Class Of Cyclic Codes Research Articles

Quantum variations of cyclotomic cosets for cyclic stabiliser codes construction

Cyclotomic cosets are intrinsically linked with the design and construction of classical cyclic codes whose properties can be inferred from the coset structures. This paper proposes some new quantum variations of cyclotomic cosets for cyclic stabiliser construction. These variations are governed by several parameters which are devoted to designing the two essential parts: the error part and the position part of these cosets. Criteria on these cosets in generating additive cyclic stabiliser are extensively studied, followed by actual implementation on stabiliser codes construction of several selected classes of length.

Study of cyclic codes from low differentially uniform functions and its consequences

Cyclic codes have many applications in consumer electronics, communication and data storage systems due to their efficient encoding and decoding algorithms. An efficient approach to constructing cyclic codes is the sequence approach. In their articles (Ding and Zhou 2014 [11]) and (Ding, 2013 [7]), Ding and Zhou constructed several classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented some open problems on cyclic codes from highly nonlinear functions. This article focuses on these exciting works by investigating new insights in this research direction. Specifically, its objective is twofold. The first is to provide a complement with some former results and present correct proofs and statements on some known ones on the cyclic codes from the APN functions. The second is studying the cyclic codes from some known functions processing low differential uniformity. Along with this article, we shall answer some open problems presented in the literature. The first concerns Open Problem 1, proposed by Ding and Zhou (2014) [11]. The others are Open Problems 5.16 and 5.25, raised by Ding (2013) [7].

The punctured codes of two classes of cyclic codes with few weights

<p style='text-indent:20px;'>The puncturing technique is one of important tools for constructing new codes from old ones. In recent years, many classes of linear codes with interesting parameters have been obtained with this technique. Based on quadratic Gauss sums, the puncturing technique and cyclotomic classes, we investigate two classes of punctured codes of cyclic codes over finite fields. The weight distribution and the parameters of the duals of these codes are studied. The parameters of these codes are new.</p>

Infinite families of 2-designs from two classes of binary cyclic codes with three nonzeros

<p style='text-indent:20px;'>Combinatorial <inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula>-designs have been an interesting topic in combinatorics for decades. It is a basic fact that the codewords of a fixed weight in a code may hold a <inline-formula><tex-math id="M2">\begin{document}$ t $\end{document}</tex-math></inline-formula>-design. Till now only a small amount of work on constructing <inline-formula><tex-math id="M3">\begin{document}$ t $\end{document}</tex-math></inline-formula>-designs from codes has been done. In this paper, we determine the weight distributions of two classes of cyclic codes: one related to the triple-error correcting binary BCH codes, and the other related to the cyclic codes with parameters satisfying the generalized Kasami case, respectively. We then obtain infinite families of <inline-formula><tex-math id="M4">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-designs from these codes by proving that they are both affine-invariant codes, and explicitly determine their parameters. In particular, the codes derived from the dual of binary BCH codes hold five <inline-formula><tex-math id="M5">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>-designs when <inline-formula><tex-math id="M6">\begin{document}$ m = 4 $\end{document}</tex-math></inline-formula>.</p>

Quantum Data-Syndrome Codes

Performing active quantum error correction to protect fragile quantum states highly depends on the correctness of error information--error syndromes. To obtain reliable error syndromes using imperfect physical circuits, we propose the idea of quantum data-syndrome (DS) codes that are capable of correcting both data qubits and syndrome bits errors. We study fundamental properties of quantum DS codes, including split weight enumerators, generalized MacWilliams identities, and linear programming bounds. In particular, we derive Singleton and Hamming-type upper bounds on degenerate quantum DS codes. Then we study random DS codes and show that random DS codes with a relatively small additional syndrome measurements achieve the Gilbert-Varshamov bound of stabilizer codes. Constructions of quantum DS codes are also discussed. A family of quantum DS codes is based on classical linear block codes, called syndrome measurement codes, so that syndrome bits are encoded in additional redundant stabilizer measurements. Another family of quantum DS codes is CSS-type quantum DS codes based on classical cyclic codes, and this includes the Steane code and the quantum Golay code.

Flag fault-tolerant error correction, measurement, and quantum computation for cyclic Calderbank-Shor-Steane codes

Flag qubits have recently been proposed in syndrome extraction circuits to detect high-weight errors arising from fewer faults. The use of flag qubits allows the construction of fault-tolerant protocols with the fewest number of ancillas known to-date. In this work, we prove some critical properties of CSS codes constructed from classical cyclic codes that enable the construction of a flag fault-tolerant error correction scheme. We then develop fault-tolerant protocols as well as a family of circuits for flag fault-tolerant error correction and operator measurement, requiring only four ancilla qubits and applicable to cyclic CSS codes of distance 3. The measurement protocol can be further used for logical Clifford gate implementation via quantum gate teleportation. We also provide examples of cyclic CSS codes with large encoding rates.

Infinite families of 2‐designs from a class of cyclic codes

AbstractCombinatorial ‐designs have wide applications in coding theory, cryptography, communications, and statistics. It is well known that the supports of all codewords with a fixed weight in a code may give a ‐design. In this paper, we first determine the weight distributions of a class of linear codes derived from the dual of some extended cyclic codes. We then obtain infinite families of 2‐designs and explicitly compute their parameters from the supports of all the codewords with a fixed weight in the codes. By a simple counting argument, we obtain exponentially many 2‐designs.

Two classes of binary cyclic codes and their weight distributions

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, two classes of cyclic codes whose duals have two zeros are presented. The weight distributions of these cyclic codes are settled with the help of Gaussian periods. The duals of one class of cyclic codes are also studied. Some of the cyclic codes presented in this paper are optimal in the sense that they meet some bounds of linear codes.

Cyclic codes with generalized cyclotomic cubic classes

This paper is study of cyclotomic classes of order 3 r for r ≥ 1 to construct a class of cyclic codes of length p m . We obtain cube-root or a similar bound on the minimum odd weight of these codes.

Some q-ary Cyclic Codes from Explicit Monomials over $$\mathbb{F}_{q}m$$

Cyclic codes as a subclass of linear codes have practical applications in communication systems, consumer electronics, and data storage systems due to their efficient encoding and decoding algorithms. The objective of this paper is to construct some cyclic codes by the sequence approach. More precisely, we determine the dimension and the generator polynomials of three classes of q-ary cyclic codes defined by some sequences with explicit polynomials over \(\mathbb{F}_{q}m\). The minimum distance of such cyclic codes is also discussed. Some of these codes are optimal according to code tables. Moreover, the third class of cyclic codes provides some answers for Open Problem 3 proposed by Ding and Zhou in [1].

The Use of Multivariate Weak-Locator Polynomials to Decode Cyclic Codes up to Actual Minimum Distance

A class of cyclic codes has recently been decoded by using the weak-locator polynomials instead of the conventional error-locator polynomials. In this paper, a generalization of weak-locator polynomials to the multivariate cases, called the multivariate weak-locator polynomials, is defined, and a new matrix whose determinant can be expressed as a multivariate weak-locator polynomial is developed for cyclic codes. Moreover, a modified matrix along with Gaussian elimination found herein enables one to determine error positions precisely. The presented matrices result in a reduction of the number of syndromes when compared with the previously known matrices. The simulation shows that, for example, with the capability of correcting more errors, a considerably sophistical scheme of decoding the (97, 49, 15) binary cyclic code based on the proposed matrices is around 115 times faster than other existing decoders. Therefore, in general, it is developed to facilitate faster decoding of a large class of cyclic codes and is naturally suitable for the software implementation.

Cyclic codes of prime power length from generalized cyclotomic classes of order 2r

In this paper, we first introduce generalized cyclotomic classes of order 2r, r ≥ 1 and then present a special class of cyclic codes with length pm. We also obtain lower bound on the minimum odd weight of these codes.

New constructions of systematic authentication codes from three classes of cyclic codes

Recently, several classes of cyclic codes with three nonzero weights were constructed. With the generic construction presented by C. Ding, T. Helleseth, T. Klove and X. Wang, we present new systematic authentication codes based on these cyclic codes. In this paper, we study three special classes of cyclic codes and their authentication codes. With the help of exponential sums, we calculate the maximum success probabilities of the impersonation and substitution attacks on the authentication codes. Our results show that these new authentication codes are better than some of the authentication codes in the literature. As a byproduct, the number of times that each element occurs as the coordinates in the codewords of the cyclic codes is settled, which is a difficult problem in general.

On the Maximum Burst-Correcting Capability of Cyclic Hsu–Kasami–Chien Codes

Hsu–Kasami–Chien (HKC) codes are a class of cyclic codes that can correct either burst or random errors. This letter studies the maximum burst-correcting capability of HKC codes. A necessary and sufficient condition for a $b$ -burst-correcting cyclic HKC code is given and simplified as a generalized one for Fire codes. An upper bound on the burst-correcting capability of HKC codes is derived, which improves upon a previous bound. Explicit examples of HKC codes are presented. A design guideline for constructing HKC codes with given burst-error and random-error correcting capabilities is presented.

The Weight Distributions of Two Classes of Nonbinary Cyclic Codes With Few Weights

The objective of this letter is to construct two classes of cyclic codes over $F_{p}$ with few weights, where $p$ is an odd prime. The weight distributions of these codes are determined. Some of the cyclic codes are optimal, which refers to meeting a certain bound on linear codes. The number of nonzero weights of these codes is not more than five and these codes can be employed to obtain secret sharing schemes.

A class of cyclic codes constructed via semiprimitive two-weight irreducible cyclic codes

We present a family of reducible cyclic codes constructed as a direct sum (as vector spaces) of two different semiprimitive two-weight irreducible cyclic codes. This family generalizes the class of reducible cyclic codes that was reported in the main result of [10]. Moreover, despite of what was stated therein, we show that, at least for the codes studied here, it is still possible to compute the frequencies of their weight distributions through the cyclotomic numbers in an easy way.

An extended characterization of a class of optimal three-weight cyclic codes over any finite field

A characterization of a class of optimal three-weight cyclic codes of dimension 3 over any finite field was recently presented in [10]. Almost immediately thereafter, several classes of cyclic codes with either optimal three weights or a few weights, were given in [3], showing at the same time that one of these classes can be constructed as a generalization of the sufficient numerical conditions of the characterization given in [10]. The main purpose of this work is to show that these numerical conditions that were found in [3], are also necessary. That is, the main contribution in this work is to present an extended characterization of a class of optimal three-weight cyclic codes, in terms of a weight distribution table. Now, note that the kind of characterizations that are given in terms of a weight distribution table are, in general, very difficult to establish. On the other hand, an interesting feature of the present work, in clear contrast with these two preceding works, is that we use some new and non-conventional methods in order to achieve our goals. In fact, through these non-conventional methods, we not only were able to extend the characterization in [10], but also present a less complex proof of such extended characterization, which avoids the use of some of the sophisticated – but at the same time complex – theorems, that are the key arguments of the proofs given in [10] and [3]. Furthermore, we also find the parameters for the dual code of any cyclic code in our extended characterization class. In fact, after the analysis of some examples, it seems that such dual codes always have the same parameters as the best known linear codes.

Cyclic codes from Whiteman’s generalized cyclotomic sequences of order 2r, r ≥ 2

In this paper, the two-prime Whiteman’s generalized cyclotomic sequences of order 2r, r ≥ 2, are employed to construct several classes of cyclic codes over the finite field GF(q) with length n1n2. The lower bounds of the minimum distance of these cyclic codes are obtained.

Quasi-Cyclic Subcodes of Cyclic Codes

We completely characterize possible indices of quasi-cyclic subcodes in a cyclic code for a very broad class of cyclic codes. We present enumeration results for quasi-cyclic subcodes of a fixed index and show that the problem of enumeration is equivalent to enumeration of certain vector subspaces in finite fields. In particular, we present enumeration results for quasi-cyclic subcodes of the simplex code and duals of certain Bose--Chaudhuri--Hocquenqhem codes. Our results are based on the trace representation of cyclic codes.

Cyclic codes from two-prime generalized cyclotomic sequences of order 6

Cyclic codes have wide applications in data storage systems and communication systems. Employing binary two-prime Whiteman generalized cyclotomic sequences of order 6, we construct several classes of cyclic codes over the finite field $\mathrm{GF}(q)$ and give their generator polynomials. And we also calculate the minimum distance of some cyclic codes and give lower bounds on the minimum distance for some other cyclic codes.