A Class of New Optimal Ternary Cyclic Codes over F3m with Minimum Distance 4

As a branch of applied mathematics, coding theory plays an important role. Among them, cyclic codes have attracted much attention because of their good algebraic structure and easy analysis performance. In this paper, we will study one class of cyclic codes over F3. Given the length and dimension, we show that it is optimal by proving its minimum distance is equal to 4, according to the Sphere Packing bound.

CYCLIC AND NEGACYCLIC CODES OF LENGTH 2p^s OVER F_(p^m) + uF_(p^m)

In this article, we focus on cyclic and negacyclic codes of length 2p^s over the ring R = Fp^m ＋ uFp^m, where p is an odd prime. On the basis of the works of Dinh （in J.Algebra 324,940-950,2010）, we use the Chinese Remainder Theorem to establish the algebraic structure of cyclic and negacyclic codes of length 2p^s over the ring Fp^m ＋ uFp^m in terms of polynomial generators. Furthermore, we obtain the number of codewords in each of those cyclic and negacyclic codes.

CYCLIC CODES OVER FORMAL POWER SERIES RINGS

In this article, cyclic codes and negacyclic codes over formal power series rings are studied. The structure of cyclic codes over this class of rings is given, and the relationship between these codes and cyclic codes over finite chain rings is obtained. Using an isomorphism between cyclic and negacyclic codes over formal power series rings, the structure of negacyclic codes over the formal power series rings is obtained.

PERIOD DISTRIBUTION OF CYCLIC CODES OVER F_q + uF_q +···+u^(m-1)F_q

In this paper, the period distribution of cyclic codes overR = F_q + uF_q +···+u^(m-1)F_q is studied, where um= 0 and q is a prime power. A necessary and sufficient condition for the existence of period of cyclic codes over R is given. The period distributions of cyclic codes over R and their dual codes are determined by employing generator polynomial. The counting formulas of the period distributions of cyclic codes over R and their dual codes are obtained.

SKEW CYCLIC CODES OVER RING F_p+vF_p

In this paper, we study skew cyclic codes over the ring Fp +vFp,where p is a odd prime and v 2=1. We give the generators of skew cyclic codes, with the consideration of the dual of skew cyclic codes.

Quaternion Integers Based Higher Length Cyclic Codes and Their Decoding Algorithm

The decoding algorithm for the correction of errors of arbitrary Mannheim weight has discussed for Lattice constellations and codes from quadratic number fields.Following these lines,the decoding algorithms for the correction of errors of n=p−12 length cyclic codes(C)over quaternion integers of Quaternion Mannheim(QM)weight one up to two coordinates have considered.In continuation,the case of cyclic codes of lengths n=p−12 and 2n−1=p−2 has studied to improve the error correction efficiency.In this study,we present the decoding of cyclic codes of length n=ϕ(p)=p−1 and length 2n−1=2ϕ(p)−1=2p−3(where p is prime integer andϕis Euler phi function)over Hamilton Quaternion integers of Quaternion Mannheim weight for the correction of errors.Furthermore,the error correction capability and code rate tradeoff of these codes are also discussed.Thus,an increase in the length of the cyclic code is achieved along with its better code rate and an adequate error correction capability.

ON THE CHARACTERIZATION OF CYCLIC CODES OVER TWO CLASSES OF RINGS

Let R be a finite chain ring with maximal ideal （7） and residue field F,and letγ be of nilpotency index t. To every code C of length n over R, a tower of codes C = （C ： γ0） C_ （C： 7） C ... C_ （C： γ2） C_ .-. C_ （C：γ^t-1） can be associated with C, where for any r C R, （C ： r） = {e C Rn I re E C}. Using generator elements of the projection of such a tower of codes to the residue field F, we characterize cyclic codes over R. This characterization turns the condition for codes over R to be cyclic into one for codes over the residue field F. Furthermore, we obtain a characterization of cyclic codes over the formal power series ring of a finite chain ring.

THE RANKS OF CYCLIC AND NEGACYCLIC CODES OVER THE FINITE RING R

The ranks of cyclic and negacyclic codes over the finite chain ring R as well as their minimal generating sets are defined, and then the expression forms we presented by studying the structures of cyclic and negacyclic codes over the finite chain ring R. Through the paper, it is assumed that the length of codes n can not be divided by the characteristic of R.

CYCLIC CODES OVER THE RING Z(_P^2) OF LENGTH p^e

The study of cyclic codes over rings has generated a lot of public interest.In this paper,we study cyclic codes and their dual codes over the ring Z P2 of length pe,and find a set of generators for these codes.The ranks and minimal generator sets of these codes are studied as well,which play an important role in decoding and determining the distance distribution of codes.

GAUSS PERIODS AND CYCLIC CODES FROM CYCLOTOMIC SEQUENCES OF SMALL ORDERS

Let p =ef +1 be an odd prime with positive integers e and f. In this paper, we calculate the values of Gauss periods of order e =3, 4, 6 over a finite field GF(q), where q is a prime with q≠p. As applications, several cyclotomic sequences of order e =3, 4, 6 are employed to construct a number of classes of cyclic codes over GF(q) with prime length. Under certain conditions, the linear complexity and reciprocal minimal polynomials of cyclotomic sequences are calculated, and the lower bounds on the minimum distances of these cyclic codes are obtained.

Binary Cyclic Codes and Minimal Codewords

Cyclic codes form an important class of codes. They have very interesting algebraic structure. Furthermore, they are equivalent to many important codes, such as binary Hamming codes, Golay codes and BCH codes. Minimal codewords in linear codes are widely used in constructing decoding algorithms and studying linear secret sharing scheme. In this paper, we show that in the binary cyclic code all of the codewords are minimal, except 0 and 1. Then, we obtain a result about the number of minimal codewords in the binary cyclic codes.

Chinese product of constacyclic and cyclic codes over finite rings

In this paper, we study the Gray images of the Chinese product of constacyclic and cyclic codes over a finite ring. We first introduce the Chinese product of constacyclic and cyclic codes over the finite ring. We then define a Gray map between codes over the finite ring and a finite field. We prove that the Gray image of the Chinese product of constacyclic codes over the finite ring is a distance-invariant quasi-cyclic code over the finite field. We also prove that each code over the finite field, which is the Gray image of the Chinese product of cyclic codes over the finite ring, is permutation equivalent to a quasi-cyclic code.

DNA cyclic codes over the ring F2[u, v]/ （u^2 - 1, v^3 - v, uv - vu）

In this paper, our main objective is to find out the necessary and sufficient conditions for a cyclic code of arbitrary length over the ring of four elements R1 = F2 ＋ u2 （u^2 = 1） to be a reversible cyclic code. We also obtain the structure of cyclic DNA codes of odd length over the ring R = F2 [u, v]/（u^2 -1, v^3 -v, uv- vu）, which plays an important role in Computational Biology. Furthermore, we establish a direct link between the elements of ring /{ and 64 codons used in the amino acids of living organisms by introducing a Gray map from R to R1. Among others, binary images of cyclic codes over R are also investigated. As applications, some cyclic DNA codes over R using the Gray map are provided.

On the Structure of Cyclic Codes over F_q+uF_q+vF_q+uvF_q

In this paper, cyclic codes over the ring R = F ＋ uF ＋ vF ＋ uvF are discussed where the ring R is not a finite chain ring. By studying the polynomial ring Kn = （F ＋ uF ＋ vF ＋ uvF ）[x] / （x^n -1）and using the corresponding relationship between the cyclic codes in R and the ideals in Kn , cyclic codes over the ring R are characterized. Finally, a Gray-map is obtained and the image of cyclic codes in R is characterized.

Construction of nonbinary quantum cyclic codes by using graph method

Using the graph method proposed by Schlingemann and Werner, this paper introduces a technique to construct nonbinary quantum cyclic codes and provides a specific example. We also construct the quantum codes [[8, 2, 4]]p and [[n, n - 2, 2]]p for all odd primes p by the graph method.

Linear recurring sequences and subfield subcodes of cyclic codes

Linear recurring sequences over finite fields play an important role in coding theory and cryptography. It is known that subfield subcodes of linear codes yield some good codes. In this paper, we study linear recurring sequences and subfield subcodes. Let Mqm(f(x)) denote the set of all linear recurring sequences over Fqm with characteristic polynomial f(x) over Fqm . Denote the restriction of Mqm(f(x)) to sequences over Fq and the set after applying trace function to each sequence in Mqm(f(x)) by Mqm(f(x)) | Fq and Tr( Mqm(f(x))), respectively. It is shown that these two sets are both complete sets of linear recurring sequences over Fq with some characteristic polynomials over Fq. In this paper, we firstly determine the characteristic polynomials for these two sets. Then, using these results, we determine the generator polynomials of subfield subcodes and trace codes of cyclic codes over Fqm .

GOOD p-ARY QUASIC-CYCLIC CODES FROM CYCLIC CODES OVER F_p+vF_p

This paper introduces a Gray map from （Fp ＋ vFp）n to F2pn, and describes the relationship between codes over Fp ＋ vFp and their Gray images. The authors prove that every cyclic code of arbitrary length n over Fp ＋ vFp is principal, and determine its generator polynomial as well as the number of cyclic codes. Moreover, the authors obtain many best-known p-ary quasic-cyclic codes in terms of their parameters via the Gray map.

(1-uv)-CONSTACYCLIC CODES OVER F_p+uF_p+vF_p+uvF_p

Constacyclic codes are an important class of linear codes in coding theory.Many optimal linear codes are directly derived from constacyclic codes.In this paper,(1 — uv)-constacyclic codes over the local ring F_p + uF_p + vF_p + uvF_p are studied.It is proved that the image of a(1 — uv)-constacyclic code of length n over F_p + uF_p + vF_p + uvF_p under a Gray map is a distance invariant quasi-cyclic code of index p2 and length p^3n over F_p.Several examples of optimal linear codes over F_p from(1 — uv)-constacyclic codes over F_p + uF_p + vF_p + uvF_p are given.

Constacyclic and cyclic codes over finite chain rings

The problem of Gray image of constacyclic code over finite chain ring is studied. A Gray map between codes over a finite chain ring and a finite field is defined. The Gray image of a linear constacyclic code over the finite chain ring is proved to be a distance invariant quasi-cyclic code over the finite field. It is shown that every code over the finite field, which is the Gray image of a cyclic code over the finite chain ring, is equivalent to a quasi-cyclic code.

A Class of Constacyclic Codes over Ring R+vR

By constructing a Gray map, a class of constacyclic codes over ring R = R＋ vR is studied. Using cyclic codes and negacyclic codes of length p^s over ring R, the structure of （1 - 2v）-constacyclic codes and dual codes of length p^s over ring R are given, the Gray images of （1 - 2v）-constacyclic codes in a particular case are also studied. It is shown that linear codes of length pS over ring R are （1 -2v）-constacyclic codes if and only if their Gray images are distance-invariant cyclic codes of length 2p^s over ring R.