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.

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.

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.

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.

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 GENERALIZED CYCLIC CODES

In this paper, we discuss how to construct a class of generalized cyclic codes, denoted by GCC. It is well known that a cyclic code is generated by a factor of x^n - 1. Clearly, any monic polynomial g(x) with degree less than n could be considered as a factor of some polynomial of degree n. Similarly the construction of cyclic codes, we explain how g(x) can generate a GCC. Meanwhile, as related to cyclic codes, experiments show that GCC can always produce a better parameter and/or give more linear codes. On the basis the of concept of GCC, we can also construct a linear code of [90,76,5] _2.

Cyclic Co des over F2＋uF2＋v F2

We study the structure of cyclic codes of an arbitrary length n over the ring F2+ uF2+ vF2, which is not a finite chain ring. We prove that the Gray image of a cyclic code length n over F2+ uF2+ vF2 is a 3-quasi-cyclic code length 3n over F2.

Cyclic Co des with Complementary Duals over Fp＋v Fp

In this paper,we characterize the necessary and sufficient conditions for a cyclic code of length n over F_p+ vF_p to be an LCD code,where p is an odd prime.

On the Characterization of Cyclic Co des over Ring F2＋uF2＋v F2

In this work, we investigate the cyclic codes over the ring F2+ uF2+ vF2. We first study the relationship between linear codes over F2+ uF2+ vF2 and that over F2.Then we give a characterization of the cyclic codes over F2+ uF2+ vF2. Finally, we obtain the number of the cyclic code over F2+ uF2+ vF2 of length n.

The Adjacency Codes of the First Yellow Graphs

The authors study the binary codes spanned by the adjacency matrices of the strongly regular graphs(SRGs)on at most two hundred vertices whose existence is unknown.The authors show that in length less than one hundred they cannot be cyclic,except for the exceptions of the SRGs of parameters(85,42,20,21)and(96,60,38,36).In particular,the adjacency code of a(85,42,20,21)is the zero-sum code.In the range[100,200]the authors find 29 SRGs that could possibly have a cyclic adjacency code.

Constructions of Sidon spaces and cyclic subspace codes

In this paper,we firstly construct several new kinds of Sidon spaces and Sidon sets by investigating some known results.Secondly,using these Sidon spaces,we will present a construction of cyclic subspace codes with cardinality τ,q^(n)-1/q-1 and minimum distance 2k-2,whereτis a positive integer.We further-more give some cyclic subspace codes with size 2τ·q^(n)-1/q-1 and without changing the minimum distance 2k-2.

On Cyclic DNA Codes Over F_(2)+uF_(2)+u^(2)F_(2)

In the present paper,we study the structure of cyclic DNA codes of even length over the ring F_(2)+uF_(2)+u^(2)F_(2)where u^(3)=0.We investigate two presentations of cyclic codes of even length over F_(2)+uF_(2)+u^(2)F_(2)satisfying the reverse constraint and the reverse-complement constraint.

A Note on Characteristic Generators of a Free Cyclic Code and Its Dual

Let C be a free cyclic code over Zp^a and dim pC = k. In the paper, we prove that if the k characteristic generators of C are p-linearly independent then the corresponding nα- k characteristic generators of C^⊥ are p-linearly independent. We then show that to any trellis that can be constructed from k p-linearly independent characteristic generators of C, there exists a trellis for C^⊥ with the same state-complexity profile, which generalizes the conjecture of Koetter and Vardy to a free cyclic code over Zpo.