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2012
A cyclic code is associated with another cyclic code to bound its minimum distance. The algebraic relation between these two codes allows the formulation of syndromes and a key equation. In this contribution, we outline the decoding approach for the case of errors and erasures and show how the Extended Euclidean Algorithm can be used for decoding.
IEEE Transactions on Information Theory, 2012
A new lower bound on the minimum distance of qary cyclic codes is proposed. This bound improves upon the Bose-Chaudhuri-Hocquenghem (BCH) bound and, for some codes, upon the Hartmann-Tzeng (HT) bound. Several Boston bounds are special cases of our bound. For some classes of codes the bound on the minimum distance is refined. Furthermore, a quadratic-time decoding algorithm up to this new bound is developed. The determination of the error locations is based on the Euclidean Algorithm and a modified Chien search. The error evaluation is done by solving a generalization of Forney's formula. Index Terms-Bose-Chaudhuri-Hocquenghem (BCH) bound, cyclic codes, Forney's formula, Hartmann-Tzeng (HT) bound, Roos bound.
2012
A new approach to bound the minimum distance of q-ary cyclic codes is presented. The connection to the BCH and the Hartmann-Tzeng bound is formulated and it is shown that for several cases an improvement is achieved.
IEEE Transactions on Information Theory, 1988
In this paper, we first investigate the distance structure of cyclic codes of composite length. A lower bound on the minimum distance for this class of codes is derived. In many cases, the lower bound gives the true minimum distance of a code. Then, we investigate the distance structure of the direct sum of two cyclic codes of composite length. We show that, under certain conditions, the direct-sum code provides two levels of error correcting capability, and hence is a two-level unequal error protection (UEP) code. Finally, a class of two-level UEP cyclic direct-sum codes and a decoding algorithm for a subclass of these codes are presented. ' at most n-1 which implies that a nonzero v(X) has at most n-1 distinct roots. Thus, v(X)=O. This implies that vl(X)=v2(X).
In forward error correction (FEC) systems, basically two mechanisms namely block coding and convolutional coding are used for error-control coding. The error-control coding processes involve encoding and decoding information symbols to get rid of errors produced by noise in communication channels. In a binary block encoding technique, message words are arranged in blocks of k bits which are converted into code words of n bits, n> k , by adding redundancy bits. Encoding is a compulsory part of error correcting codes and without knowledge of encoding structure the decoding of code words is impossible. In this paper a very simple and most effective technique for systematic encoding of linear binary cyclic codes is used which improves the encoding speed by reducing the computational complexity. In addition to encoding speed, the encoding can be done by hand using XOR (modulo-2) operations without involving computers after generating an encoding table for the code.
2012
A new bound on the minimum distance of q-ary cyclic codes is proposed. It is based on the description by another cyclic code with small minimum distance. The connection to the BCH bound and the Hartmann-Tzeng (HT) bound is formulated explicitly. We show that for many cases our approach improves the HT bound. Furthermore, we refine our bound for several families of cyclic codes.
2011
A new bound on the distance of binary cyclic codes is proposed. The approach is based on the representation of a subset of the roots of the generator polynomial by a rational function. A new bound on the minimum distance is proven and several classes of binary cyclic codes are identified. For some classes of codes, this bound is better than the known bounds (e.g. BCH or Hartmann-Tzeng bound). Furthermore, a quadratic-time decoding algorithm up to this new bound is developed.
2010 Information Theory and Applications Workshop (ITA), 2010
Among the classes of LDPC codes that have been constructed and designed, the only class of LDPC codes that are cyclic is the class of codes constructed based on the incidence vectors of lines of finite geometries, called finite geometry LDPC codes. Cyclic finite geometry LDPC codes are known to have large minimum distances and can provide good error performance with very low error-floors using iterative decoding based on belief propagation. Their cyclic structure allows them to be efficiently and systematically encoded with simple shift-registers in linear time with linear complexity. An obvious question is whether, besides cyclic finite geometry LDPC codes, there are other cyclic codes with large minimum distances that can be efficiently decoded iteratively using channel soft information. In this paper, we present one such class of cyclic codes. Codes in this class are two-step majority-logic decodable and they are also constructed based on finite geometries. Two iterative decoding algorithms are devised for this class of cyclic codes and they provide significant coding gain over the two-step majority-logic decoding of codes in this class.
IEEE Access
Let p be an odd prime, q = p m , R = F q +uF q with u 2 = 1, and S = F q +uF q +vF q +uvF q with u 2 = 1, v 2 = 1, uv = vu. In this paper, F q RS-cyclic codes over F q RS are studied. As an application, we present a construction of quantum error-correcting codes (QECCs) from the F q RS-cyclic codes over F q RS, which provides new QECCs. We also consider linear complementary dual (LCD) codes from the F q RS-cyclic codes over F q RS. Among others, we construct a Gray map over F q RS and discuss the Gray images of F q RS-cyclic codes over F q. INDEX TERMS Cyclic codes, Mixed alphabet codes, QECCs, LCD codes.
arXiv (Cornell University), 2023
The Euclidean hull of a linear code C is defined as C ∩ C ⊥ , where C ⊥ denotes the dual of C under the Euclidean inner product. A linear code with zero hull dimension is called a linear complementary dual (LCD) code. A pair (C, D) of linear codes of length n over F q is called a linear complementary pair (LCP) of codes if C ⊕ D = F n q. In this paper, we give a characterization of LCD and LCP of cyclic codes of length q m − 1, m ≥ 1, over the finite field F q in terms of their basic dual zeros and their trace representations. We also formulate the hull dimension of a cyclic code of arbitrary length over F q with respect to its basic dual zero. Moreover, we provide a general formula for the dimension of the intersection of two cyclic codes of arbitrary length over F q based on their basic dual zeros.
Indian journal of science and technology, 2012
In forward error correction (FEC) systems, basically two mechanisms namely block coding and convolutional coding are used for error-control coding. The error-control coding processes involve encoding and decoding information symbols to get rid of errors produced by noise in communication channels. In a binary block encoding technique, message words are arranged in blocks of k bits which are converted into code words of n bits, n> k , by adding redundancy bits. Encoding is a compulsory part of error correcting codes and without knowledge of encoding structure the decoding of code words is impossible. In this paper a very simple and most effective technique for systematic encoding of linear binary cyclic codes is used which improves the encoding speed by reducing the computational complexity. In addition to encoding speed, the encoding can be done by hand using XOR (modulo-2) operations without involving computers after generating an encoding table for the code.
Applicable Algebra in Engineering, Communication and Computing
Cyclic codes are among the most important families of codes in coding theory for both theoretical and practical reasons. Despite their prominence and intensive research on cyclic codes for over a half century, there are still open problems related to cyclic codes. In this work, we use recent results on the equivalence of cyclic codes to create a more efficient algorithm to partition cyclic codes by equivalence based on cyclotomic cosets. This algorithm is then implemented to carry out computer searches for both cyclic codes and quasi-cyclic (QC) codes with good parameters. We also generalize these results to repeated-root cases. We have found several new linear codes that are cyclic or QC as an application of the new approach, as well as more desirable constructions for linear codes with best known parameters. With the additional new codes obtained through standard constructions, we have found a total of 14 new linear codes.
AIP Conference Proceedings, 2019
The codes generated using tensor product and called tensor codes have properties and composition similar to Linear Error Block codes (LEB codes). In this paper we study in depth the construction of new LEB codes using tensor product (TP). We also show that the TP code formed by two LEB codes is also an LEB code. We prove that the TP of two Hamming codes is not a Hamming code with minimum distance 3, besides, it's a non-perfect LEB code. We show that the TP code formed by two π-cyclic codes (resp. simplex LEB codes) is a π-cyclic code (resp. simplex LEB code).
2013
Two generalizations of the Hartmann-Tzeng (HT) bound on the minimum distance of q-ary cyclic codes are proposed. The first one is proven by embedding the given cyclic code into a cyclic product code. Furthermore, we show that unique decoding up to this bound is always possible and outline a quadratic-time syndrome-based error decoding algorithm. The second bound is stronger and the proof is more involved.
Computers & Mathematics with Applications, 2012
The structure of DNA is used as a model for constructing good error correcting codes and conversely error correcting codes that enjoy similar properties with DNA structure are also used to understand DNA itself. Recently, naturally four element sets are used to model DNA by some families of error correcting codes. Hence the structure of such codes has been studied. In this paper, the authors first relate DNA pairs with a special 16 element ring. Then, the so-called cyclic DNA codes of odd length that enjoy some of the properties of DNA are studied. Their algebraic structure is determined. Further, by introducing a map, a family of cyclic codes over this ring is mapped to DNA codes. Hamming minimum distances are also studied. The paper concludes with some DNA examples obtained via this family of cyclic codes.
IEEE Transactions on Information Theory, 2013
Known properties of cyclic codes are used to give a unified description of many classical decoding algorithms for Reed-Solomon codes up to half the minimum distance. This description allows also simplified proofs for these decoders. Further, a novel decoding algorithm is derived using these properties directly and variants of a new error/erasure decoding algorithm are given. For decoding beyond half the minimum distance, a basis of all solutions for decoding is derived. This basis allows to use side information in order to decode beyond half the minimum distance. Other methods where this basis can be used are power decoding, also known as virtual syndrome extension, where additional equations are generated by taking powers of the received symbols, and interleaved Reed-Solomon codes. The extended Euclidean algorithm, which calculates the greatest common divisor, plays an essential role in many presented methods.
There exist a need to develop decoding algorithms that have the ability to quickly recover from erasures only. One application of such a decoder can be in a packet-switched network transporting real-time data. At the higher network layers of these networks, all the data packets are received errorfree, or one or more packets are lost. With real-time data, the delay introduced while recovering from these losses can have a great influence on the quality of service of the application. In this paper the authors introduce a generic erasures-only decoding algorithm, based on simple linear algebra and arithmetic, to enable fast recovery from burst erasures using cyclic codes of the form g(x) · P (x) from GF (2 m ).
Journal of the Franklin Institute, 2013
The ring in the title is the first non commutative ring to have been used as alphabet for block codes. The original motivation was the construction of some quaternionic modular lattices from codes. The new application is the construction of space time codes obtained by concatenation from the Golden code. In this article, we derive structure theorems for cyclic codes over that ring, and use them to characterize the lengths where self dual cyclic codes exist. These codes in turn give rise to formally self dual quaternary codes.
1989
The class of Quasi-Cyclic Error Correcting Codes is investigated. It is shown that they contain many of the best known binary and nonbinary codes. Tables of rate 1/p and (p − 1)/p Quasi-Cyclic (QC) codes are constructed, which are a compilation of previously best known codes as well as many new codes constructed using exhaustive, and other more sophisticated search techniques. Many of these binary codes attain the known bounds on the maximum possible minimum distance, and 13 improve the bounds. The minimum distances and generator polynomials of all known best codes are given. The search methods are outlined and the weight divisibility of the codes is noted. The weight distributions of some s-th Power Residue (PR) codes and related rate 1/s QC codes are found using the link established between PR codes and QC codes. Subcodes of the PR codes are found by deleting certain circulant matrices in the corresponding QC code. They are used as a starting set of circulants for other techniques. Nonbinary Power Residue codes and related QC codes are constructed over GF(3), GF(4), GF(5), GF(7) and GF(8). Their subcodes are also used to find good nonbinary QC codes. A simple and efficient algorithm for constructing primitive polynomials with linearly independent roots over the Galois Field of q elements, GF(q), is developed. Tables of these polynomials are presented. These Tables are unknown for polynomials with nonbinary coefficients, and the known binary Tables are incomplete. The polynomials are employed in such diverse areas as construction of error correcting codes, efficient VLSI implementation of multiplication and inverse operations over Galois Fields, and digital testing of integrated circuits. Using the link established between generalized tail biting convolutional codes and binary QC codes, good QC codes are constructed based on iii Optimum Distance Profile (ODP) convolutional codes. Several best rate 2/3 systematic codes up to circulant size 20 are constructed in this manner.
2010
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Designs, Codes and Cryptography
Recently, a new algorithm to test equivalence of two cyclic codes has been introduced which is efficient and produced useful results. In this work, we generalize this algorithm to constacyclic codes. As an application of the algorithm we found many constacyclic codes with good parameters and properties. In particular, we found 23 new codes that improve the minimum distances of BKLCs.
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