of all possible linear codes D j of length l over F q t j for all j = 1, 2, . . ., r II .This technique is clear in the following corollary.Hereafter, the numbers s i , t j , and t j will appear frequently in the succeeding results.If needed, the reader is referred back to Remark 2.2 for the definitions of s i , t j , and t j .
Corollary 3.1.Let H ≤ G be finite abelian groups such that gcd(|H|, q) = 1 and l = [G : H].Assume that F q [H] contains r I (resp., 2r II ) primitive idempotents of type I (resp., II).Assume further that the primitive idempotents of type I are induced by q-cyclotomic classes of size s i for each i = 1, 2, . . ., r I and the primitive idempotents of type II are induced by q-cyclotomic classes of sizes t j and t j , pair-wise, for each j = 1, 2, . . ., r II .Then the number of Hermitian self-dual We note that S q (0) is a q-cyclotomic class of H of type I. Then r I ≥ 1, and hence, the product i=1 N H (q si , l) = 0 for all odd positive integers l.Hence, there are no Hermitian self-dual H-quasi-abelian codes if l = [G : H] is odd.Therefore, we have the following result derived from ( 6) and (7).Lemma 3.2.There exists a Hermitian self-dual H-quasi-abelian code in F q [G] if and only if the index l = [G : H] is even.Remark 3.3.From Lemma 3.2, it is apparent that given a finite abelian group G and q = q 2 0 , the existence of Hermitian self-dual H-quasi-abelian codes in F q [G] depends only on the choice of H, particularly on index l being even.
In the theory of quasi-cyclic codes, it is practical to use a relatively small fixed value of the index l mainly for the purpose of efficient decoding [3].Moreover, this case contains the known case of double circulant codes (see [10, Section VI.A] and [12, Section II.A]).Since the theory of quasi-abelian codes generalizes that of quasi-cyclic codes, we can adopt those concepts.Note that a quasi-cyclic code is cyclic when l = 1.Thus l = 2 is the smallest index such that a code is quasi-cyclic.Specifically for l = 2, one can talk about self-dual 1-generator quasi-abelian codes (see Section 4).Consider the example below for the number of quasi-abelian codes of index 2.
Example 3.4.Let H ≤ G be finite abelian groups such that gcd(|H|, q) = 1 and l = [G : H] = 2. Assume that F q [H] contains r I (resp., 2r II ) primitive idempotents of type I (resp., II).Assume further that the primitive idempotents of type I are induced by q-cyclotomic classes of size s i for each i = 1, 2, . . ., r I and the primitive idempotents of type II are induced by q-cyclotomic classes of sizes t j and t j , pair-wise, for each j = 1, 2, . . ., r II .Then the number of Hermitian self-dual H-quasi-abelian codes of index 2 in F q [G] is r II j=1 (q tj + 3).
In the next two subsections, we consider the case where the subgroups H of G are some p-groups.It is interesting to see that for this particular case, the cardinality and the number of q-cyclotomic classes of H can be explicitly determined.Hence, one can obtain the actual number of resulting Hermitian self-dual H-quasi-abelian codes.In this regard, we offer sufficient and necessary conditions for a q-cyclotomic class of H to be of type I or type II.
The succeeding discussion is instrumental in determining the explicit forms of r I and r II .Let H ∼ = (Z p k ) s , where k and s are positive integers, and p is prime such that gcd(p, q) = 1.Define Consequently, we have |H 1 | = 1 and, via inclusion-exclusion principle, for each i = 1, 2, . . ., k. Recall that q = q 2 0 where q 0 is a prime power.Hereafter, let ν p i := ord p i (q) and µ p i := ord p i (q 0 ), Now, consider the case where q is odd and p = 2, i.e., H ∼ = (Z 2 k ) s .Suppose h ∈ H 2 .Since ord(h) = 2 for all h ∈ H 2 , q ≡ ±1 (mod ord(h)) and q 0 ≡ ±1 (mod ord(h)), then we have h = q•h = q 0 •h = q 0 •(−h) = −q 0 • h.Then S q (h) = S q (−q 0 • h) is of type I and having cardinality equal to 1.For the case where h ∈ H 2 i , 2 ≤ i ≤ k, we have the same result.Suppose h ∈ H 2 i , for a given 2 ≤ i ≤ k, and assume S q (h) is of type I. Then |S q (h)| = ν 2 i is odd (see [7, Remark 2.6 (2)]).Moreover, the elements of H 2 i are partitioned into q-cyclotomic classes of the same type and size (see [7, Remark 2.5 (ii)]).Thus, Furthermore, it can be shown that µ 2 i = 2 for all i = 2, 3, . . ., k.Note that 2 i | (q − 1) since ν 2 i = 1 and thus, 2 i | (q 2 0 − 1).We show that indeed, µ 2 i = 2. Suppose contrary, i.e., µ 2 i = 1 = ν 2 i .It implies that q 0 • h = h and −h = −q 0 • h = q • h = h, since S q (h) is assumed to be of type I.It implies that h = 0 or ord(h) = 2 which contradicts that h ∈ H 2 i , i = 2, 3, . . ., k.We state these observations in the following lemma.
In the next proposition, we give the necessary and sufficient conditions for a q-cyclotomic class of H to be of type I or type II.Since all q-cyclotomic classes in H 2 i are of the same type and size, we characterize the q-cyclotomic classes of H through its subsets H 2 i , for 0 ≤ i ≤ k, keeping in mind that S q (h) is always of type I, for all h ∈ H 1 ∪ H 2 .Proposition 3.6.Let h ∈ H 2 i , for a given 0 ≤ i ≤ k.Then S q (h) is of type I if and only if q 0 ≡ −1 (mod 2 i ).Equivalently, S q (h) is of type II if and only if q 0 ≡ −1 (mod 2 i ).
On the other hand, assume q 0 ≡ −1 (mod 2 i ).Thus, for each h ∈ H 2 i , −q 0 • h = h ∈ S q (h).Hence, S q (h) is of type I. Remark 3.7.Using Proposition 3.6, we can completely classify the sets H 2 i , 0 ≤ i ≤ k, that contain q-cyclotomic classes of type I or type II.Choose the largest integer 0 ≤ r ≤ k such that 2 r |(q 0 + 1).Hence, by Proposition 3.6 H 2 i contains q-cyclotomic classes of type I for all i = 0, 1, . . ., r and the rest of the sets H 2 j contain elements of type II, for j = r + 1, . . ., k.This will lead to a decomposition of Let r be a positive integer as described in Remark 3.7.Since ν 2 i = 1 for all 0 ≤ i ≤ r , then Thus, from (1), this will give the following decomposition, Similar with (3), every H-quasi-abelian code C in F q [G] can be written as where C i , D r,j and D r,j are linear codes of length l over F q , F q ν 2 r and F q ν 2 r , respectively, for i = 1, 2, . . ., 2 r s , r = r + 1, . . ., k, and j = 1, 2, . . ., (2 rs − 2 (r−1)s )/2ν 2 r .Given the decomposition of C in (8), we deduce the next proposition.

The number of Hermitian self-dual H-quasi-abelian codes in
Table 3.1 illustrates Proposition 3.8 when q = 9, l = 2, for k = 1, 2 and s = 1, 2. Note that in the last column, A • B gives the number of the resulting codes.Moreover, since the value of k ≤ 2 and q 0 = 3, then r = k, for k = 1, 2. Hence, the second factor in the formula given by B is empty and set to be 1.In other words, all cyclotomic classes of H is of type I, for k = 1, 2. In this case, the numbers in the last column of the table also gives the number of Hermitian self-dual 1-generator H-quasi-abelian codes as presented in Corollary 4.5 (i).

Table 1. Number of Hermitian self-dual H-quasi-abelian codes in Fq
and q = 9.To complete our characterization, consider H ∼ = (Z p k ) s , k, s > 0, where p is an odd prime and gcd(p, q) = 1.Recall that in the case p = 2, there is a chance that the q-cyclotomic classes of H are divided exactly into classes of type I and type II.It is interesting to note that it is a totally different situation when p is odd.Specifically, we show that all non-zero elements in H belong to just one type of q-cyclotomic classes.Moreover, the necessary and sufficient conditions for them to be of type I or type II are determined.Recall that H p i is the set containing all elements of H of order p i , i = 0, 1, . . ., k and We start with H p the characterization of q-cyclotomic classes of H. Proposition 3.10.Let h ∈ H p .Then S q (h) is of type I if and only if ord p (q) is odd and ord p (q 0 ) is even.Equivalently, S q (h) is of type II if and only if ord p (q) is even or ord p (q 0 ) is odd.
Proof.Following the notation introduced above, let which implies that h = 0, a contradiction.Hence, ord p (q 0 ) = 2ν p , which is even.
Next, we show that all q-cyclotomic classes of H \ {0} are of the same type.Because of this, the q-cyclotomic classes of H are completely characterized.Proposition 3.11.Let a ∈ H p and b ∈ H p i , for any given 1 ≤ i ≤ k.Then, S q (a) is of type I if and only if S q (b) is of type I. Equivalently, S q (a) is of type II if and only if S q (b) is of type II.
Proof.Let a ∈ H p and assume that S q (a) is of type I.Then, by Proposition 3.10, ν p = ord p (q) is odd and µ p = ord p (q 0 ) = 2ν p is even.We show that p i | q νp•p i−1 − 1 by induction on i.It is clear when i = 1.Now, assume p i−1 | q νp•p i−2 − 1 , for 1 < i ≤ k.Then, q νp•p i−2 ≡ 1 (mod p i−1 ) and hence, q νp•p i−2 •j ≡ 1 (mod p i−1 ) for all j ≥ 0. Thus, and means ν p i is odd.Note that µ p i ∈ {ν p i , 2ν p i }.Since µ p is even, ν p i is odd and µ p | µ p i hence, µ p i = 2ν p i .Hence, In other words, q 0 (q S q (a) and S q (p i−1 • b) are of the same type.
Combining Propositions 3.10 and 3.11, the corollary below follows immediately.Corollary 3.12.Let h be a non-zero element in H ∼ = (Z p k ) s , p is odd and gcd(p, q) = 1.Then S q (h) is of type I if and only if ord p (q) is odd and ord p (q 0 ) is even.Equivalently, S q (h) is of type II if and only if ord p (q) is even or ord p (q 0 ) is odd.
We are now ready to obtain a decomposition for F q [H].This entails computing for r I and r II .If there exists h ∈ H \ {0} such that S q (h) is of type I, then by Corollary 3.12, r II = 0 and where ν p 0 = ν 1 = 1 and p is − p (i−1)s is equal to 1 when i = 0. On the other hand, if there exists h ∈ H \ {0} such that S q (h) is of type II, then Corollary 3.12 implies that r I = |H 1 | = 1 and Recall that ν p := ord p (q) and µ p := ord p (q 0 ).From the above calculations, together with Corollary 3.12 and (1), we have if ν p is odd and µ p is even, It also implies that an H-quasi-abelian code C in F q [G] can be decomposed as if ν p is odd and µ p is even, where C 1 and C i,j are linear codes of length l over F q and F q ν p i , respectively, for i = 1, 2, . . ., k and j = 1, 2, . . ., (2 is − 2 (i−1)s )/ν p i .Similarly, both D i,j and D i,j are linear codes of length l over F q ν p i , for i = 1, 2, . . ., k and j = 1, 2, . . ., (2 is − 2 (i−1)s )/2ν p i .The above decomposition of the code C will lead us to the following proposition.
Proposition 3.13.Let H ≤ G be finite abelian groups such that Proof.Apply the same arguments as in the proof of Proposition 3.8 to (9).
An example is given when H ∼ = (Z p ) s is an elementary p-group.
Example 3.14.Let H ≤ G be finite abelian groups such that H ∼ = (Z p ) s , p is odd, gcd(|H|, q) = 1 and the index l = [G : H] is even.Then the number of Hermitian self-dual H-quasi-abelian codes in if ν p is even or µ p is odd.
See Table 3.2 for the number of Hermitian self-dual H-quasi-abelian codes when p = 3, q = 4, l = 2, for k = 1, 2 and s = 1, 2. In this case, ν p = 1 and µ p = 2. Then the q-cyclotomic classes of H are all of type I, and hence, this table also illustrates the 1-generator case given in Corollary 4.5 (ii), type I case.

Hermitian self-dual 1-generator quasi-abelian codes
In this section, we study 1-generator H-quasi-abelian codes in where H ≤ G are finite abelian groups such that gcd(|H|, q) = 1.The main idea here is to use [6, Theorem 6.1] and combine it with the characterization of Hermitian self-dual H-quasi-abelian codes obtained in  Proposition 2.3.We also consider the case where H ∼ = (Z p k ) s , for p = 2 or p is odd, and obtain explicit enumeration.
From [6], we have the following characterization of 1-generator quasi-abelian codes.
Theorem 4.1 ([6, Theorem 6.1]).Let q be a prime power and let H ≤ G be finite abelian groups with l = [G : H] and gcd(|H|, q) = 1.Let e 1 , e 2 , . . ., e t be the primitive idempotents of F q [H].In the light of (3), let be an H-quasi-abelian code in F q [G], where C i is a linear code of length l over L i ∼ = F q [H]e i .Then C is 1-generator if and only if the L i -dimension of C i is at most 1, for each i = 1, 2, . . ., t.
Since the F q -dimension of a 1-generator H-quasi-abelian code C in F q [G] cannot exceed |H|, C ⊥ H could never be a 1-generator if [G : H] > 2. In the case where [G : H] = 2, we have the following characterization.
Corollary 4.2.Assume the notation in Theorem 4.1.In addition, we assume that [G : H] = 2.If C is a 1-generator H-quasi-abelian code in F q [G], then the following statements are equivalent.i) C ⊥H is a 1-generator H-quasi-abelian code.
iii) The F q -dimension of C is |H|.ii) D j is a linear code of dimension 1 and length 2 over K j for all j = 1, 2, . . ., r II .i) If p = 2 and q is odd, then the number of Hermitian self-dual 1-generator H-quasi-abelian codes in F q [G] is ii) If p is odd and gcd(p, q) = 1, then the number of Hermitian self-dual 1-generator H-quasi-abelian codes in F q [G] is    (q 0 + 1)(q νp 0 + 1) if ν p is odd and µ p is even, (q 0 + 1)(q νp + 1) if ν p is even or µ p is odd.

Summary
Characterization and enumeration of Hermitian self-dual quasi-abelian codes were established based on the well-known decomposition of quasi-abelian codes.Necessary and sufficient conditions for the existence of Hermitian self-dual 1-generator quasi-abelian codes were also given.For special cases where the underlying groups are some p-groups, complete classification of cyclotomic classes has been done.As a result, the actual number of resulting Hermitian self-dual quasi-abelian codes has been determined.It is interesting to note that the results in this work is restricted to F q [H] being a semi-simple group algebra, i.e., the characteristic of F q and |H| are coprime, where H is a finite abelian group.

Corollary 4 . 3 .
A 1-generator H-quasi-abelian code C in F q [G] is Hermitian self-dual if and only if [G : H] = 2 (i.e., G = Z 2 × H)and, in (3), C is decomposed as i is Hermitian self-dual of length 2 over E i for all i = 1, 2, . . ., r I , and

Table 2 .
Number of Hermitian self-dual H-quasi-abelian codes in Fq