Class Of Cyclic Codes Research Articles

Several Classes of Cyclic Codes With Either Optimal Three Weights or a Few Weights

Cyclic codes with a few weights are very useful in the design of frequency hopping sequences and the development of secret sharing schemes. In this paper, we mainly use Gauss sums to represent the Hamming weights of cyclic codes whose duals have two zeroes. A lower bound of the minimum Hamming distance is determined. In some cases, we give the Hamming weight distributions of the cyclic codes. In particular, we obtain a class of three-weight optimal cyclic codes achieving the Griesmer bound, which generalizes a Vega’s result, and several classes of cyclic codes with a few weights, which solve an open problem proposed by Vega.

Cyclic codes from the second class two-prime Whiteman’s generalized cyclotomic sequence with order 6

Let n1=df+1$n_{1}=df+1$ and n2=dfź+1$n_{2}=df^{\prime }+1$ be two distinct odd primes with positive integers d,f,fź$d, f, f^{\prime }$ and gcd(f,fź)=1$\gcd (f,f^{\prime })=1$. In this paper, we compute the linear complexity and the minimal polynomial of the two-prime Whiteman's generalized cyclotomic sequence of order d=6$d=6$ over GF(q)$\text {GF}(q)$, where q=pm$q=p^{m}$ and p is an odd prime and m is an integer. We employ this sequence of order 6 to construct several classes of cyclic codes over GF(q)$\text {GF}(q)$ with length n1n2$n_{1}n_{2}$. We also obtain lower bounds on the minimum distance of these cyclic codes.

Some cyclic codes with prime length from cyclotomy of order 4

Cyclic codes are a subclass of linear codes and are widely used in many applications as they have efficient encoding and decoding algorithms. In this paper, we investigate two classes of cyclic codes over źźq$\mathbb {F}_{q}$ with prime length using cyclotomy of order 4. Both the dimensions and the generator polynomials are explicitly determined. The method serves as a connection between cyclic codes and sequences over źźq$\mathbb {F}_{q}$ whose supports are the unions of certain cyclotomic classes of order 4. The main results thus partially answer two open problems in Ding (2015).

On the weight distributions of some cyclic codes

Finding the weight distributions of specific cyclic codes has been an interesting area of research. Although the class of cyclic codes has been studied for many years, their weight distributions are known only for a few cases. In this paper, we use matrix method to determine the weight distribution of a family of cyclic codes.

A class of $p$-ary cyclic codes and their weight enumerators

Let $\mathbb{F}_{p^m}$ be a finite field with $p^m$ elements, where $p$ is an odd prime, and $m$ is a positive integer. Let $h_1(x)$ and $h_2(x)$ be minimal polynomials of $-\pi^{-1}$ and $\pi^{-\frac{p^k+1}{2}}$ over $\mathbb{F}_p$, respectively, where $\pi $ is a primitive element of $\mathbb{F}_{p^m}$, and $k$ is a positive integer such that $\frac{m}{\gcd(m,k)}\geq 3$. In [23], Zhou et al. obtained the weight distribution of a class of cyclic codes over $\mathbb{F}_p$ with parity-check polynomial $h_1(x)h_2(x)$ in the following two cases: &nbsp • $k$ is even and $\gcd(m,k)$ is odd; &nbsp • $\frac{m}{\gcd(m,k)}$ and $\frac{k}{\gcd(m,k)}$ are both odd. In this paper, we further investigate this class of cyclic codes over $\mathbb{F}_p$ in other cases. We determine the weight distribution of this class of cyclic codes.

Pseudo-cyclic Codes and the Construction of Quantum MDS Codes

Constacyclic codes which generalize the classical cyclic codes have played important roles in recent constructions of many new quantum maximum distance separable (MDS) codes. However, the mathematical mechanism may not have been fully understood. In this paper, we use pseudo-cyclic codes, which is a further generalization of constacyclic codes, to construct the quantum MDS codes. We can not only provide a unified explanation of many previous constructions, but also produce some new quantum MDS codes.

Hasse–Weil bound for additive cyclic codes

We obtain a bound on the minimum distance of additive cyclic codes via the number of rational points on certain algebraic curves over finite fields. This is an extension of the analogous bound in the case of classical cyclic codes. Our result is the only general bound on such codes aside from Bierbrauer's BCH bound. We compare our bounds' performance against the BCH bound for additive cyclic codes in a special case and provide examples where it yields better results.

An association between primitive and non-primitive BCH codes using monoid rings

BCH codes are one of the most important classes of cyclic codes for error correction. In this study, we generalize BCH codes using monoid rings instead of a polynomial ring over the binary field F 2. We show the existence of a non-primitive binary BCH code C bn of length bn, corresponding to a given length n binary BCH code C n . The value of b is investigated for which the existence of the non-primitive BCH code C bn is assured. It is noticed that the code C n is embedded in the code C bn . Therefore, encoding and decoding of the codes C n and C bn can be done simultaneously. The data transmitted by C n can also be transmitted by C bn . The BCH code C bn has better error correction capability whereas the BCH code C n has better code rate, hence both gains can be achieved at the same time.

Complete weight distributions of two classes of cyclic codes

Complete weight distribution can be used to study authentication codes and the Walsh transform of monomial functions over finite fields. Also, the Hamming weight distribution of a code can be obtained from its complete weight distribution. In this paper, we investigate the complete weight distributions of two classes of cyclic codes. We explicitly present the complete weight enumerators of the cyclic codes. Particularly, we partly solve an open problem proposed in Luo and Feng (IEEE Trans. Inf. Theory 54(12), 5345---5353 (2008)).

On cyclotomic cosets and code constructions

New properties of q-ary cyclotomic cosets modulo n=qm−1, where q≥3 is a prime power, are investigated in this paper. Based on these properties, the dimension as well as bounds for the designed distance of some families of classical cyclic codes can be computed. As an application, new families of nonbinary Calderbank–Shor–Steane (CSS) quantum codes as well as new families of convolutional codes are constructed in this work. These new CSS codes have parameters better than the ones available in the literature. The convolutional codes constructed here have free distance greater than the ones available in the literature.

A class of cyclic codes whose duals have five zeros

In this paper, a family of cyclic codes over $${\mathbb {F}}_{p}$$Fp whose duals have five zeros is presented, where p is an odd prime. Furthermore, the weight distribution of these cyclic codes is determined.

Two classes of cyclic codes and their weight enumerator

Let p be an odd prime, and m, k and d be positive integers such that $$2 \le k\le \frac{m+1}{2}$$2≤k≤m+12 and $$\hbox {gcd}(m,d)=1. \pi $$gcd(m,d)=1.ź is a primitive element of the finite field $${\mathbb {F}}_{p^{m}}$$Fpm. The weight enumerator of cyclic codes over $${\mathbb {F}}_{p}$$Fp whose duals have 2k zeros $$\pi ^{-(p^{jd}+1)/2}$$ź-(pjd+1)/2 and $$-\pi ^{-(p^{jd}+1)/2} (j=0,1,\ldots ,k-1)$$-ź-(pjd+1)/2(j=0,1,ź,k-1) is determined in the present paper. The weight enumerator of cyclic codes over $${\mathbb {F}}_{p}$$Fp whose duals have $$2k-1$$2k-1 zeros $$\pi ^{-(p^{(k-1)d}+1)/2}, \pi ^{-(p^{jd}+1)/2}$$ź-(p(k-1)d+1)/2,ź-(pjd+1)/2 and $$-\pi ^{-(p^{jd}+1)/2} (j=0,1,\ldots ,k-2)$$-ź-(pjd+1)/2(j=0,1,ź,k-2) is also determined when $$2\not \mid \frac{m}{gcd(m,k-1)}$$2źmgcd(m,k-1) holds.

A class of cyclic codes from two distinct finite fields

Let Fq be a finite field with q elements and m1, m2 two distinct positive integers such that gcd⁡(m1,m2)=d. Suppose that α1 and α2 are two primitive elements of Fqm1 and Fqm2, respectively. Let n=(qm1−1)(qm2−1)/(qd−1) and Ti denote the trace function from Fqmi to Fq for i=1,2. We define a cyclic codeC(q,m1,m2)={c(a,b):a∈Fqm1,b∈Fqm2}, wherec(a,b)=(T1(aα10)+T2(bα20),T1(aα11)+T2(bα21),…,T1(aα1n−1)+T2(bα2n−1)). In this paper, we use Gauss sums to investigate the weight distribution of C(q,m1,m2) and prove that it has at most four nonzero weights if d=2 and gcd⁡(m1−m22,q−1)=1. Furthermore, we get a class of three-weight cyclic codes if |m1−m2|=2. Some optimal or nearly optimal cyclic codes are presented.

Some new classes of cyclic codes with three or six weights

In this paper, a class of three-weight cyclic codes over prime fields $\mathbb{F}_p$ of odd order whose duals have two zeros, and a class of six-weight cyclic codes whose duals have three zeros are presented. The weight distributions of these cyclic codes are derived.

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.

The weight distributions of a class of non-primitive cyclic codes with two nonzeros

Recently, a class of non-primitive cyclic codes with two nonzeros have received much attention of researchers and their weight distributions have been obtained for several cases of two key parameters related to the nonzeros. In this paper, by evaluating certain Jacobi sums, we determine the weight distributions of this class of cyclic codes for one more special case.

The Weight Distributions of Several Classes of Cyclic Codes From APN Monomials

Let $m\geq 3$ be an odd integer and $p$ be an odd prime. % with $p-1=2^rh$, where $h$ is an odd integer. In this paper, many classes of three-weight cyclic codes over $\mathbb{F}_{p}$ are presented via an examination of the condition for the cyclic codes $\mathcal{C}_{(1,d)}$ and $\mathcal{C}_{(1,e)}$, which have parity-check polynomials $m_1(x)m_d(x)$ and $m_1(x)m_e(x)$ respectively, to have the same weight distribution, where $m_i(x)$ is the minimal polynomial of $\pi^{-i}$ over $\mathbb{F}_{p}$ for a primitive element $\pi$ of $\mathbb{F}_{p^m}$. %For $p=3$, the duals of five classes of the proposed cyclic codes are optimal in the sense that they meet certain bounds on linear codes. Furthermore, for $p\equiv 3 \pmod{4}$ and positive integers $e$ such that there exist integers $k$ with $\gcd(m,k)=1$ and $\tau\in\{0,1,\cdots, m-1\}$ satisfying $(p^k+1)\cdot e\equiv 2 p^{\tau}\pmod{p^m-1}$, the value distributions of the two exponential sums $T(a,b)=\sum\limits_{x\in \mathbb{F}_{p^m}}\omega^{\Tr(ax+bx^e)}$ and $ S(a,b,c)=\sum\limits_{x\in \mathbb{F}_{p^m}}\omega^{\Tr(ax+bx^e+cx^s)}, $ where $s=(p^m-1)/2$, are settled. As an application, the value distribution of $S(a,b,c)$ is utilized to investigate the weight distribution of the cyclic codes $\mathcal{C}_{(1,e,s)}$ with parity-check polynomial $m_1(x)m_e(x)m_s(x)$. In the case of $p=3$ and even $e$ satisfying the above condition, the duals of the cyclic codes $\mathcal{C}_{(1,e,s)}$ have the optimal minimum distance.

Weight distributions of a class of cyclic codes with two nonzeros

Weight distributions of a class of cyclic codes with two nonzeros

On the Weight Distribution of Cyclic Codes With Niho Exponents

Recently, there has been intensive research on the weight distributions of cyclic codes. In this paper, we compute the weight distributions of three classes of cyclic codes with Niho exponents. More specifically, we obtain two classes of binary three-weight and four-weight cyclic codes and a class of nonbinary four-weight cyclic codes. The weight distributions follow from the determination of value distributions of certain exponential sums. Several examples are presented to show that some of our codes are optimal and some have the best known parameters.

Weight Distributions of Two Classes of Cyclic Codes With Respect to Two Distinct Order Elements

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Cyclic codes have been studied for many years, but their weight distributions are known only for a few cases. In this paper, let F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> be an extension of a finite field F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> and r = q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> , we determine the weight distributions of the cyclic codes C={c(a, b): a, b ∈ F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> }, c(a, b)= Tr <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r/q</sub> (ag <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> +bg <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> ),...,Tr <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r/q</sub> (ag <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n-1</sup> +b <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n-1</sup> )), g <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , g <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ∈ F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> , in the following two cases: 1) ord(g <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> )=n, n|r-1 and g <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> =1 and 2) ord(g <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> )=n, g <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> =g <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , ord(g <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> )=n/2, m=2, and 2(r-1)/n|(q+1).