The homogeneous weight for RkR_kRk, related Gray map and new binary quasicyclic codes (original) (raw)
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Applicable Algebra in Engineering, Communication and Computing, 2013
We consider quasi-cyclic codes over the ring F 2 + uF 2 + vF 2 + uvF 2 , a finite non-chain ring that has been recently studied in coding theory. The Gray images of these codes are shown to be binary quasi-cyclic codes. Using this method we have obtained seventeen new binary quasi-cyclic codes that are new additions to the database of binary quasi-cyclic codes. Moreover, we also obtain a number of binary quasi-cyclic codes with the same parameters as best known binary linear codes that otherwise have more complicated constructions. Keywords Lee weight • Gray maps • Cyclic codes • Quasi-cyclic codes Mathematics Subject Classification (2000) 94B15 • 94B05 • 13P05 1 Introduction Codes over finite rings have received considerable attention in recent years. One of the rings that have been considered recently in coding theory is the ring R 2 = F 2 + uF 2 +
Some new binary quasi-cyclic codes from codes over the ring F2 + uF2+vF2+uvF2
2013
We consider quasi-cyclic codes over the ring F 2 + uF 2 + vF 2 + uvF 2 , a finite non-chain ring that has been recently studied in coding theory. The Gray images of these codes are shown to be binary quasi-cyclic codes. Using this method we have obtained seventeen new binary quasi-cyclic codes that are new additions to the database of binary quasi-cyclic codes. Moreover, we also obtain a number of binary quasi-cyclic codes with the same parameters as best known binary linear codes that otherwise have more complicated constructions. Keywords Lee weight • Gray maps • Cyclic codes • Quasi-cyclic codes Mathematics Subject Classification (2000) 94B15 • 94B05 • 13P05 1 Introduction Codes over finite rings have received considerable attention in recent years. One of the rings that have been considered recently in coding theory is the ring R 2 = F 2 + uF 2 +
On bounds for codes over Frobenius rings under homogeneous weights
Discrete Mathematics, 2004
Homogeneous weight functions were introduced by Heise and Constantinescu (Lineare Codes über Restklassenringen ganzer Zahlen und ihre Automorphismen bezüglich einer verallgemeinerten Hamming-Metrik, Ph.D. Thesis, Technische Universität München, 1995; Problemy Peredachi Informatsii 33(3) (1997) 22-28). They appear as a natural generalization of the Hamming weight on finite fields and the Lee weight on Z 4 and have proven to be important in further papers (J. Combin. Theory 92 (2000) 17-28). This article develops a Plotkin and an Elias bound for (not necessarily linear) block codes on finite Frobenius rings that are equipped with this weight.
Applied Mathematics Letters, 2008
We extend the results of [J.F. Qian, L.N. Zhang, S.X. Zhu, (1 + u)-constacyclic and cyclic codes over F 2 + uF 2 , Appl. Math. Lett. 19 (2006) 820-823. [3]] to codes over the commutative ring R = F p k + uF p k , where p is prime, k ∈ N and u 2 = 0. In particular, we prove that the Gray image of a linear (1 − u)-cyclic code over R of length n is a distance-invariant quasicyclic code of index p k−1 and length p k n over F p k. We also prove that if (n, p) = 1, then every code of length p k n over F p k which is the Gray image of a linear cyclic code of length n over R is permutation-equivalent to a quasicyclic code of index p k−1 .
Codes over an infinite family of rings with a Gray map
Designs, Codes and Cryptography, 2013
Codes over an infinite family of rings which are an extension of the binary field are defined. Two Gray maps to the binary field are attached and are shown to be conjugate. Euclidean and Hermitian self-dual codes are related to binary self-dual and formally self-dual codes, giving a construction of formally self-dual codes from a collection of arbitrary binary codes. We relate codes over these rings to complex lattices. A Singleton bound is proved for these codes with respect to the Lee weight. The structure of cyclic codes and their Gray image is studied. Infinite families of self-dual and formally self-dual quasi-cyclic codes are constructed from these codes.
On gray images of constacyclic codes over the finite ring F₂ + u₁F₂ + u₂F₂
Işık University Press, 2019
We introduce the finite ring F2 + u1F2 + u2F2, u1 2 = u1 , u2 2 = 0 , u1.u2 = u2.u1 = 0 which is not a finite chain ring. We focus on (1 + u2)-constacyclic codes over F2 + u1F2 + u2F2 of odd length. We prove that the Gray image of a linear (1 + u2)-constacyclic code over F2 + u1F2 + u2F2 of odd length n is a quasi-cyclic code of index 4 and length 4n over F2.
Optimal, Divisible binary Codes from Gray-Homogeneous Images of Codes over R_{k,m}
European Journal of Pure and Applied Mathematics, 2017
In this work, we find a form for the homogeneous weight over the ring R_{k,m}, using the related theoretical results from the literature. We then use the first order Reed-Muller codes to find a distance-preserving map that takes codes over R_{k,m} to binary codes. By considering cyclic, constacyclic and quasicyclic codes over R_{k,m} of different lengths for different values of k and m, we construct a considerable number of optimal binary codes that are divisible with high levels of divisibility. The codes we have obtained are also quasicyclic with high indices and they are all self-orthogonal when km\geq 4 The results, which have been obtained by computer search are tabulated.
Quasi-cyclic codes over Z/sub 4/ and some new binary codes
IEEE Transactions on Information Theory, 2002
Recently, (linear) codes over and quasi-cyclic (QC) codes (over fields) have been shown to yield useful results in coding theory. Combining these two ideas we study-QC codes and obtain new binary codes using the usual Gray map. Among the new codes, the lift of the famous Golay code to produces a new binary code, a (92 2 28)-code, which is the best among all binary codes (linear or nonlinear). Moreover, we characterize cyclic codes corresponding to free modules in terms of their generator polynomials.
Quasi-Cyclic Codes over the Field F p
DergiPark (Istanbul University), 2018
In this study, images of cyclic codes in two variable rings with coefficient field p F are detected. A special ring in two variables is defined under certain conditions. The Gray images of the cyclic codes over this ring are investigated. Relations between the codes over this ring and the codes over a finite chain ring in one variable are obtained via a Gray map. Another Gray map from the finite chain ring to a finite field is defined and then the images of cyclic codes are obtained. It is obtained that the Gray image of a cyclic code over R with length n .
Binary quasi-cyclic codes of index 2 and skew polynomial rings
Finite Fields and Their Applications, 2012
We present a study of the factorization of the polynomial X m − 1 in M 2 (F 2 )[X] and we determine the period of any reversible polynomial of this polynomial ring by using skew polynomial rings. These results are applied to the construction of the class of quasicyclic codes Ω(P ) introduced by Cayrel et al. Furthermore, we present a new construction of the self dual subclass.