Irfan Siap

Quadratic residue codes over Fp+vFp and their Gray images

Abstract In this paper quadratic residue codes over the ring F p + v F p are introduced in terms of their idempotent generators. The structure of these codes is studied and it is observed that these codes enjoy similar properties as quadratic residue codes over finite fields. For the case p = 2 , Euclidean and Hermitian self-dual families of codes as extended quadratic residue codes are considered and two optimal Hermitian self-dual codes are obtained as examples. Moreover, a substantial number of good p-ary codes are obtained as images of quadratic residue codes over F p + v F p in the cases where p is an odd prime. These results are presented in tables.

Cyclic codes over F2[u]/(u4- 1) and applications to DNA codes

The structure of DNA is used as a model for constructing good error correcting codes and conversely error correcting codes that enjoy similar properties with DNA structure are also used to understand DNA itself. Recently, naturally four element sets are used to model DNA by some families of error correcting codes. Hence the structure of such codes has been studied. In this paper, the authors first relate DNA pairs with a special 16 element ring. Then, the so-called cyclic DNA codes of odd length that enjoy some of the properties of DNA are studied. Their algebraic structure is determined. Further, by introducing a map, a family of cyclic codes over this ring is mapped to DNA codes. Hamming minimum distances are also studied. The paper concludes with some DNA examples obtained via this family of cyclic codes.

Structure and reversibility of 2D hexagonal cellular automata

Cellular automata are used to model dynamical phenomena by focusing on their local behavior which depends on the neighboring cells in order to express their global behavior. The geometrical structure of the models suggests the algebraic structure of cellular automata. After modeling the dynamical phenomena, it is sometimes an important problem to be able to move backwards in order to understand it better. This is only possible if cellular automata is reversible. In this paper, 2D finite cellular automata defined by local rules based on hexagonal cell structure are studied. Rule matrix of the hexagonal finite cellular automaton is obtained. The rank of rule matrices representing the 2D hexagonal finite cellular automata via an algorithm is computed. It is a well known fact that determining the reversibility of a 2D cellular automata is a very difficult problem in general. Here, the reversibility problem of this family of 2D hexagonal cellular automata is also resolved completely.

MacWilliams identity for m-spotty Lee weight enumerators

It is a well established fact that m-spotty byte error control codes provide a good source for detecting and correcting errors in semiconductor memory systems using high-density RAM chips with wide I/O data (e.g. 8, 16 or 32 bits). Recently, a MacWilliams identity that establishes an important relation between an m-spotty weight enumerator of a binary code and its dual has been proven in Kazuyoshi Suzuki et al. (2007) [5]. In this paper, we introduce the m-spotty Lee weights and the m-spotty Lee weight enumerator of a quaternary code and prove a MacWilliams type identity.

Quadratic Residue Codes over F_p+vF_p and their Gray Images

Abstract In this paper quadratic residue codes over the ring F p + v F p are introduced in terms of their idempotent generators. The structure of these codes is studied and it is observed that these codes enjoy similar properties as quadratic residue codes over finite fields. For the case p = 2 , Euclidean and Hermitian self-dual families of codes as extended quadratic residue codes are considered and two optimal Hermitian self-dual codes are obtained as examples. Moreover, a substantial number of good p-ary codes are obtained as images of quadratic residue codes over F p + v F p in the cases where p is an odd prime. These results are presented in tables.

Characterization of two-dimensional cellular automata over ternary fields☆

Abstract The set of papers [3] , [4] , [6] , [7] (Chattopadhyay et al., 1999; Dihidar and Choudhury, 2004; Khan et al., 1997, 1999) deals with the behavior of the uniform two-dimensional cellular automata over binary fields ( Z 2 ). Some structural properties and precise mathematical models using matrix algebra over the field Z 2 are reported for characterizing the behavior of two-dimensional nearest neighborhood linear cellular automata with null and periodic boundary conditions [3] , [4] , [6] , [7] (Chattopadhyay et al., 1999; Dihidar and Choudhury, 2004; Khan et al., 1997, 1999). In this paper, we characterize two-dimensional linear cellular automata transformations by using matrix algebra built on Z 3 . We analyze some results for two-dimensional CA with rule numbers 2460N and 2460P. Finally, we investigate the dimension of the kernel of two-dimensional cellular automata defined by the rule number 2460N.

On ℤ2ℤ2[u]-additive codes

In this paper, a new class of additive codes which is referred to as ℤ2 ℤ2[u]-additive codes is introduced. This is a generalization towards another direction of recently introduced ℤ2 ℤ4-additive codes [J. Borges, C. Fernández-Córdoba, J. Pujol, J. Rif´a, and M. Villanueva, ℤ2 ℤ4-linear codes: Generator matrices and duality, Designs Codes Cryptogr. 54(2) (2010), pp. 167–179]. ℤ2 ℤ4-additive codes have shown to provide a promising class of codes with their algebraic structure and applications such as steganography. The standard generator matrices are established and by introducing orthogonality the parity-check matrices are also obtained. A MacWilliams-type identity that relates the weight enumerator of a code with its dual is proved. Furthermore, a Gray map that maps these codes to binary codes is defined and some examples of optimal codes which are the binary Gray images of ℤ2 ℤ2[u]-additive codes are presented.

Structure of codes over the ring Z_{3}[v]/\langle v^{3}-v\rangle

In this paper, we study the structure of linear codes over the non chain ring

On ℤprℤps-additive codes

Z_{3}[v]/\langle v^{3}-v\rangle