 # Construction and enumeration for self-dual cyclic codes of even length over F_2^m + uF_2^m

Let F_2^m be a finite field of cardinality 2^m, R=F_2^m+uF_2^m (u^2=0) and s,n be positive integers such that n is odd. In this paper, we give an explicit representation for every self-dual cyclic code over the finite chain ring R of length 2^sn and provide a calculation method to obtain all distinct codes. Moreover, we obtain a clear formula to count the number of all these self-dual cyclic codes. As an application, self-dual and 2-quasi-cyclic codes over F_2^m of length 2^s+1n can be obtained from self-dual cyclic code over R of length 2^sn and by a Gray map preserving orthogonality and distances from R onto F_2^m^2.

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## 1 Introduction

The class of self-dual codes is an interesting topic in coding theory duo 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 many instances, self-dual codes have been found by first finding a code over a ring and then mapping this code onto a code over a subring (subfield) through a map that preserves duality. In the literatures, the mappings typically map to codes over , and since codes over these rings have had the most use (cf. s3 , s4 , s19 s27 ).

Let be a finite field of elements, where is a prime number, and denote Then is a finite chain ring and every invertible element in is of the form: , where and . Let be a fixed positive integer, and denote which is an -free module with the usual componentwise addition and scalar multiplication by elements of . In coding theory, an -submodule of is called a linear code over of length .

The Euclidean inner product on is defined by

for all vectors

. Then the (Euclidean) dual code of a linear code over of length is defined by , which is also a linear code over of length . In particular, is said to be (Euclidean) self-dual if .

Let in which the arithmetic is done modulo . As usual, in this paper we identify cyclic codes over of length with ideals of the ring under the identification map defined by for all . Moreover, is called a simple-root cyclic code if , and called a repeated-root cyclic code otherwise.

Let and where . As in s3 , we define and define the Lee weight of by , where is the Hamming weight of the vector . Then is an isomorphism of -linear spaces from onto , and can be extended to an isomorphism of -linear spaces from onto by:

 ϕ(ξ)=(b0,b1,…,bN−1,a0+b0,a1+b1,…,aN−1+bN−1),

for all , where with and .

The following conclusion is derived from Corollary 14 of s3 : Let be an ideal of and set . Then

(i) is a -quasi-cyclic code over of length .

(ii) The Hamming weight distribution of is exactly the same as the Lee weight distribution of .

(iii) is a self-dual code over of length if is a self-dual code over of length .

Hence it is an effective way to obtain self-dual and -quasi-cyclic codes over of length from self-dual cyclic codes over of length .

There were a lot of literatures on linear codes, cyclic codes and constacyclic codes of length over rings () for various prime , positive integers and some positive integer . For example, s4 , s5 , s8 , s11 , s13 s27

. The classification of self-dual codes plays an important role in studying their structures and encoders. However, it is a very difficult task in general, and only some codes of special lengths over certain finite fields or finite chain rings are classified.

For example, all constacyclic codes of length over the Galois extension rings of was classified and their detailed structures was established in s13 . Dinh s14 classified all constacyclic codes of length over . Dinh et al. s15 studied negacyclic codes of length over . Chen et al. s11 investigated constacyclic codes of length over . Dinh et al. s16 studied constacyclic codes of length over when (mod ). These papers mainly used the methods in s13 and s14 , and the main results and their proofs depend heavily on the code lengths , and . It is particularly important to note that the representation and enumeration for self-dual cyclic codes were not studied in these papers.

Dinh et al. s17 determined the algebraic structures of all cyclic and negacyclic codes of length over , established the duals of all such codes and gave some special subclass of self-dual negacyclic codes of length over . But the representation and enumeration for all self-dual negacyclic codes and all self-dual cyclic codes were not obtained.

Chen et al. s12 gave some new necessary and sufficient conditions for the existence of nontrivial self-dual simple-root cyclic codes over finite commutative chain rings and studied explicit enumeration formulas for these codes, but self-dual repeated-root cyclic codes over finite commutative chain rings were not considered.

In s8 we gave an explicit representation for every self-dual cyclic code over of length and obtained an exact formula to count the number of all these self-dual cyclic codes. Especially, we provided an efficient method for the construction of all distinct self-dual cyclic codes with length over by use of properties for Kronecker product of matrices and calculation for linear equations over .

Recently, in s5 we provided a new way different from the methods used in s11 and s13 s18 to study -constacyclic codes over of length , where arbitrary and is an arbitrary positive integer satisfying . In particular, we obtained the following:

Determined the algebraic structure and generators for each code. On that basis, we obtained many clear enumeration results for all codes.

Gave an explicit representation for the dual code of each -constacyclic code over of length .

Provided a clear distinguish condition (criteria) for the (Euclidean) self-duality of each cyclic code and negacyclic code (corresponds to and , respectively).

Based on this, we consider further to give an explicit representation and enumeration for self-dual cyclic codes and self-dual negacyclic codes over . In this paper, we focus on the case .

The present paper is organized as follows. In Section 2, we provide the necessary notation and preparatory conclusions. Based on this, we give an explicit representation for all distinct self-dual cyclic codes of length over by Theorem 2.6. At the end, we obtain a clear formula to count the number of these self-dual cyclic codes. In most cases, the representation for each code in Theorem 2.6 is strongly dependent to determine a special kind of subsets in the residue class ring , for all irreducible and self-reciprocal divisor of in with degree and integers , . In Section 3, we give an effective algorithm to calculate these sets and obtain a precise representation for them by use of trace functions. In Section 4, we provide a proof for Theorem 2.6. In Section 5, we give a way to calculate the number of self-dual cyclic codes over of length from the positive integers directly. As an application, we list all self-dual cyclic codes over of length precisely. Section 6 concludes the paper.

## 2 Explicit representation for self-dual cyclic codes over R

In this section, we give the result for representing and enumerating all distinct self-dual cyclic codes of length over (). To do this, we introduce the necessary notation and conclusions first.

As is odd, there are distinct monic irreducible polynomials in such that . This implies

 x2sn−1=(xn−1)2s=f1(x)2sf2(x)2s…fr(x)2s.

For any integer , , we assume and denote . Then and . Hence there exist such that . This implies

 vj(x)2sFj(x)2s+wj(x)2sfj(x)2s=(vj(x)Fj(x)+wj(x)fj(x))2s=1.

In this paper, we adopt the following notations, where .

• in which the arithmetic is done modulo .

• Let be defined by  (mod ).

Then , and for all .

• in which the arithmetic is done modulo .

• in which the arithmetic is done modulo . It is well known that is an extension field of with degree , and hence .

• ().

• ().

Remark is a finite field with operations defined by the usual polynomial operations modulo , is a finite chain ring with operations defined by the usual polynomial operations modulo and is a principal ideal ring with operations defined by the usual polynomial operations modulo . In this paper, we adopt the following points of view:

 Fj⊆Kj⊆A and Kj+uKj⊆A+uA as sets.

Obviously, is not a subfield of , is not a subring of and is not a subring of when .

Now, we consider how to determine cyclic codes over of length , i.e. ideals of the ring .

Lemma 2.1 (cf. s5 Lemma 2.2) For any , where with , we define

 Ψ(ξ)=a(x)+ub(x)=∑0≤j≤2sn−1(aj+ubj)xj.

Then is a ring isomorphism from onto .

From now on, we will identify with under the ring isomorphism defined in Lemma 2.1. Then in order to determine all distinct cyclic codes over of length , it is sufficient to give all distinct ideals of the ring . For the latter, we have the following conclusion.

Lemma 2.2 (cf. s5 Theorem 2.7 and Corollary 3.9) Let . Then is an ideal of if and only if for each integer , , there is a unique ideal of the ring such that

 C=ε1(x)C1⊕ε2(x)C2⊕…⊕εr(x)Cr (mod x2sn−1),

where for all . In this case, the number of codewords in is .

All distinct ideals of had been listed by Theorem 3.8 of  for all . Here, we give the structure of its subring .

Lemma 2.3 (cf. s6 Lemma 3.7 and s7 Example 2.1) The ring have the following properties:

(i) is a finite chain ring, generates the unique maximal ideal of , is the nilpotency index of and the residue class field of modulo is .

(ii) We regard as a subset of in the sense of Remark before Lemma 2.1. Then every element of has a unique -adic expansion:

(iii) All distinct ideals of are given by: , . Moreover, for .

(iv) Let . Then , and hence .

(v) For any , we have

 fj(x)l(Kj/⟨fj(x)t⟩)={t−1∑k=lbk(x)f(x)k∣bk(x)∈Fj, k=l,…,t−1}

and , where we set for convenience.

Remark For any integer , , by Lemma 2.3(iv) we can identify with up to a natural ring isomorphism. We will take this view in the rest of this paper. Then for any , we stipulate

 fj(x)l⋅F2m[x]⟨fj(x)t⟩=fj(x)l(Kj/⟨fj(x)t⟩).

For any polynomial of degree , the reciprocal polynomial of is defined as . is said to be self-reciprocal if for some . It is known that if , and for any polynomials with positive degrees satisfying .

As , where are pairwise coprime monic irreducible polynomials in , it follows that

 xn−1=xn+1=˜(xn+1)=˜f1(x)˜f2(x)…˜fr(x),

where are pairwise coprime monic irreducible polynomials in as well. Hence after a rearrangement of , there are integers and such that

;

for some , for all ;

for some , for all .

The following lemma can be easily verified from the definition of self-reciprocal polynomials over .

Lemma 2.4 (cf. s9 Lemma 3.2) Using the notation above, we have the following conclusions, where .

(i) and , for all .

(ii) is even for all .

Let and be matrices over of sizes and respectively. Recall that the Kronecker product of and is defined by which is a matrix over of size . Then we denote

 G2=(1011), G2λ=G2⊗G2λ−1=(G2λ−10G2λ−1G2λ−1), λ=2,3,….

Denote by

the identity matrix of order

. For any , let be the submatrix in the upper left corner of , i.e.,

• , where is a matrix over of size .

Especially, we have .

For any matrix over , let be the transpose of . In the rest of this paper, we adopt the following notation.

• , .
Then and (cf. s8 Theorem 2(i)).

• For any integers and , and , denote

 Ωj,ν={β(x)∈F2m[x]⟨fj(x)ν⟩∣β(x)+x−νdjβ(x−1)≡0 (mod fj(x)ν)}.

We will give an effective algorithm to determine the subset of and prove that in the next section of this paper.

A recursive algorithm to determine the subspace of was given by Theorem 1 in s8 . Moreover, we have the following conclusion for .

Lemma 2.5 (cf. s8 Theorem 2(i)) Using the notations above, let and . Then.

, in which is a fixed -linear combination of , , for all .

, , in which is a fixed -linear combination of , , for all .

, in which is a fixed -linear combination of , , for all .

and .

Now is the time to list self-dual cyclic codes of length over .

Theorem 2.6 Using the notations above, all distinct self-dual cyclic codes of length over are given by

 C=r⨁j=1εj(x)Cj (mod x2sn−1),

where is an ideal of given by the following three cases:

(I) is given by one of the following three subcases.

If , there are ideals:

; where .

If , there are ideals:

;

,
where with
;

where .

If , there are ideals:

.

, where with , .

where .

,
where , and
.

, where , and
.

(II) If