Weight Distribution Of Cyclic Codes Research Articles

Complete b-symbol weight distribution of some irreducible cyclic codes

Recently, b-symbol codes are proposed to protect against b-symbol errors in b-symbol read channels. It is an interesting subject of study to consider the complete b-symbol weight distribution of cyclic codes since b-symbol metric is a generalization for Hamming metric. The complete b-symbol Hamming weight distribution of irreducible cyclic codes is known in only a few cases. In this paper, we give a complete b-symbol Hamming weight distribution of a class of irreducible cyclic codes with two nonzero b-symbol Hamming weights.

Weight distribution of cyclic codes defined by quadratic forms and related curves

We consider cyclic codes $\mathcal{C}_\mathcal{L}$ associated to quadratic trace forms in $m$ variables $Q_R(x) = \operatorname{Tr}_{q^m/q}(xR(x))$ determined by a family $\mathcal{L}$ of $q$-linearized polynomials $R$ over $\mathbb{F}_{q^m}$, and three related codes $\mathcal{C}_{\mathcal{L},0}$, $\mathcal{C}_{\mathcal{L},1}$ and $\mathcal{C}_{\mathcal{L},2}$. We describe the spectra for all these codes when $\mathcal{L}$ is an even rank family, in terms of the distribution of ranks of the forms $Q_R$ in the family $\mathcal{L}$, and we also compute the complete weight enumerator for $\mathcal{C}_\mathcal{L}$. In particular, considering the family $\mathcal{L} = \langle x^{q^\ell} \rangle$, with $\ell$ fixed in $\mathbb{N}$, we give the weight distribution of four parametrized families of cyclic codes $\mathcal{C}_\ell$, $\mathcal{C}_{\ell,0}$, $\mathcal{C}_{\ell,1}$ and $\mathcal{C}_{\ell,2}$ over $\mathbb{F}_q$ with zeros $\{ \alpha^{-(q^\ell+1)} \}$, $\{ 1,\, \alpha^{-(q^\ell+1)} \}$, $\{ \alpha^{-1},\,\alpha^{-(q^\ell+1)} \}$ and $\{ 1,\,\alpha^{-1},\,\alpha^{-(q^\ell+1)}\}$ respectively, where $q = p^s$ with $p$ prime, $\alpha$ is a generator of $\mathbb{F}_{q^m}^*$ and $m/(m,\ell)$ is even. Finally, we give simple necessary and sufficient conditions for Artin-Schreier curves $y^p-y = xR(x) + \beta x$, $p$ prime, associated to polynomials $R \in \mathcal{L}$ to be optimal. We then obtain several maximal and minimal such curves in the case $\mathcal{L} = \langle x^{p^\ell}\rangle$ and $\mathcal{L} = \langle x^{p^\ell}, x^{p^{3\ell}} \rangle$.

The minimum Hamming distances of repeated-root cyclic codes of length $$6p^s$$ and their MDS codes

In this paper, let p be a prime with $$p\ge 7$$ . We determine the weight distributions of cyclic codes of length 6 over $$\mathbb {F}_q$$ and the minimum Hamming distances of all repeated-root cyclic codes of length $$6p^s$$ over $$\mathbb {F}_q$$ , where q is a power of p and s is an integer. Furthermore, we find all maximum distance separable codes of length $$6p^s$$ .

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 note on the weight distribution of some cyclic codes

Let Fq be the finite field with q elements and Cn be the cyclic group of order n, where n is a positive integer relatively prime to q. Let H,K be subgroups of Cn such that H is a proper subgroup of K. In this note, the weight distributions of the cyclic codes of length n over Fq with generating idempotents Kˆ and eH,K=Hˆ−Kˆ are explicitly determined, where Kˆ=1/|K|∑g∈Kg and Hˆ=1/|H|∑g∈Hg. Our result naturally gives a new characterization of a theorem by Sharma and Bakshi [18] that determines the weight distribution of all irreducible cyclic codes of length pm over Fq, where p is an odd prime and q is a primitive root modulo pm. Finally, two examples are presented to illustrate our results.

Recent progress on weight distributions of cyclic codes over finite fields

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. In coding theory it is often desirable to know the weight distribution of a cyclic code to estimate the error correcting capability and error probability. In this paper, we present the recent progress on the weight distributions of cyclic codes over finite fields, which had been determined by exponential sums. The cyclic codes with few weights which are very useful are discussed and their existence conditions are listed. Furthermore, we discuss the more general case of constacyclic codes and give some equivalences to characterize their weight distributions.

Weight distributions of cyclic codes of length [formula omitted

Let Fq be a finite field with q elements, l an odd prime, and t,v positive integers such that lv|(q−1) and gcd(t,l)=1. In this paper, we give a combinatorial result and use it to determine the weight distribution of a cyclic code of length tlm with t|(q−1), which is an open question in Yang et al. (2013). Moreover we compute the weight distributions of cyclic codes of length tlm, where q≡3(mod4) and t=4 or 8.

Weight distribution of cyclic codes with arbitrary number of generalized Niho type zeroes

Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. Most previous results obtained so far were for cyclic codes with no more than three zeroes. Inspired by the works of Li et al. (Sci China Math 53:3279---3286, 2010; IEEE Trans Inf Theory 60:3903---3912, 2014), we study two families of cyclic codes over $${\mathbb F}_p$$Fp with arbitrary number of zeroes of generalized Niho type, more precisely $${\mathcal {C}_{(d_0,d_1,\ldots ,d_t)}^{(1)}}$$C(d0,d1,?,dt)(1) (for $$p=2$$p=2) of $$t+1$$t+1 zeroes, and $${\mathcal {C}_{(\widetilde{d}_1,\ldots ,\widetilde{d}_t)}^{(2)}}$$C(d~1,?,d~t)(2) (for any prime $$p$$p) of $$t$$t zeroes for any $$t$$t. We find that the first family has at most $$(2t+1)$$(2t+1) non-zero weights, and the second has at most $$2t$$2t non-zero weights. Their weight distribution are also determined in the paper.

Hamming Weights of the Duals of Cyclic Codes With Two Zeros

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. In this paper, let \(\Bbb F_r\) be a finite field with \(r=q^m\) . Suppose that \(g_1, g_2 \in \Bbb F_r^*\) are not conjugates over \(\Bbb F_q\) , \( \operatorname {ord}(g_1)=n_1\) , \( \operatorname {ord}(g_2)=n_2\) , \(d=\gcd (n_1, n_2)\) , and \(n={n_1n_2}/ d\) . Let \(\Bbb F_q(g_1)=\Bbb F_{q^{m_1}}, \Bbb F_q(g_2)=\Bbb F_{q^{m_2}}\) , and \(\text{T}_i\) denote the trace function from \(\Bbb F_{q^{m_i}}\) to \(\Bbb F_q\) for \(i=1, 2\) . We define a cyclic code \(\mathcal C_{(q, m, n_1, n_2)}=\{c(a,b) : a \in \Bbb F_{q^{m_1}}, b \in \Bbb F_{q^{m_2}}\},\) where \(c(a,b)=(\text {T}_1(ag_1^0)+\text {T}_2(bg_2^0), \text {T}_1(ag_1^1)+\text {T}_2(bg_2^1), \ldots , \text {T}_1(ag_1^{n-1})+\text {T}_2(bg_2^{n-1})).\) We mainly use Gauss periods to present the weight distribution of the cyclic code \(\mathcal C_{(q, m, n_1, n_2)}\) . As applications, we determine the weight distribution of cyclic code \(\mathcal C_{(q, m,q^{m_1}-1,q^{m_2}-1)}\) with \(\gcd (m_1,m_2)=1\) ; in particular, it is a three-weight cyclic code if \(\gcd (q-1,m_1-m_2)=1\) . We also explicitly determine the weight distributions of some classes of cyclic codes including several classes of four-weight cyclic codes.

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 cyclic codes with respect to pairwise coprime order elements

Let Fr be an extension of a finite field Fq with r=qm. Let each gi be of order ni in Fr⁎ and gcd⁡(ni,nj)=1 for 1⩽i≠j⩽u. We define a cyclic code over Fq byC(q,m,n1,n2,…,nu)={C(a1,a2,…,au):a1,a2,…,au∈Fr}, whereC(a1,a2,…,au)=(Trr/q(∑i=1uaigi0),…,Trr/q(∑i=1uaigin−1)) and n=n1n2⋯nu. In this paper, we present a method to compute the weights of C(q,m,n1,n2,…,nu). Further, we determine the weight distributions of the cyclic codes C(q,m,n1,n2) and C(q,m,n1,n2,1).

The Weight Distributions of a Class of Cyclic Codes with Three Nonzeros over𝔽3

Cyclic codes have efficient encoding and decoding algorithms. The decoding error probability and the undetected error probability are usually bounded by or given from the weight distributions of the codes. Most researches are about the determination of the weight distributions of cyclic codes with few nonzeros, by using quadratic forms and exponential sums but limited to low moments. In this paper, we focus on the application of higher moments of the exponential sums to determine the weight distributions of a class of ternary cyclic codes with three nonzeros, combining with not only quadratic forms but also MacWilliams’ identities. Another application of this paper is to emphasize the computer algebra system Magma for the investigation of the higher moments. In the end, the result is verified by one example using Matlab.

The Weight Enumerator of Three Families of Cyclic Codes

Cyclic codes are a subclass of linear codes and have wide applications in consumer electronics, data storage systems, and communication systems due to their efficient encoding and decoding algorithms. Cyclic codes with many zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a very small number of cases. In general, the calculation of the weight distribution of cyclic codes is heavily based on the evaluation of some exponential sums over finite fields. Very recently, Li studied a class of p-ary cyclic codes of length p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m</sup> -1, where p is a prime and m is odd. They determined the weight distribution of this class of cyclic codes by establishing a connection between the involved exponential sums with the spectrum of Hermitian forms graphs. In this paper, this class of p-ary cyclic codes is generalized and the weight distribution of the generalized cyclic codes is settled for both even m and odd m along with the idea of Li The weight distributions of two related families of cyclic codes are also determined.

Weight Distribution of a Class of Cyclic Codes With Arbitrary Number of Zeros

Cyclic codes have been widely used in digital communication systems and consumer electronics as they have efficient encoding and decoding algorithms. The weight distribution of cyclic codes has been an important topic of study for many years. It is in general hard to determine the weight distribution of linear codes. In this paper, a class of cyclic codes with any number of zeros is described and their weight distributions are determined.

The Weight Distribution of a Class of Cyclic Codes Related to Hermitian Forms Graphs

The determination of weight distribution of cyclic codes involves evaluation of Gauss sums and exponential sums. Despite of some cases where a neat expression is available, the computation is generally rather complicated. In this note, we determine the weight distribution of a class of reducible cyclic codes whose dual codes may have arbitrarily many zeros. This goal is achieved by building an unexpected connection between the corresponding exponential sums and the spectrums of Hermitian forms graphs.

The Weight Distributions of Cyclic Codes and Elliptic Curves

Cyclic codes with two zeros and their dual codes as a practically and theoretically interesting class of linear codes have been studied for many years and find many applications. The determination of the weight distributions of such codes is an open problem. Generally, the weight distributions of cyclic codes are difficult to determine. Utilizing a class of elliptic curves, this paper determines the weight distributions of dual codes of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -ary cyclic codes with two zeros for a few more cases, where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> is an odd prime power.

The weight distributions of cyclic codes with two zeros and zeta functions

We consider the weight distribution of the binary cyclic code of length 2 n − 1 with two zeros α a , α b . Our proof gives information in terms of the zeta function of an associated variety. We carry out an explicit determination of the weight distribution in two cases, for the cyclic codes with zeros α 3 , α 5 and α , α 11 . These are the smallest cases of two infinite families where finding the weight distribution is an open problem. Finally, an interesting application of our methods is that we can prove that these two codes have the same weight distribution for all odd n .

A commutative algebra approach to linear codes

Recently some methods have been proposed to find the distance and weight distribution of cyclic codes using Gröbner bases. We identify a class of codes for which these methods can be generalized. We show that this class contains all interesting linear codes and we provide variants and improvements. This approach sometimes reveals an unexpected algebraic structure in the code. We also investigate the decoding for a subclass, proving the existence of general error locator polynomials.

GRÖBNER BASIS TECHNIQUES TO COMPUTE WEIGHT DISTRIBUTIONS OF SHORTENED CYCLIC CODES

Using Gröbner techniques, we can exhibit a method to get the distance and weight distribution of cyclic codes and shortened cyclic codes, improving earlier similar results for the distance of cyclic codes.

Weight distribution of some reducible cyclic codes

Let q = p m where p is an odd prime, m ⩾ 3 , k ⩾ 1 and gcd ( k , m ) = 1 . Let Tr be the trace mapping from F q to F p and ζ p = e 2 π i p . In this paper we determine the value distribution of following two kinds of exponential sums ∑ x ∈ F q χ ( α x p k + 1 + β x 2 ) ( α , β ∈ F q ) and ∑ x ∈ F q χ ( α x p k + 1 + β x 2 + γ x ) ( α , β , γ ∈ F q ) , where χ ( x ) = ζ p Tr ( x ) is the canonical additive character of F q . As an application, we determine the weight distribution of the cyclic codes C 1 and C 2 over F p with parity-check polynomial h 2 ( x ) h 3 ( x ) and h 1 ( x ) h 2 ( x ) h 3 ( x ) , respectively, where h 1 ( x ) , h 2 ( x ) and h 3 ( x ) are the minimal polynomials of π −1 , π −2 and π − ( p k + 1 ) over F p , respectively, for a primitive element π of F q .