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On the (1+u^2+u^3)-Constacyclic and Cyclic Codes Over the Finite Ring F_2+uF_2+u^2F_2+u^3F_2+vF_2
Abstract
In this paper, a new finite ring is introduced with its algebraic properties
and a new Gray map is defined on the ring. It is obtained that the Gray images of
both the cyclic and the (1+u^2 +u^3)- constacyclic codes over the finite ring are permutation equivalent to binary quasicyclic codes.
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LETTER Linear Codes and (1 + uu)-Constacyclic Codes over R[u]/(u 2 + u) * Jian GAO †a) , Student Member and Fang-Wei FU †b) , Nonmember SUMMARY In this short correspondence, (1 + uv)-constacyclic codes over the finite non-chain ring R[v]/(v 2 + v) are investigated, where R = F 2 +uF 2 with u 2 = 0. Some structural properties of this class of constacyclic codes are studied. Further, some optimal binary linear codes are obtained from these constacyclic codes. key words: finite non-chain ring, Gray map, (1 + uv)-constacyclic codes, optimal binary linear codes
We study (1 + 2v)-constacyclic codes overR + vR and their Gray images, where v2 + v = 0 and R is a finite chain ring with maximal ideal λ> and nilpotency index e. It is proved that the Gray map images of a (1 + 2v)-constacyclic codes of length n over R + vR are distance-invariant linear cyclic codes of length 2n over R. The generator polynomials of this kind of codes for length n are determined, where n is relatively prime to p, p is the character of the field R/λ> . Their dual codes are also discussed.
This paper studies (1 + u)-constacyclic codes over the ring F 2 + uF 2 + vF 2 + uvF 2. It is proved that the image of a (1+u)-constacyclic code of length n over F 2+uF 2+vF 2+uvF 2 under a Gray map is a distance invariant binary quasi-cyclic code of index 2 and length 4n. A set of generators of such constacyclic codes for an arbitrary length is determined. Some optimal binary codes are obtained directly from (1 + u)-constacyclic codes over F 2 + uF 2 + vF 2 + uvF 2. © 2012 Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg.
Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson, Kerdock, Preparata, Goethals, and Delsarte-Goethals. It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z_4, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the Z_4 domain implies that the binary images have dual weight distributions. The Kerdock and "Preparata" codes are duals over Z_4 -- and the Nordstrom-Robinson code is self-dual -- which explains why their weight distributions are dual to each other. The Kerdock and "Preparata" codes are Z_4-analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z_4, which greatly simplifies encoding and decoding. An algebraic hard-decision decoding algorithm is given for the "Preparata" code and a Hadamard-transform soft-decision decoding algorithm for the Kerdock code. Binary first- and second-order Reed-Muller codes are also linear over Z_4, but extended Hamming codes of length n >= 32 and the Golay code are not. Using Z_4-linearity, a new family of distance regular graphs are constructed on the cosets of the "Preparata" code.
In this work, we study constacyclic codes of odd length over finite commutative local Frobenius non-chain rings of order 16. We give the structure of λi-constacyclic codes over each local Frobenius non-chain ring via cyclic codes over these rings where λi acts as a weight preserving unit in the corresponding ring. We also obtain binary optimal codes which are images of these constacyclic codes under a gray map.
A result of J.-F. Qian, L.-N. Zhang and S.-X. Zhu [Constacyclic and cyclic codes over F 2 +uF 2 +u 2 F 2 , IEICE Trans. Fundamentals, E89-A, No. 6 (2006)] is extended to codes over the commutative ring F 2 k +uF 2 k +u 2 F 2 k where k∈ℕ and u 3 =0. A new Gray map between codes over F 2 k +uF 2 k +u 2 F 2 k and F 2 k is defined. It is proved that the Gray image of the linear (1-u 2 )-cyclic code over the commutative ring F 2 k +uF 2 k +u 2 F 2 k of length n is a distance-invariant quasicyclic code of index 2 2k-1 and the length 2 2k n over F 2 k .
We introduce a Gray map from Fp+vFpFp+vFp to Fp2 and study (1−2v)(1−2v)-constacyclic codes over Fp+vFpFp+vFp, where v2=vv2=v. It is proved that the image of a (1−2v)(1−2v)-constacyclic code of length nn over Fp+vFpFp+vFp under the Gray map is a distance-invariant linear cyclic code of length 2n2n over FpFp. The generators of such constacyclic codes for an arbitrary length are determined and their dual codes are also discussed.
Given an integer m which is a product of distinct primes pi, a method is given for constructing codes over the ring of integers modulo m from cyclic codes over GF(pi). Specifically, if we are given a cyclic (n, ki) code over GF(pi) with minimum Hamming distance di, for each i, then we construct a code of block length n over the integers modulo m with πi pkii codewords, which is both linear and cyclic and has minimum Hamming distance minidi.
Linear codes over the ring of integers modulo q = pr, p a prime, are considered. Natural analogs to Hamming, Reed—Solomon, and BCH codes over finite fields are defined and their properties investigated. Some ring theoretic problems encountered are discussed.
- N Aydin
- Y Cengellenmis
- A Dertli
Aydin N., Cengellenmis Y., Dertli A., 2017, On some constacyclic codes over Z 4 [u]/(u 2 − 1),
their Z 4 images, and new codes, Des. Codes Cryptogr., DOI 10.1007/s10623-017-0392-y.
On (1 + u)−cyclic and cyclic codes over F 2 + uF 2 + vF 2
- A Dertli
- Y Cengellenmis
Dertli A., Cengellenmis Y., 2016, On (1 + u)−cyclic and cyclic codes over F 2 + uF 2 + vF 2,
European J. of Pure and Applied Math., 9: 305-313.
+ u)−constacyclic and cyclic codes over F 2 + uF 2
- J F Qian
- L N Zang
- S X Zhu
Qian J.F., Zang L.N., Zhu S.X., 2006, (1 + u)−constacyclic and cyclic codes over F 2 + uF 2,
Appl. Math. Lett., 19: 820-823.
Constacyclic and cyclic codes over F 2 + uF 2 + u 2 F 2
- J F Qian
- L N Zang
- S X Zhu
Qian J.F., Zang L.N., Zhu S.X., 2006, Constacyclic and cyclic codes over F 2 + uF 2 + u 2 F 2,
IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences,
2011, E89-A(6): 1863-1865.