Two-dimensional Cyclic Codes Research Articles

On generating sets of 2D cyclic codes of length 2ks and corrections to some previous results

In this paper, we first study when two-dimensional cyclic codes of length [Formula: see text] have generating sets of a certain form. Then we deduce that several of the main results of the papers [Z. Rajabi and K. Khashyarmanesh, Repeated-root two-dimensional constacyclic codes of length [Formula: see text], Finite Fields Appl. 50 (2018) 122–137] and [Z. Sepasdar and K. Khashyarmanesh, Characterizations of some two-dimensional cyclic codes correspond to the ideals of [Formula: see text], Finite Fields Appl. 41 (2016) 97–112] are not accurate. We also present counterexamples for these results and find other conditions needed for them to hold.

A note on two-dimensional cyclic and constacyclic codes

During the study of the two-dimensional cyclic (TDC) codes of length $n=ls$ over a finite field $\mathbb{F}_q$ where $s=2^k$, Sepasdar and Khashyarmanesh (2016, [11]) arose a problem that the technique used by them to characterize TDC codes of length $n=ls$ does not work for TDC codes of length $3l$. It naturally motivates us to study the TDC codes of other lengths together with $3l$. Further, $(\lambda_1,\lambda_2)$-constacyclic codes are the generalization of constacyclic codes. Thus, we study two-dimensional cyclic codes of length $3l$ and $(\lambda_1,\lambda_2)$-constacyclic codes of length $2l$, respectively over finite fields. Here, the generating set of polynomials for these two-dimensional codes and their duals are obtained. Finally, with the help of our derived results, we have constructed many MDS codes corresponding to the two-dimensional codes.

1-generator two-dimensional quasi-cyclic codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$

In this paper, we obtain generating set of polynomials of two-dimensional cyclic codes over the ring $R=\mathbb{Z}_4[u]/\langle u^2-1\rangle$, where $u^2=1$. Moreover, we find generator polynomials for two-dimensional quasi-cyclic codes and two-dimensional generalized quasi-cyclic codes over $R$ and specify a lower bound on minimum distance of free 1-generator two-dimensional quasi-cyclic codes and two-dimensional generalized quasi-cyclic codes over $R$.

Finding a generator matrix of a multidimensional cyclic code

We generalize Sepasdar's method for finding a gene-\rator matrix of two-dimensional cyclic codes to find a generating set and a linearly independent subset of a general multicyclic code. From these sets, a basis of the code as a vector subspace can be deduced or constructed. A generator matrix can be then deduced from this basis.

Characterizations of some two-dimensional cyclic codes correspond to the ideals of [formula omitted

Two-dimensional cyclic code is one of the natural generalizations of cyclic code. In this paper we study the algebraic structure of some two-dimensional cyclic codes and their dual codes.

Two-Dimensional Error-Pattern-Correcting Codes

Two-dimensional (2D) cyclic codes are presented which correct any single occurrence of known 2D error patterns within a 2D array of bits. Applications for this type of codes include storage and display devices. The code construction begins with a generation of distinct syndrome sets for all targeted 2D error patterns. A method to refine the syndrome sets is then presented for making each syndrome set to contain distinct members, thereby guaranteeing full correction capability for the given list of known error patterns. Using an example construction, the effectiveness of the proposed coding approach is demonstrated versus the maximum-distance-separable (MDS) random-error-correcting code and known 2D burst-correcting codes for a 2D intersymbol interference (ISI) channel that yields a few dominant, but relatively large error patterns.

Information sets in abelian codes: defining sets and Groebner basis

In Bernal and Simon (IEEE Trans Inf Theory 57(12):7990---7999, 2011) we introduced a technique to construct information sets for every semisimple abelian code by means of its defining set. This construction is a non trivial generalization of that given by Imai (Inf Control 34:1---21, 1977) in the case of binary two-dimensional cyclic (TDC) codes. On the other hand, Sakata (IEEE Trans Inf Theory IT-27(5):556---565, 1981) showed a method for constructing information sets for binary TDC codes based on the computation of Groebner basis which agrees with the information set obtained by Imai. Later, Chabanne (IEEE Trans Inf Theory 38(6):1826---1829, 1992) presents a generalization of the permutation decoding algorithm for binary abelian codes by using Groebner basis, and as a part of his method he constructs an information set following the same ideas introduced by Sakata. In this paper we show that, in the general case of q-ary multidimensional abelian codes, both methods, that based on Groebner basis and that defined in terms of the defining sets, also yield the same information set.

On toric codes and multivariate Vandermonde matrices

Toric codes are a class of m-dimensional cyclic codes introduced recently by Hansen (Coding theory, cryptography and related areas (Guanajuato, 1998), pp 132–142, Springer, Berlin, 2000; Appl Algebra Eng Commun Comput 13:289–300, 2002), and studied in Joyner (Appl Algebra Eng Commun Comput 15:63–79, 2004) and Little and Schenck (SIAM Discrete Math, 2007). They may be defined as evaluation codes obtained from monomials corresponding to integer lattice points in an integral convex polytope $$P \subseteq {\mathbb{R}}^m$$. As such, they are in a sense a natural extension of Reed–Solomon codes. Several articles cited above use intersection theory on toric varieties to derive bounds on the minimum distance of some toric codes. In this paper, we will provide a more elementary approach that applies equally well to many toric codes for all $$m \ge 2$$. Our methods are based on a sort of multivariate generalization of Vandermonde determinants that has also been used in the study of multivariate polynomial interpolation. We use these Vandermonde determinants to determine the minimum distance of toric codes from simplices and rectangular polytopes. We also prove a general result showing that if there is a unimodular integer affine transformation taking one polytope P 1 to a second polytope P 2, then the corresponding toric codes are monomially equivalent (hence have the same parameters). We use this to begin a classification of two-dimensional cyclic toric codes with small dimension.

Artin–Schreier curves and weights of two-dimensional cyclic codes

Let F q be the finite field with q elements of characteristic p, F q m be the extension of degree m>1 and f( x) be a polynomial over F q m . The maximum number of affine F q m -rational points that a curve of the form y q − y= f( x) can have is q m+1 . We determine a necessary and sufficient condition for such a curve to achieve this maximum number. Then we study the weights of two-dimensional (2-D) cyclic codes. For this, we give a trace representation of the codes starting with the zeros of the dual 2-D cyclic code. This leads to a relation between the weights of codewords and a family of Artin–Schreier curves. We give a lower bound on the minimum distance for a large class of 2-D cyclic codes. Then we look at some special classes that are not covered by our main result and obtain similar minimum distance bounds.

A BOUND ON THE NUMBER OF RATIONAL POINTS OF CERTAIN ARTIN-SCHREIER FAMILIES

ABSTRACT Let for some and consider the extension with . We consider families of curves of the form . We call such families Artin-Schreier families, even though not every curve in need be an Artin-Schreier curve. It is easy to see that the members of such a family can have at most affine -rational points. Using a well-known coding theory technique, we determine the condition under which can attain this bound and we obtain some simple, but interesting, corollaries of this result. One of these consequences shows the existence of maximal curves of Artin-Schreier type. Our main result is important for minimum distance analysis of certain two-dimensional cyclic codes.

Doubly-periodic sequences and a class of two-dimensional cyclic codes

In this paper we continue our study of doubly-periodic sequences. It is shown that the set of eventually doubly-periodic sequences over a finite field has the structure of a Hopf algebra and that it factors as the tensor product of the Hopf algebra of eventually singly periodic sequences with itself. As an application of this result, we generalize a previous theorem by introducing a coalgebra structure for a space of doubly-periodic sequences (which solve a two-dimensional recurrence) and decomposing it as a sum of products of coalgebras of singly periodic sequences. The result is further applied to the well-known construction of two-dimensional cyclic product codes using cyclic ordering yielding an induced coalgebra structure for these codes.

Decoding binary 2-D cyclic codes by the 2-D Berlekamp-Massey algorithm

A method of decoding two-dimensional (2-D) cyclic codes by applying the 2-D Berlekamp-Massey algorithm is proposed. To explain this decoding method, the author introduces a subclass of 2-D cyclic codes, which are called 2-D BCH codes due to their similarity with BCH codes. It is shown that there are some short 2-D cyclic codes with a better cost parameter value. The merit of the approach is verified by showing several simple examples of 2-D cyclic codes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Cycle representatives of quasi-irreducible two-dimensional cyclic codes

The author presents a method of finding the cycle representatives of any quasi-irreducible (QIR) 2-D cyclic code by extending A.P. Kurdjukov's (Probl. Peredach. Inform., vol.12, no.4, p.107-8, 1976) result on quasi-irreducible (i.e. nonsquare-free) 10D cyclic codes. The algorithm is not strictly deterministic in the sense that it is necessary to obtain a set of representative arrays for the code by a trial-and-error method. The result is useful for finding the cycle representatives of any 2D cyclic code by combining QIR components with the aid of G. Sequin's (1974) method to the case where the symbol field is the binary Galois field GF(2). In particular, the result is useful for determining the weight distribution of any two-dimensional cyclic code. >

The concatenated structure of cyclic and Abelian codes

It is shown that a linear two-dimensional cyclic code of length nN can be factorized into a direct sum of concatenated codes, with cyclic inner and outer codes and, conversely, thai a two-dimensional cyclic code can be constructed in this way. This result is extended, and it appears that the Abelian codes are obtainable by taking a direct sum of several concatenations of cyclic codes. Codes are constructed that are better than any previously known. In particular, low-rate cyclic codes superior to the duals of high-rate BCH codes are constructed.

On determining the independent point set for doubly periodic arrays and encoding two-dimensional cyclic codes and their duals

For the purpose of encoding two-dimensional cyclic (TDC) codes, an effective algorithm for finding the independent point set of an arbitrary module of doubly periodic (DP) arrays is proposed. In addition, a method for determining the characteristic ideal of a given set of DP arrays is exhibited. With the aid of these methods it is possible to specify the structure of the generator and check ideals of TDC codes. By applying this algorithm to nonsemisimple binary TDC codes with small areas, several optimal linear codes have been found and their previously unknown TDC structures have been exhibited.

A theory of two-dimensional cyclic codes

A basic theory of two-dimensional cyclic codes (TDC codes) is presented. This theory is applicable to the binary TDC codes of odd area, i.e., the TDC codes with an odd number of symbols from GF(2). It is shown that such TDC codes are completely characterized by the common zeros, which correspond to the roots for ordinary cyclic codes. The fundamental structure of the codes such as the positions of the check symbols is examined via the common zeros. Using the results, we show that encoding of TDC codes are readily implemented by two-dimensional feedback shift registers.

Two-dimensional Fire codes

To improve the reliability of two-dimensional information, codes that can correct two-dimensional bursts (or spots) may be useful. In this paper a class of two-dimensional burst-correcting codes, called two-dimensional Fire codes, is proposed. The definition of these codes is a natural extension of that of the conventional Fire codes. The two-dimensional Fire code is a two-dimensional cyclic code designed for single two-dimensional burst correction. Several important properties such as the burst-correcting capability and the positions of the check symbols are presented. Also, encoding and decoding methods are presented. It is shown that the encoding and decoding are easily implemented by using two-dimensional feedback shift registers.

A theory of two-dimensional linear recurring arrays

In this paper, two-dimensional arrays of elements of an arbitrary finite field are examined, especially arrays having maximum-area matrices. We first define two-dimensional linear recurring arrays. In order to study the characteristics of two-dimensional linear recurring arrays, we also define two-dimensional linear cyclic codes. A systematic method of constructing two-dimensional linear recurring arrays having maximum-area matrices is given using the theory of two-dimensional cyclic codes. These arrays, here called \gamma \beta -arrays, may be said to be two-dimensional analogs of M -sequences. A \gamma \beta -array of area N_x \times N_y exists over GF(q) if and only if N_x N_y is equal to q^N _ 1 for some positive integer N . Many interesting characteristics of the \gamma \beta -array, such as the properties of its autocorrelation function and the properties of the characteristic arrays, are deduced and explained.