Skew Generalized Cyclic Code over R[x_1;σ_1,δ_1][x_2;σ_2,δ_2]

07/13/2019
by   Shikha Patel, et al.
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This article studies two dimensional skew generalized cyclic (SGC) codes, i.e., SGC codes over the ring R[x_1;σ_1,δ_1][x_2;σ_2,δ_2], where R is a ring with unity. Here, we discuss the structural properties of SGC codes, and a sufficient number of examples of MDS codes are given to show the importance of our results. Also, we improve BCH lower bounds for the minimum distance of SGC codes with non-zero derivation.

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1 Introduction

Coding theory is mainly the study of the methods for efficient and accurate transfer of information from one place to another place. In coding theory, linear codes have been studying for the last six decades. Initially, linear codes were studied over the binary field. In , the study of codes over finite rings was initiated by Blake [1]. The cyclic codes are the most important class of linear codes from an implementation point of view and also playing a preminent role in the development of algebraic coding theory. In , Hartmann and Tzeng [7] studied the BCH lower bounds for the minimum distance of a cyclic code.

In , Boucher et al. [3] introduced skew cyclic codes which is a generalization of the cyclic codes over a non-commutative ring, namely skew polynomial ring , where is a field, and is an automorphism on . Besides the commutative case, skew polynomial rings have many applications in the construction of algebraic codes with better parameters. In , Boucher and Ulmer [6] obtained some skew codes with Hamming distance more significant than the known linear codes with the same parameters. Again, in , they [5] revisited the linear codes by using the concept of the skew polynomial ring with derivation. Later on, in , Cuitio and Tironi [13] studied the structural properties of skew generalized cyclic (SGC) codes over finite fields and also obtained some BCH type lower bounds for their minimum distance.

The two-dimensional cyclic code was introduced by Ikai et al. and Imai [8, 10] in . It is a generalization of the cyclic code. Inspired by these works, in this article, we study structural properties of the two-dimensional skew generalized cyclic (SGC) codes and obtain the BCH lower bound for the minimum distance of a skew generalized cyclic code for non-zero derivation. The two-dimensional theory is beneficial for the analysis and generation of two-dimensional periodic arrays. It gives a construction method for the two-dimensional feedback shift register with a minimum number of storage devices that generates a given two-dimensional periodic array.

Presentation of the manuscript is as follows: Section contains some preliminaries while Section discusses SGC codes and their properties. In Section , we give some BCH lower bounds which generalize the known result of [13] for non-zero derivation. To show the importance of our results, in Section , we provide some examples of MDS codes in and also MDS codes for SGC codes over , respectively. Section concludes the work.

2 Preliminaries

Let be a ring with unity and an endomorphism of . A map is said to be a -derivation if and for all . Let be a Galois field with elements where for some prime and . Consider the Frobenius automorphism defined by , which fixes the subfield with elements and its order is .

Definition 2.1.

[9, 11, 12] A pseudo-linear map is an additive map defined by

(1)

where , , is an matrix with coordinates in and . If , then is called a semi-linear map.

Consider . Then is a noncommutative ring under usual addition of polynomials and multiplication is defined with respect to for any . This ring is also known as skew polynomial ring.

We consider the iterated skew polynomial ring over the ring . Let be a skew polynomial ring over the ring , where is an automorphism and is a -derivation of . If is an endomorphism and is a -derivation of , then the skew polynomial ring is called an iterated skew polynomial ring over . It is denoted by .
Let and we consider the finite sets where is a endomorphism of and , where is the -derivation of , for . Then by induction, we define the iterated skew polynomial ring over . We denote it by . If has only identity automorphism, then is denoted by (iterated skew polynomial ring of derivation type over ) and if has only zero derivation, then the iterated skew polynomial ring is .

Let be the lexicographical order on . Now, for any , we have if and only if or , . Further, if and , then we write and is a total order on . For any polynomial , we define

contains a term .

Recall that the lex-degree of , denoted by , is the greatest element of with respect to the total order on . The lexicographical order on is defined as follows:
For any , if and only if . The lex-leading term of a polynomial is the term of corresponding to its lex-degree. The lex-leading coefficient of is the coefficient of its lex-leading term.

Now, we present some basic results over iterated skew polynomial ring.

Lemma 2.1.

[15] Let be a ring and a derivation on . Let be the skew polynomial ring of derivation type over and be another derivation of . Then can be extended to a derivation of by if and only if commutes with .

Theorem 2.1.

[15] Let be a ring and a finite set of derivations of . Let be the set of all polynomials in indeterminates with coefficients in , for , where . In , addition is usual and multiplication is defined with respect to for all in , for . Then is a skew polynomial ring (of derivation type) over , for all if and only if , for all .

Definition 2.2.

[8] A binary two-dimensional code of area is the set of arrays over , called codewords or code arrays. A two-dimensional code is said to be linear if and only if form a subspace of the -dimensional space of the arrays over . A two-dimensional cyclic code is defined as a two dimensional linear code such that for each code-array of , all the arrays obtained by permuting the columns or the rows of cyclically are also code-arrays.

We fix a monic polynomial over the ring

(2)

where .

Define a map the

-linear map which maps a vector

to the class , i.e., for all , where

, and is an -arrays, i.e., is written as

The map is -linear isomorphism between -modules. Now, we give some definitions about codes in .

Definition 2.3.

Let be a nonempty set. Then

  1. is a -linear code if is a left -submodule of the left -module .

  2. Let be a linear code over of length whose codewords are viewed as arrays, i.e., is written as

    If is closed under row -skew-shift and column -skew-shift of codewords, then is a -dimensional skew-cyclic code of size over under and .

  3. is an -skew code for if is an -linear code such that , where is given in equation and , is of order and is of order . Moreover, , and , , where and are of the following form:

    and

  4. is an -skew code if is an -linear code such that , where is given in equation with is of order and is of order defined as above and .

Definition 2.4.

Let . The Hamming weight of a codeword is the number of non-zero components. For any two codewords and of the Hamming distance is defined as . The Hamming distance for the code is

Let and be two elements of . Then the Euclidean inner product of and in is . The dual code of is . A code is called self-orthogonal if and self dual if .

3 Skew generalized cyclic (SGC) codes and its properties

In this section, we give some algebraic properties of two-dimensional skew generalized cyclic (SGC) codes. In , Cuitio and Tironi [13] studied the structural properties of skew generalized cyclic codes over . We generalize these results over .

Remark 3.1.

[13] Let be any pseudo-linear transformation on . If is a polynomial, then is not in general a pseudo-linear transformation. Also, we have for every and such that . This implies that the map is a left -module isomorphism when is a -module with the product for any and is endowed with the left -module structure given by the left action of , i.e., for any .

Based on above facts, we have the following characterization of -skew code.

Theorem 3.1.

Let be a non-empty subset of . Then is a -skew code if and only if is a left -submodule of .

Now, we give the definition of skew generalized cyclic code over .

Definition 3.1.

Let be a nonempty set. Then is said to be a -skew generalized cyclic code if , where is a monic polynomial such that . The polynomial is called the generator polynomial of . Notationally, we have .

Let be a -skew generalized cyclic code with . We know that is a free left -module of dimension with basis

where and if , then its generator matrix is given by

(3)

where , i.e., is an arrays which is given as

It is well-known that . Therefore, . Now, we have the following proposition for the dual code of -skew generalized cyclic code .

Proposition 3.1.

Let be as in equation and let be its associated pseudo-linear transformation as in equation . Suppose that . If for some monic skew polynomials and , then linearly independent columns of the matrix given by

form a basis of , where , and .

Proof.

Let , where

and and also , for some . Then . This implies . Therefore, . This shows that for any . Further, we note that

where lexdeg and lexdeg. Therefore, are linearly independent. ∎

When is commutative, we have the following theorem related to the dual code of -skew generalized cyclic code for .

Theorem 3.2.

Suppose is commutative and is a -derivation, where for . If is a -skew generalized cyclic code, then is a -skew code with -derivation for , where for a matrix we denote and its transpose matrix.

Proof.

We define two pseudo-linear maps and by and for for every , respectively. For any and , we have This implies that , i.e., , for . Therefore, we have , i.e., for all , and . ∎

4 BCH lower bounds for the minimum distance of a SGC code with non-zero derivation

Let be the Galois field, for is a prime and . In , Cuitio and Tironi [13] investigated for the BCH type lower bounds for the minimum distance of a skew generalized cyclic (SGC) code for , i.e, for zero derivation, where is an automorphism.

The following definition [5] generalizes the classical notion of the norm of a field element: for is recursively defined as

In particular, if , we get the classical norm

In this section, we generalize this BCH lower bound for the minimum distance of the SGC codes [13] under certain conditions for , where is a -derivation. Further, we generalize the result and improve this lower bound for the SGC codes. We consider Frobenius automorphism by , where . We assume that , where and . Here, we consider a map by for all , where

, i.e., . The map is an -linear isomorphism between -modules. We have the following lemmas that are used in proof of the main result.

Lemma 4.1.

In for non-zero derivation, we have and , where and