Chapter

Cyclic Codes

January 2003

In book: Block Error-Correcting Codes (pp.141-177)

Authors:

Polytechnic University of Catalonia

Cyclic codes are linear codes that are invariant under cyclic permutations of the components of its vectors. These codes have a nice algebraic structure (after reinterpreting vectors as univariate polynomials) which favors its study and use in a particularly effective way.

Development of Automated Identification Technology Objects During Their Movement Along Not Typed Routes Using Multi-Code Labeling

Conference Paper

Sep 2018

ResearchGate has not been able to resolve any references for this publication.

Conference Paper

Cyclotomic cosets and group algebra codes define over cyclic groups of prime order

May 2015

Denis C.K. Wong

Explicit formula for all primitive idempotents in F[G], the group algebra defined over all cyclic groups G of order less than or equal to 100 over the binary field F = F 2 are obtained. The minimum distance, the dimension, and the generating polynomial of the minimal codes generated by the primitive idempotents are also obtained.

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Explicit Cyclic and Quasi-Cyclic Codes With Optimal, Best Known Parameters, and Large Relative Minim...

December 2024 · IEEE Transactions on Information Theory

In this paper, we construct many infinite families of distance-optimal codes with new parameters, some of which are BCH codes and quasi-cyclic codes. In particular, we report the first infinite family of binary distance-optimal BCH codes with the minimum distance 8. Secondly, several infinite families of binary BCH codes and quasi-cyclic codes are presented. Many codes in these families have ... [Show full abstract] optimal or best known parameters. Thirdly, we construct infinite families of binary cyclic [ n ,≥ n +1/2, d ] ₂ codes with minimum distances d ≥ ⎡ n −1/Π ^s _{i =1} p_i ⎤, n = (2 ^{p 1} − 1)(2 ^{p 2} − 1) . . . (2 ^{ps} − 1), p ₁ , . . . , p_s are different primes. Our construction extends the main result of a recent paper published in IEEE Trans. Inf. Theory, 2023, by Sun, Li and Ding, to much more general binary cyclic codes with various lengths.We also construct an infinite family of binary quasi-cyclic codes with the rate around 1/2 and relative minimum distance lower bounded by O (1/log ₂ log ₂ n ).

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Abelian group factorization from cyclic and quasi-cyclic codes

October 2020 · Discrete Mathematics Algorithms and Applications

In this paper, we use the algebraic structures of cyclic codes and algorithmic techniques to find factorizations of abelian groups from cyclic codes. We construct specific subclasses of quasi-cyclic codes and provide the conditions with which we obtain a normalized factorization of certain abelian groups. The factorization, in both cases, is constituted by two sets, one corresponding to the ... [Show full abstract] cyclic code and the other corresponding to the words that represent all possible error polynomials of the cyclic code besides the zero vector.

Article

Algebraic theory of block codes detecting independent errors

February 1977 · Journal of Soviet Mathematics

V. A. Zinov'ev

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(Sigma-Delta) Codes

April 2013

In this paper we introduce the notion of cyclic (

f(t),\sigma,\delta

)-codes for f(t)\in \Ore. These codes generalize the

\theta

-codes as introduced by D. Boucher, F. Ulmer, W. Geiselmann \cite{BGU}. We construct generic and control matrices for these codes. As a particular case the (\si,\de)-W-code associated to a Wedderburn polynomial are defined and we show that their control matrices ... [Show full abstract] are given by generalized Vandermonde matrices. All the Wedderburn polynomials of

\mathbb F_q[t;\theta]

are described and their control matrices are presented. A key role will be played by the pseudo-linear transformations.

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On skew cyclic codes over $ M_{2}(\mathbb{F}_{2})

August 2023 · AIMS Mathematics

The algebraic structure of skew cyclic codes over

M_{2}

(

\mathbb{F}_2

), using the

\mathbb{F}_4

-cyclic algebra, is studied in this work. We determine that a skew cyclic code with a polynomial of minimum degree d(x) is a free code generated by d(x) . According to our findings, skew cyclic codes of odd and even lengths are cyclic and 2 -quasi-cyclic over $ M_{2}(\mathbb{F}_{2}) ... [Show full abstract]

, respectively. We provide the self-dual skew condition of Hermitian dual codes of skew cyclic codes. The generator polynomials of Euclidean dual codes are obtained. Furthermore, a spanning set of a double skew cyclic code over

M_{2}(\mathbb{F}_{2}) $ is considered in this paper.

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Skew-constacyclic codes over $\frac{\mathbb{F}_q[v]}{\langle\,v^q-v\,\rangle}

February 2019

In this paper, the investigation on the algebraic structure of the ring

\frac{\mathbb{F}_q[v]}{\langle\,v^q-v\,\rangle}

and the description of its automorphism group, enable to study the algebraic structure of codes and their dual over this ring. We explore the algebraic structure of skew-constacyclic codes, by using a linear Gray map and we determine their generator polynomials. Necessary and ... [Show full abstract] sufficient conditions for the existence of self-dual skew cyclic and self-dual skew negacyclic codes over

\frac{\mathbb{F}_q[v]}{\langle\,v^q-v\,\rangle}

are given.

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Conference Paper

Unknown syndrome calculation in algebraic decoding of a class of cyclic codes

September 2015

Recently, two novel matrices whose some entries are not syndromes and other entries are known syndromes have been presented to generate weak-locator polynomials needed in decoding the ternary quadratic residue code of length 61. This paper proposes a new unknown syndrome calculation for a class of cyclic codes and a completely algebraic decoding of the (23, 11, 9) ternary quadratic residue code ... [Show full abstract] up to actual minimum distance. For exactly four errors, the decoding algorithm developed here has to use the unknown syndrome, which appears in an entry of the above-mentioned matrix. The actual value of such an unknown syndrome can be determined precisely by the ratio of two determinants of matrices obtained from any one of the above-mentioned matrices.

Article

On row-cyclic codes with algebraic structure

April 1994 · Designs Codes and Cryptography

Roberta Evans Sabin

In this article, some row-cyclic error-correcting codes are shown to be ideals in group rings in which the underlying group is metacyclic. For a given underlying group, several nonequivalent codes with this structure may be generated. Each is related to a cyclic code generated in response, to the metrics associated with the underlying metacyclic group. Such codes in the same group ring are ... [Show full abstract] isomorphic as vector spaces but may vary greatly in weight distributions and so are nonequivalent. If the associated cyclic code is irreducible, examining the structure of its isomorphic finite field yields all nonequivalent codes with the desired structure. Several such codes have been found to have minimum distances equalling those of the best known linear codes of the same length and dimension.

Article

A commutative algebra approach to linear codes

April 2009 · Journal of Algebra

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 ... [Show full abstract] the decoding for a subclass, proving the existence of general error locator polynomials.

Conference Paper

On the automorphism groups of affine-invariant codes

January 1994 · Designs Codes and Cryptography

T.P. Berger

Using the classification of finite simple groups, the author obtains a condition for a permutation group on a finite field GF(pm ) which is necessary to have the affine group AGL(1,pm) as a subgroup. Applying this result to the automorphism groups of non trivial affine-invariant codes, the author proves that the automorphism groups discussed are always subgroups of the general affine group ... [Show full abstract] AGL(m,p)

Conference Paper

On quasi-cyclic codes

September 1998

Many binary quasi-cyclic codes have a better minimum distance than other known linear codes with the same parameters. We present 43 new quasi-cyclic codes, which improve on the parameters of previously found quasi-cyclic codes. Five of them even improve the lower bound on the minimum distance for linear codes. Furthermore we present a decoding algorithm for the binary quasi-cyclic codes of rate ... [Show full abstract] 1/m

Cyclic Codes

Abstract

No full-text available

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