The class of self-dual codes is an interesting topic in coding theory due to their connections to other fields of mathematics such as Lattices, Cryptography, Invariant Theory, Block designs, etc. A common theme for the construction of self-dual codes is the use of a computer search. In order to make this search feasible, special construction methods have been used to reduce the search field. In recent years, one of the important construction methods is to use left ideals in a finite group algebra over finite fields and finite rings (see , for an example).
A linear code is said to be self-dual if . Binary self-dual codes are called Type II if the weights of all codewords are multiple of and Type I otherwise. Type II codes are said to have weights that are doubly-even as well. It is well-known that the upper bound for minimum distance of a binary self-dual code of length is
A self-dual binary code is called extremal if it meets the bound.
Let be a finite field of elements and be a finite group. The group algebra is an -algebra with basis . Addition, multiplication with scalars and multiplication are defined by:
for any and . Then is a noncommutative ring with identity where and is the identity elements of and respectively. It is known that is semisimple if and only if .
In this paper, let
be a dihedral group of order . For any , , we define
Then is an isomorphism of -linear spaces from onto . As a natural generalization of Dutra et al. , a nonempty subset of the -linear space is called a left dihedral code (or left -code for more clear) over if is a left ideal of . We will equate with in this paper.
There have been many research results on codes as two-sided ideals and left ideals in a finite group algebra over finite fields. For example, Dutra et al  investigated codes that are two-sided ideals in a semisimple finite group algebra , and given a criterion to decide if these ideals are all the minimal two-sided ideals of when is a dihedral group. Brochero Martínez  shown all central irreducible idempotents and their Wedderburn decomposition of the semisimple dihedral group algebra when every divisor of divides . Moreover, we gave a system theory for left -codes over finite fields in  where , and obtained a complete description for left -codes over Galois rings in  were .
McLoughlin  provided a new construction of the self-dual, doubly-even and extremal [48,24,12] binary linear block code using a zero divisor in the dihedral group algebra . Dougherty et al.  and  gave constructions of self-dual and formally self-dual codes from group rings where the ring is a finite commutative Frobenius ring. They shown that several of the standard constructions of self-dual codes are found within this general framework. Additionally, they shown precisely which groups can be used to construct the extremal Type II codes of length 24 and 48.
One of the most studied open questions in coding theory is to ask whether there is an extremal doubly-even binary self-dual codes of length a multiple of . There are still many problems worth studying in this field. For examples,
For which does there exists a doubly-even self-dual binary code (Open Question 7.7 in )?
Now, we study binary self-dual codes in a new way different from the methods used in ,  and . We give an explicit construction and enumeration for all distinct self-dual binary left -codes in this paper. These codes make up an important class of binary self-dual codes of length a multiple of . In future work, we will try to determine extremal doubly-even binary self-dual codes of length a multiple of among these codes.
In this paper, let be a binary field and , where and are nonnegative integers such that is odd.
is odd. Thenwhere
The present paper is organized as follows. In section 2, we give a concatenated structure of any binary left -code. Precisely, we show that every binary left -code can be decomposed into a direct sum of concatenated codes, where the inner code is a binary cyclic code of length and the outer code is a special linear code over finite commutative chain rings of length . In Section 3, we provide an explicit representation and enumeration for all distinct binary left -codes. Specifically, we give a unique generator matrix of the outer code for each concatenated code in the decomposition of any binary left -code. In Section 4, we determine the dual code for each code and give an explicit representation for all distinct self-dual binary left -codes. In Section 5, we give recursive algorithms to solve problems in the construction of self-dual binary left -codes and obtain a clear formula to count the number of all these self-dual codes. In Section 6, we list all distinct self-dual binary left -codes for , and provide extremal self-dual binary codes with parameters , , , , respectively. Section 7 concludes the paper.
Ii Concatenated structure of left -codes
In this section, we introduce the necessary notations and sketch the results of finite commutative chain rings first. Then we give a concatenated structure of any binary left -code.
For any nonzero polynomial of degree , the reciprocal polynomial of is defined by
and is said to be self-reciprocal if .
As is an odd positive integer, we have that
where are pairwise coprime irreducible polynomials in such that
There are nonnegative integers and such that
with degree .
is self-reciprocal and of degree for all .
is not self-reciprocal, and for all .
It is clear that and
This implies . In this paper, we denote
where we regard elements of as polynomials in of degree and the arithmetic is done modulo .
where we regard elements of as polynomials in of degree and the arithmetic is done modulo , for all .
The rings () are important roles in this paper and their structures can be found in many dispersive literature.
(cf.  Example 2.1) Using the notations above, denote for any and . Then we have the following conclusions.
(i) is a finite chain ring, is the unique maximal ideal of , the nilpotency index of is and .
(ii) Let . Then every element of has a unique -expansion:
(iii) We can regard elements of as polynomials in of degree and the arithmetic is done modulo . Hence .
For each , denote . Then and are coprime polynomials. Hence there are polynomials such that
This implies and
In the rest of this paper, let satisfying
From classical ring theory and the Chinese remainder theorem, we deduce the following lemma (cf.  Lemma 3.2).
(i) , and for all in the ring .
(ii) , where with as its multiplicative identity. Moreover, this decomposition is a ring direct sum in that for all .
(iii) For each , the map
is an isomorphism of rings from onto .
(iv) For any , , define
Then is a ring isomorphism from the direct product ring onto .
As usual, we equate each vectorwith . Then binary cyclic codes of length are identified with ideals of the ring . In particular, we have the following properties for the ideal of .
Let . Then
(i) is a binary cyclic code of length with parity check polynomial and generating idempotent .
(ii) As a binary linear code of length ,
is a basis of . Hence .
(ii) Since is an -linear space with a basis , by Lemma II.2(iii) we see that , is an -basis of .
Now, let be a linear code of length over , i.e. is an -submodule of . For each , we denote by
the Hamming weight of and define the minimum Hamming distance of as
As a natural generalization of the concept for concatenated codes over finite field (cf. , Definition 2.1), using the notations of Lemma II.2(iii) we define the concatenated code of the inner code and the outer code by
By Lemma II.2(iii), we conclude that is a binary quasi-cyclic code of length and index and the number of codewords is equal to . This implies
and the minimum Hamming distance of satisfies
where is the minimum Hamming weight of as a binary linear code of length .
As the end of this section, we consider the structure of binary left -codes. As
is a cyclic subgroup of with order generated by . It is obvious that the group algebra is equal to the residue class ring . Hence is a subring of and
in which , . Now, we define a map by
Then one can easily verify that is an -module isomorphism from onto .
Let be a nonempty subset of . Then is a left ideal of , i.e. is a binary left -code, if and only if is an -submodule of and for any . From this and by
we deduce that
Let . Then . Hence is a binary left -code if and only if there is a unique -submodule of satisfying the following condition:
such that . We will equate with in this paper.
Every binary left -code can be uniquely decomposed as the following:
where , , is a linear code of length over the finite chain ring satisfying the following condition
(i) If , satisfies
(ii) If , the pair of linear codes is given by
where is an arbitrary linear code of length over .
Moreover, the number of codewords in is .
For any integer , , denote
Now, we claim that
In fact, by we have
and . From these, we deduce that and
By Equation (1), it follows that
From these and by Equation (2), we deduce that
Therefore, Equation (4) was proved.
Using the notations in Lemma II.2(iii) and (iv), for any , , we define
It is clear that is an -module isomorphism from onto . Now, let be an -submodule of . Then for each integer , , there is a unique -submodule of such that
It is obvious that .
Moreover, for any integer , , and , let where
By Equation (4), in the ring we have
From this and by , we deduce that
Therefore, we have one of the following two cases:
(i) Let . In this case, we have and hence satisfies Condition (3).
(ii) Let . we have and . In this case, and satisfy the above conditions if and only if and is an arbitrary linear code over of length .
In the rest of this paper, we call the canonical form decomposition of the binary left -code .
By Theorem II.4, in order to list all distinct binary left -codes we just need to solve the following questions:
Question 1. Determine all linear codes of length over for each .
Question 2. Determine all linear codes of length over satisfying Condition (3) for each .
We will solve these two questions in the next section.
Iii Representation and enumeration for binary left -codes
In this section, we solve the two questions at the end of Section II first. Then we obtain an explicit representation and enumeration for binary left -codes.
Let and . By Lemma II.1(ii), has a unique -adic expansion:
If , we define the -degree of as the least index for which and denote as