Construction Of Cyclic Codes Research Articles

Constructions of cyclic codes and extended primitive cyclic codes with their applications

Linear codes with a few weights have many nice applications including combinatorial design, distributed storage system, secret sharing schemes and so on. In this paper, we construct two families of linear codes with a few weights based on special polynomials over finite fields. The first family of linear codes are extended primitive cyclic codes which are affine-invariant. The second family of linear codes are reducible cyclic codes. The parameters of these codes and their duals are determined. As the first application, we prove that these two families of linear codes hold t-designs, where t=2,3. As the second application, the minimum localities of the codes are also determined and optimal locally recoverable codes are derived.

Cyclic LRC-LCD Codes With Hierarchical Locality

Locally recoverable (LRC) codes and linear complementary dual (LCD) codes are both useful in distributed storage systems. In this letter, we establish a connection between LRC codes and LCD codes. We derive some conditions on the construction of cyclic codes with $(r,\delta)$ locality and codes with hierarchical locality so that they are also LCD codes. Some constructed codes are optimal. These codes may be useful in simultaneously providing local recoverability and security against certain attacks in distributed storage systems.

On Squares of Cyclic Codes

The square C* <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> of a linear error correcting code C is the linear code spanned by the component-wise products of every pair of (non-necessarily distinct) words in C. Squares of codes have gained attention for several applications mainly in the area of cryptography, and typically in those applications, one is concerned about some of the parameters (dimension and minimum distance) of both C* <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and C. In this paper, motivated mostly by the study of this problem in the case of linear codes defined over the binary field, squares of cyclic codes are considered. General results on the minimum distance of the squares of cyclic codes are obtained, and constructions of cyclic codes C with a relatively large dimension of C and minimum distance of the square C* <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> are discussed. In some cases, the constructions lead to codes C such that both C and C* <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> simultaneously have the largest possible minimum distances for their length and dimensions.

Self-dual skew codes and factorization of skew polynomials

The construction of cyclic codes can be generalized to so-called “module θ-codes” using noncommutative polynomials. The product of the generator polynomial g of a self-dual “module θ-code” and its “skew reciprocal polynomial” is known to be a noncommutative polynomial of the form Xn−a, reducing the problem of the computation of all such codes to the resolution of a polynomial system where the unknowns are the coefficients of g. We show that a must be ±1 and that over F4 for n=2s the factorization of the generator g of a self-dual θ-cyclic code has some rigidity properties which explains the small number of self-dual θ-cyclic codes with length n=2s. In the case θ of order two, we present a construction of self-dual codes, based on the least common multiples of noncommutative polynomials, that allows to reduce the computation to polynomial systems of smaller sizes than the original one. We use this approach to construct a [78,39,19]4 self-dual code and a [52,26,17]9 self-dual code which improve the best previously known minimal distances for these lengths.

A q-polynomial approach to cyclic codes

Cyclic codes have been an interesting topic of both mathematics and engineering for decades. They are prominently used in consumer electronics, data transmission technologies, broadcast systems, and computer applications. Three classical approaches to the study and construction of cyclic codes are those based on the generator matrix, the generator polynomial and the idempotent. The objective of this paper is to develop another approach – the q-polynomial approach. Fundamental theory of this approach will be developed, and will be employed to construct a new family of cyclic codes in this paper.

Encoding through generalized polynomial codes

This paper introduces novel constructions of cyclic codes using semigroup rings instead of polynomial rings. These constructions are applied to define and investigate the BCH, alternant, Goppa, and Srivastava codes. This makes it possible to improve several recent results due to Andrade and Palazzo [1].

Construction of cyclic codes over [formula omitted] for DNA computing

In this paper, we develop the theory for constructing linear and additive cyclic codes of odd length over GF ( 4 ) that are suitable for DNA computing. We call this class of codes reversible complement cyclic codes. We use this theory to study all such codes of lengths 7 , 9 , 11 and 13 . We list the codes that have the largest number of codewords for a given minimum Hamming distance. We show that some of these codes have more codewords than previously known codes with the same minimum Hamming distance.

Construction and decomposition of cyclic codes of composite length (Corresp.)

A construction of cyclic codes in which an "inner" cyclic code is concatenated with appropriately chosen "outer" cyclic codes is presented. An example of the construction that improves on BCH codes of low rate is provided. Conversely, any cyclic code of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n_1 n_2</tex> can be constructed by this construction, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n_1</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n_2</tex> are relatively prime.