Additive codes were first introduced by Delsarte in 1973 as subgroups of the underlying abelian group in a translation association scheme. In the case where the association scheme is the Hamming scheme, that is, when the underlying abelian group is of order 2 n , the additive codes are of the form Z 2 α × Z 4 β with α + 2 β = n . In 2010, Borges et al. introduced Z 2 Z 4 -additive codes which they defined them as the subgroups of Z 2 α × Z 4 β . In this chapter we introduce Z 2 Z 2[u]-linear and Z 2 Z 2[u]-cyclic codes where Z 2 = 0 1 is the binary field and Z 2 u = 0 1 u 1 + u is the ring with four elements and u 2 = 0 . We give the standard forms of the generator and parity-check matrices of Z 2 Z 2[u]-linear codes. Further, we determine the generator polynomials for Z 2 Z 2 u -linear cyclic codes. We also present some examples of Z 2 Z 2[u]-linear and Z 2 Z 2[u]-cyclic codes.
- Z2Z2[u]-linear codes
- cyclic codes
- generator matrix
- parity-check matrix
- minimal spanning set
In coding theory, the most important class of error-correcting codes is the family of linear codes, because the encoding and decoding procedures for a linear code are faster and simpler than those for arbitrary nonlinear codes. Many practically important linear codes have also an efficient decoding. Specifically, a linear code of length
2. 2 2[
Let be the finite field and , be the finite ring of four elements. Since is a subring of , we define the following set:
This set is not well defined with respect to the usual multiplication by . So it is not an -module. Hence the set cannot be endowed with an algebraic structure directly. Therefore we introduce a new multiplication to make it well defined and enriched with an algebraic structure.
Let ; then
as . It is easy to see that the mapping is a ring homomorphism. Now, using this map we define the following -scalar multiplication on . For any element :
This new multiplication is well defined and also can be extended over as follows. Let and ; then define
Note that the ring is isomorphic to as an additive group. Therefore, any -linear code is isomorphic to a group of the form , for some . Now let us consider the following sets:
where if , then is called free over β:
Now, denote the dimension of , , and as
where . The Gray map is an isometry which transforms the Lee distance in to the Hamming distance in . The Hamming and the Lee distance between two codewords is the Hamming weight and the Lee weight of their differences, respectively. The Hamming weight of a codeword is defined as the number of its non-zero entries, and the Lee weights of the elements of are defined as . It is worth mentioning that the Gray map is linear, i.e., for a -linear code , we have as a binary linear code which is not the case for -additive codes in general. We can extend the definition of the Lee weight of a codeword in to the Lee weight of a codeword as follows:
where is the Hamming weight of and is the Lee weight of . Further, the minimum distance of the -linear code , denoted by , is naturally defined as
where . If is a -linear code of type , then Gray image is a binary linear code of length and size . It is also called a -linear code.
2.1 Generator matrices of [
A generator matrix for a linear code is the matrix
where and are matrices with all entries from and and are identity matrices with given sizes. Further has codewords.
with all binary entries. By applying the necessary row operations to the above matrix, we have the desired form.
Then multiplying the second row by
is of type .
Moreover, the Gray image of is a simplex code of length 7 with parameters [7, 3, 4] which is the dual of the well-known [7, 4, 3] Hamming code.
2.2 Duality on [
u]-linear codes and parity-check matrices
In the literature, there is a very well-known concept for the duals of the codes over finite fields and rings. If is a linear code over , the dual code of in is the set of all codewords that are orthogonal to every codeword of . A generator matrix for is called a parity-check matrix of . In this part, we determine the standard form of the parity-check matrix of a -linear code . Let us begin with the definition of an inner product over .
Further, the dual code of a -linear code is defined in the usual way with respect to this inner product as
Hence, if is a -linear code, then is also a -linear code. It is worth mentioning that any two codewords of a -linear code may be orthogonal to each other, but the binary parts of the codewords may not be orthogonal. For example, are orthogonal to each other, whereas the binary or -components are not orthogonal. Moreover, the Gray map preserves the orthogonality.
We give the standard form of the parity-check matrices of -linear codes with the following theorem.
Therefore, is of type and has codewords. The Gray image is a well-known Hamming code with parameters .
3. 2 2[
u]-linear cyclic codes
Cyclic codes form a very small but highly structured and important subset of the set of linear codes. In general, these codes are much easier to implement, and hence they have a very rich algebraic structure that allows them to be encoded and decoded in a relatively easier way. Since cyclic codes can be identified as ideals in a certain ring, they are also of considerable interest from an algebraic point of view. Cyclic codes over finite fields were first introduced by E. Prange in 1957 and 1959 with two Air Force Cambridge Research Laboratory reports. In this section we study the structure of 2 2[
Now, let . Since is cyclic, then , and . Therefore,
Hence, and so is also cyclic.
Let and . can be identified with a module element consisting of two polynomials each from different rings in such that
This identification gives a one-to-one correspondence between elements in and elements in .
This multiplication is well defined, and moreover, is a -module with respect to this multiplication.
The codewords of may be represented as polynomials in by using the above identification. Thus, if is a cyclic code, then the element can be viewed as
Further, the property translates to
Hence we give the following theorem.
3.1 The generators and the spanning sets of [
u]-linear cyclic codes
Let be a -linear code. We know that both and [
It is clear that is a module homomorphism where the is an [
Further the kernel of is
Now, define the set
It is clear that
Now, let . So, we have , and hence for some polynomial . Therefore , and this implies that is a submodule of generated by one element of the form with . Then by the First Isomorphism Theorem, we have
Let such that . This discussion shows that any -linear cyclic code can be generated as a [
where . Since the polynomial can be restricted to a polynomial in , we can write
with binary polynomials and where and .
Therefore . On the other hand,
So, and hence .
Note that for a -linear cyclic code , if , then is a free -module. However, if and , then it is not a free -module. But we can still present with the minimal spanning sets. The following theorem determines the minimal spanning sets for a -linear cyclic code .
where and . Then forms a minimal spanning set for as an -module. Furthermore, has codewords.
Therefore, we have and . Hence by using the minimal spanning sets in Theorem 3.1.3, we can write the generator matrix for the -linear cyclic code as follows:
It is worth mentioning that the Gray image of is a linear binary code with the parameters [21, 5, 10], which are optimal. If the code has the best minimum distance compared to the existing bounds for fixed length and the size, then is called optimal or good parameter code.
Again by using the minimal spanning sets in the above theorem, we have the following generator matrix for :
The Gray image of is a [25, 11, 4] linear binary code. Moreover we can write the standard form of this generator matrix as
Hence is of type and has codewords.
In this chapter we introduced 2 2[
The author would like to thank professors Irfan Siap and Taher Abualrub for their valuable comments and suggestions to improve the quality of the chapter.
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