Class Of Cyclic Codes Research Articles

Seven Classes of Three-Weight Cyclic Codes

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, compared with linear block codes. In this paper, seven classes of three-weight cyclic codes over \gf(p) whose duals have two zeros are presented, where p is an odd prime. The weight distributions of the seven classes of cyclic codes are settled. Some of the cyclic codes are optimal in the sense that they meet certain bounds on linear codes. The application of these cyclic codes in secret sharing is also considered.

Evaluation of the Weight Distribution of a Class of Cyclic Codes Based on Index 2 Gauss Sums

The duals of cyclic codes with two zeros have been extensively studied, and their weight distributions have recently been evaluated in some cases. In this paper, we determine the weight distribution of a certain new class of such codes by computations involving index 2 Gauss sums.

The weight distributions of some cyclic codes with three or four nonzeros over $$\mathbb F _3$$ F 3

Because of efficient encoding and decoding algorithms comparing with linear block codes, cyclic codes form an important family and have applications in communications and storage systems. However, their weight distributions are known only for a few cases mainly on the codes with no more than three nonzeros. In this paper, the weight distributions of two classes of cyclic codes with three or four nonzeros are determined.

FPGA Implementation of Reed-Solomon Encoder and Decoder for Wireless Network 802. 16

new class of cyclic codes that is Reed-Solomon codes are discussed for IEEE 802.16 wireless networks. Reed-Solomon codes are used for the error detection and correction in communication systems. This is important in information theory and coding to correct burst errors. Here Reed-Solomon code for wireless network 802.16 is synthesized using VHDL on Xilinx and simulated on ISE simulator. The Reed-Solomon encoder has been checked for different error-correcting capabilities that is 4, 6, 8 etc. Reed-Solomon decoder for IEEE 802.16 network is synthesized on VHDL for error detection and correction. Here pipelining is introduced in Reed-Solomon decoder to improve the performance. The performance of Reed-Solomon encoder RS (255,239) for IEEE 802.16 is shown and Reed-Solomon decoder is checked for both RS(255,243) and RS(255,239) and synthesizable on FPGA.

Cyclic Codes From the Two-Prime Sequences

Cyclic codes are a subclass of linear codes and have wide applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, the two-prime sequence is employed to construct several classes of cyclic codes over GF(q). Lower bounds on the minimum weight of these cyclic codes are developed. Some of the codes obtained are optimal or almost optimal. The p-ranks of the twin-prime difference sets and a class of almost difference sets are computed.

Constructions of new families of nonbinary asymmetric quantum BCH codes and subsystem BCH codes

Two code constructions generating new families of good nonbinary asymmetric quantum BCH codes and good nonbinary subsystem BCH codes are presented in this paper. The first one is derived from q-ary Steane’s enlargement of CSS codes applied to nonnarrow-sense BCH codes. The second one is derived from the method of defining sets of classical cyclic codes. The asymmetric quantum BCH codes and subsystem BCH codes here have better parameters than the ones available in the literature.

Iterative Algorithms for Decoding a Class of Two-Step Majority-Logic Decodable Cyclic Codes

Codes constructed based on finite geometries form a large class of cyclic codes with large minimum distances which can be decoded with simple majority-logic decoding in one or multiple steps. In 2001, Kou, Lin and Fossorier showed that the one-step majority-logic decodable finite geometry codes form a class of cyclic LDPC codes whose Tanner graphs are free of cycles of length 4. These cyclic finite geometry LDPC codes perform very well over the AWGN channel using iterative decoding based on belief propagation (IDBP) and have very low error-floors. However, the standard IDBP is not effective for decoding other cyclic finite geometry codes because their Tanner graphs contain too many short cycles of length 4 which severely degrade the decoding performance. This paper investigates iterative decoding of two-step majority-logic decodable finite geometry codes. Three effective algorithms for decoding these codes are proposed. These algorithms are devised based on the orthogonal structure of the parity-check matrices of the codes to avoid or reduce the degrading effect of the short cycles of length 4. These decoding algorithms provide significant coding gains over the standard IDBP using either the sum-product or the min-sum algorithms.

Some cyclic codes of prime-power length

A new class of cyclic codes of length 2n over GF(q) is proposed, where q is a prime of the form 8m ± 3 and n > 3 is an integer. These codes are defined in terms of their generator polynomials. These codes have many properties analogous to those of duadic codes. Generator polynomials of some duadic codes of length pn over GF(q) are also discussed, where p is an odd prime, n is an integer and q = ρ or ρ2 for some prime ρ.

Cyclic Codes and Sequences: The Generalized Kasami Case

In this paper, the large family of generalized Kasami sequences has been studied. In particular, the cross-correlation distribution among these sequences has been explicitly calculated. Meanwhile, the weight distributions of two classes of cyclic codes could also be determined. This paper generalizes the results from several previous papers.

CONSTRUCTIONS OF SUBSYSTEM CODES OVER FINITE FIELDS

Subsystem codes protect quantum information by encoding it in a tensor factor of a subspace of the physical state space. They generalize all major quantum error protection schemes, and therefore are especially versatile. This paper introduces numerous constructions of subsystem codes. It is shown how one can derive subsystem codes from classical cyclic codes. Methods to trade the dimensions of subsystem and co-subsystem are introduced that maintain or improve the minimum distance. As a consequence, many optimal subsystem codes are obtained. Furthermore, it is shown how given subsystem codes can be extended, shortened, or combined to yield new subsystem codes. These subsystem code constructions are used to derive tables of upper and lower bounds on the subsystem code parameters.

Error-Pattern-Correcting Cyclic Codes Tailored to a Prescribed Set of Error Cluster Patterns

A new class of cyclic codes is discussed which is highly tailored to a prescribed set of dominant error cluster patterns. The cyclic code construction is based on a generator polynomial that produces a distinct syndrome set for each error pattern in the target set. By tailoring the generator polynomial specifically to the set of dominant error patterns, the code becomes highly effective in handling single and multiple occurrences of dominant error patterns at a very high code rate. A list decoding strategy based on a set of test word-error events is developed for the proposed codes, which efficiently utilizes both the algebraic information from the captured syndrome and the reliability measures provided by the local correlators matched to the dominant error patterns. By forcing a decoder to correct a single-pattern event for each test input word, multiple decoders running in parallel on the list of test words can effectively correct multiple error-pattern occurrences within the channel detector output word.

On the Weight Distributions of Two Classes of Cyclic Codes

Let q=pm where p is an odd prime, mges2, and 1lesklesm-1. Let Tr be the trace mapping from Fq to Fp and zetap=e2pii/p be a primitive pth root of unity. In this paper, we determine the value distribution of the following exponential sums: SigmaxisinF qchi(alphaxp k +1+betax2) (alpha, betaisinFq) where chi(x)=zetap Tr(x) is the canonical additive character of Fq. As applications, we have the following. 1) We determine the weight distribution of the cyclic codes C1 and C2 over Fpt with parity-check polynomial h2(x)h3(x) and h1(x)h2(x)h3(x), respectively, where t is a divisor of d=gcd(m, k), and h1(x), h2(x) , and h3(x) are the minimal polynomials of pi-1, pi-2, and pi-(p k +1) over Fpt, respectively, for a primitive element pi of Fq. 2) We determine the correlation distribution between two m-sequences of period q-1. Moreover, we find a new class of p-ary bent functions. This paper extends the results in Feng and Luo (2008).

Cyclic Codes and Reducible Additive Equations

We prove a Weil-Serre type bound on the number of solutions of a class of reducible additive equations over finite fields. Using the trace representation of cyclic codes, this enables us to write a general estimate for the weights of cyclic codes. We extend Wolfmann's weight bound to a larger classes of cyclic codes. In particular, our result is applicable to any cyclic code over Fp and Fp 2 , where p is an arbitrary prime. Examples indicate that our bound performs very well against the Bose-Chaudhuri-Hocquenghem (BCH) bound and that it yields the exact minimum distance in some cases

The lengths of Hermitian self-dual extended duadic codes

Duadic codes are a class of cyclic codes that generalize quadratic residue codes from prime to composite lengths. For every prime power q , we characterize integers n such that there is a duadic code of length n over F q 2 with a Hermitian self-dual parity-check extension. We derive asymptotic estimates for the number of such n as well as for the number of lengths for which duadic codes exist.

Transitive and self-dual codes attaining the Tsfasman-Vla/spl breve/dut$80-Zink bound

A major problem in coding theory is the question of whether the class of cyclic codes is asymptotically good. In this correspondence-as a generalization of cyclic codes-the notion of transitive codes is introduced (see Definition 1.4 in Section I), and it is shown that the class of transitive codes is asymptotically good. Even more, transitive codes attain the Tsfasman-Vlabrevedut$80-Zink bound over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> , for all squares q=l <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . It is also shown that self-orthogonal and self-dual codes attain the Tsfasman-Vlabrevedut$80-Zink bound, thus improving previous results about self-dual codes attaining the Gilbert-Varshamov bound. The main tool is a new asymptotically optimal tower E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> subeE <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1 </sub> subeE <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> sube middotmiddotmiddot of function fields over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (with q=lscr <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ), where all extensions E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> /E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> are Galois

Improvements on Generalized Hamming Weights of Some Trace Codes

We obtain improved bounds for the generalized Hamming weights of some trace codes which include a large class of cyclic codes over any finite field. In particular, we improve the corresponding bounds of Stichtenoth and Voss [8] using various methods altogether.

Is the class of cyclic codes asymptotically good?

There is the long-standing question whether the class of cyclic codes is asymptotically good. By an old result of Lin and Weldon, long Bose-Chaudhuri-Hocquenhem (BCH) codes are asymptotically bad. Berman proved that cyclic codes are asymptotically bad if only finitely many primes are involved in the lengths of the codes. We investigate further classes of cyclic codes which also turn out to be asymptotically bad. Based on reduction arguments we give some evidence that there are asymptotically good sequences of binary cyclic codes in which all lengths are prime numbers provided there is any asymptotically good sequence of binary cyclic codes.

A solution of the isomorphism problem for circulant graphs

The isomorphism problem for circulant graphs (Cayley graphs over the cyclic group) which has been open since 1967 is completely solved in this paper. The main result of the paper gives an efficient isomorphism criterion for circulant graphs of arbitrary order. This result also solves an isomorphism problem for colored circulant graphs and some classes of cyclic codes. 2000 Mathematics Subject Classification 05C60 (primary), 05C25, 05E30, 20B25 (secondary).

Minimum distance bounds for cyclic codes and Deligne's theorem

At the present time, there are very good methods to obtain bounds for the minimum distance of BCH codes and their duals. On the other hand, there are few other bounds suitable for general cyclic codes. Therefore, research Problem 9.9 of MacWilliams and Sloane (1977), The Theory of Error-Correcting Codes, asks if the bound of Deligne (1974) for exponential sums in several variables or the bound of Lang and Weil (1954), can be used to obtain bounds on the minimum distance of codes. This question is answered in the affirmative by showing how Deligne's theorem can be made to yield a lower bound on the minimum distance of certain classes of cyclic codes. In the process, an infinite family of binary cyclic codes is presented for which the bound on minimum distance so derived is as tight as possible. In addition, an infinite family of polynomials of degree 3 in 2 variables over a field of characteristic 2, for which Deligne's bound is tight, is exhibited. Finally, a bound is presented for the minimum distance of the duals of the binary subfield subcodes of generalized Reed-Muller codes as well as for the corresponding cyclic codes. It is noted that these codes contain examples of the best binary cyclic codes. >

Memory Package Error Detection and Correction

Single error correcting-double error detecting (SEC-DED) codes have been successfully used in computer memories for reliability. In the present-day technology of very large scale integration storage arrays bit error correction as well as byte error detection/byte error correction become extremely important. We devise here classes of cyclic codes with generator of the form (xb -1) p(x) for some suitable irreducible polynomial p(x) which provide 1) single byte error correction (SBEC), and 2) single error correction, double error detection, and byte error detection (SEC-DED-BED) codes. The codes have high information rate and are easy to implement. This general class of codes combines elegantly the results of several researchers and in some cases extends their results.