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 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.
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: , and . Let be a fixed positive integer and
be an -free module with the usual componentwise addition and scalar multiplication by elements of . Then a nonempty subset of is an -submodule of if , for all and . Let be an invertible element in . The -cyclic shift operator on is defined by
for all . Then is an invertible. An -submodule is said to be -invariant if
In coding theory, an -submodule of is called a linear code over of length and every -invariant -submodule of of is called a -constacyclic code. In particular, a -constacyclic code is called a negacyclic code when , and is called a cyclic code when .
Let in which the arithmetic is done modulo . In this paper, -constacyclic codes over of length are identified with ideals of the ring , under the identification map defined by for all and .
The Euclidean inner product on is defined by for all . 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 where . As in , we define and define the Lee weight of by , where
is the Hamming weight of the vector. Then is an isomorphism of -linear space from onto , and can be extended to an isomorphism of -linear spaces from onto by the rule:
for all , where with and .
The following conclusion is derived from Corollary 14 of : 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 .
Therefore, it is an effective way to obtain self-dual and -quasi-cyclic codes over of length from self-dual code over of length .
There were a lot of literatures on linear codes, cyclic codes and constacyclic codes of length over rings () for various prime and positive integers and some positive integer . For example, s1 –s4 , s6 –s17 and s19
. 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 also established in s8 . Dinh s9 classified all constacyclic codes of length over . Dinh et al. s10 studied negacyclic codes of length over the ring . Chen et al. s7 investigated constacyclic codes of length over . Dinh et al. s11 studied constacyclic codes of length over when (mod ). These papers mainly used the methods in s8 and s9 , 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. s12 determined the algebraic structures of all cyclic and negacyclic codes of length over , established the duals of all such codes and given some subclass of self-dual negacyclic codes of length over by Theorems 4.2, 4.4 and 4.9 of s12 . But the representation and enumeration for all self-dual negacyclic codes and all self-dual cyclic codes were not obtained.
In Section 4 of s5 , Choosuwan et al. given an alternative characterization for cyclic codes over of length for any positive integer and all primes . The paper only provided a form for the Euclidean dual code of any cyclic code over of length (see Page 9, Theorem 19 in ). On that basis, the paper provided formulas to count the number of Euclidean self-dual cyclic codes and Hermitian self-dual cyclic codes over of length respectively. For example, in Corollary 22(ii) of , the number of Euclidean self-dual cyclic codes of length over () was
But this statement requires correction. In fact, the above formula is missing a term (which proves to be quite important). In Theorem 2 of this paper, we prove the following correction: the number of Euclidean self-dual cyclic codes of length over () is
It seems that the error was caused by a mistake in taking the geometric sum. But the fundamental reason is that Choosuwan et al.  were not given an explicit representation for each Euclidean self-dual cyclic code of length over .
Recently, in s6 we provided a new way different from the methods used in s7 –s13 to determine the algebraic structures, the generators and enumeration of -constacyclic codes over of length , where is a positive integer satisfying and . Then we given an explicit representation for the dual code of every cyclic code and every negacyclic code. Moreover, we provided a discriminant condition for the self-duality of each cyclic code and negacyclic code over of length .
On the basis of the results obtained in , we can consider to give an explicit representation and enumeration for self-dual cyclic codes and self-dual negacyclic codes over . In this paper, we focus on (Euclidean) self-dual cyclic codes over of length .
The present paper is organized as follows. In Section 2, we review the known results for self-dual cyclic codes of length over first. Then we prove that almost all of these self-dual cyclic codes are determined by a special kind of subsets in the residue class ring for certain integers , . In Section 3, we study the calculation and representation of the set by use of properties for Kronecker product of matrices and calculation of linear equations over . In Section 4, we give an explicit representation and enumeration for self-dual cyclic codes over of length . As an application, we list precisely all distinct self-dual cyclic codes over of length for in Section 5. Section 6 concludes the paper.
In this section, we list the necessary notations and some known results for cyclic codes of length over the ring () needed in the following sections.
Let be a finite field of elements, where is a prime number, and satisfying . Then . As in Section 7 of , we denote ,
and for all integers .
Denote in the following lemma.
Lemma 1 (cf.  Corollary 7.1) Denote , and . Then all distinct -constacyclic codes over of length and their dual codes are given by the following table:
in which and .
Moreover, the number of -constacyclic codes over of length is equal to
Now, we consider how to determine the self-dual cyclic codes of length over the ring (). We set and . Then and . This implies the following:
(), where .
for any .
From this and by Lemma 1, we list all cyclic codes over of length and their dual codes as follows.
Case I. codes:
with and , where .
Case II. codes:
with and , where and .
Case III. codes:
with and , where .
Case IV. codes:
with and , where and .
Case V. codes:
with and , where , and .
As , every self-dual cyclic code over of length must contain codewords. From this, we deduce that there is no self-dual codes in Cases II and IV.
Let be a code in Case I. Then if and only if satisfying , i.e., (mod ).
Let be a code in Case III. Then if and only if , i.e., .
Let be a code in Case III. Then if and only if and satisfying . The latter is equivalent to (mod ), and the former is equivalent to and . When this condition is satisfied, we have .
In the light of the above discussion, we have the following conclusion.
Lemma 2 For any integer , , we denote
Then all distinct self-dual cyclic codes over of length are given by the following three cases:
(ii) , where .
(iii) , where and .
In order to present all cyclic codes over of length explicitly, by Lemma 2 we need to determine the subset of for and where .
3 Calculation and representation of the set
In this section, we consider how to calculate effectively and represent the subset of defined in Lemma 2, where .
For any matrix over , let be the transposition of . In the rest of this paper, we adopt the following notation
Then we have
In order to present the set of polynomial, it is sufficiency to determine the set .
Let and be matrices over of sizes and respectively. Recall that the Kronecker product of and is defined by which is a a matrix over of size . Then we denote
Theorem 1 Using the notation above, we have the following conclusions.
(i) We have with .
(ii) Let and assume be the least positive integer such that Let
be the identity matrix of order, and be the submatrix in the upper left corner of , i.e.,
, where is a matrix over of size .
Especially, we have . Then is the solution space over of the following homogeneous linear equations:
Moreover, we have , i.e., .
(iii) Let and , where . Then each vector can be calculated recursively by the following three steps:
Step 1. Choose column vector and set .
Step 2. Solving the following linear equations over :
where and , we obtain solution vectors
Step 3. Set
Therefore, for any integer we have the following conclusions:
, where is a fixed -linear combination of for all .
, where is a fixed -linear combination of for all .
Proof. Let be an integer with . By (mod ), we obtain
For simplification and clarity of the expression, in the following we denote