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. An effective way for the construction of self-dual codes is the use of some specific algebraic structures.
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 Then is an -free module with the usual componentwise addition and scalar multiplication by elements of . Let be an -submodule of and be an invertible element in . Then is called a linear code over of length . Moreover, is called a -constacyclic code if
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 .
There were a lot of literature on linear codes, cyclic codes and constacyclic codes of length over rings () for various prime , positive integer and some positive integer (see s1 , s2 , s3 , s4 and s6 –s18 , for examples).
Specifically, all constacyclic codes of length over the Galois extension rings of
were classified and their detailed structures was also established ins9 . Dinh s10 classified all constacyclic codes of length over . Then negacyclic codes of length , constacyclic codes of length and constacyclic codes of length ( (mod )) over were investigated by Dinh et al. s11 , Chen et al. s8 and Dinh et al. s12 , respectively. We note that the representation and enumeration for self-dual cyclic codes and self-dual negacyclic codes were not studied in these papers.
Dinh et al. s13 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 by Theorems 4.2, 4.4 and 4.9 of s13 . But the representation and enumeration for all self-dual negacyclic codes and all self-dual cyclic codes were not given.
Choosuwan et al. s5 done the following:
In pages and , they proved that every (Euclidean) self-dual cyclic code over of length is given by
where and satisfying
in which is an matrix over defined by
Also Dinh et al. determined the number of self-dual cyclic codes of length over () by Section 4 of DM2018 .
But they didn’t give a method how to solve the equation and didn’t obtain an representation for solutions of this equation in s5 and the equation (2.1) in DM2018 . So they didn’t provide an explicit representation for all distinct self-dual cyclic codes over of length .
In s6 , we provided a new way different from the methods used in s8 –s14 to determine the algebraic structures, generators and enumeration of -constacyclic codes over of length , where is an arbitrary positive integer satisfying and . Then we gave 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 s6 , we can consider to give an explicit representation for self-dual cyclic codes and self-dual negacyclic codes over .
Recently, by a new way different from that of s5 , we s7 gave an efficient method for the construction of all distinct self-dual cyclic codes with length over . In particular, we provide an exact formula to count the number of all these self-dual cyclic codes and corrected a mistake in Corollary 22(ii) of s5 . However, the methods and results of s7 depend heavily on that the characteristic of the field is . They can’t be used directly to the case for self-dual cyclic codes with length over where is odd. Hence we need to develop a new approach to the latter situation.
The present paper is organized as follows. In Section 2, we review the known results for self-dual cyclic codes of length over and prove that 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 give an explicit representation of the set by studying properties for Kronecker product of matrices over with a specific type. In Section 4, we provide an efficient method to construct and represent all distinct self-dual cyclic codes of length over precisely. As an application, we list all distinct self-dual cyclic codes over of length for in Section 5. Section 6 concludes the paper.
In this section, we list some known results for cyclic codes of length over the ring () needed in the following sections.
By Corollary 7.1 in s6 , every cyclic code over of length and its dual code are given by the following five cases.
Case I. codes:
with and , where .
Case II. codes:
where and .
Case III. codes:
with and , where .
Case IV. codes:
where and .
Case V. codes:
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, III and IV.
Let be a code in Case I. Then if and only if satisfying , i.e.,
Let be a code in Case V. Then if and only if and satisfying . The latter is equivalent to
The former is equivalent to that
In the light of the above discussion, we have the following conclusion.
Proposition 1 For any integer , , we denote
Then all distinct self-dual cyclic codes over of length are given by the following two cases:
(i) , where .
where and .
In order to present all self-dual cyclic codes over of length explicitly, by Proposition 1 we need to determine the following subsets of :
Let and be matrices over of sizes and respectively. The Kronecker product of and is defined by which is a a matrix over of size . For any positive integer , we define a lower triangular matrix over as follows
In fact, we have (cf. s5 ). Precisely, we have
(mod ), where . Moreover, we have the following property for the matrix .
Proposition 2 Let be any positive integer and set . Then
Proof. Let and , where . As is odd, we have . Then by Lucas’s Theorem for a combinatorial identity in number theory (see Benjamin , for examples), we have
From these, by Equation (2),
and , we deduce
This implies that .
For examples, we have where , and
3 Calculation and representation of the set
In this section, we consider how to calculate and represent the subset of defined by Equation (1), where .
For any matrix over and positive integer , let be the transpose of and
be the identity matrix of order. In the rest of this paper, we adopt the following notation
Then we have
In order to present the subset of polynomials in , it is equivalent to determine the set .
Theorem 1 Let and assume be the least positive integer such that Let be the submatrix in the upper left corner of defined by
where is a lower triangular matrix over of size . Then we have the following conclusions.
(ii) , and .
(iii) Denote (mod ). For any , we denote . Then
be the column vectors of, i.e., for all and . Then is an -basis of . Precisely, we have and
Proof. (i) First, we prove that over . Since is a lower triangular matrix of order , is a lower triangular matrix and its -entry is
by Equation (2) in Section 2. Now, let and where . By the definition of and the following combinatorial identities
and , the -entry of is equal to
As stated above, we conclude that . Now, let and assume that (mod ). Then by Propostion 2, it follows that
(ii) By (i) and Equation (4) for the definition of , it follows that
This implies , and hence . From this and by linear algebra theory, we deduce that
Since is a lower triangular matrix of order with diagonal entries:
by Equation (4) we see that is a lower triangular matrix with nonzero diagonal entries:
and is a lower triangular matrix with nonzero diagonal entries:
These imply and . From these and by , we deduce that and .
(iii) In the following, we denote
for any integer : . Then . By (mod ) in , it follows that
From this and by Equation (2), we deduce that
for all . These imply
by the definition of the matrix in Section 2.