Nuh Aydin

The Structure of 1-Generator Quasi-Twisted Codes and New Linear Codes

One of the most important problems of coding theory is to construct codes with best possible minimum distances. Recently, quasi-cyclic (QC) codes have been proven to contain many such codes. In this paper, we consider quasi-twisted (QT) codes, which are generalizations of QC codes, and their structural properties and obtain new codes which improve minimum distances of best known linear codes over the finite fields GF(3) and GF(5). Moreover, we give a BCH-type bound on minimum distance for QT codes and give a sufficient condition for a QT code to be equivalent to a QC code.

Quasi-cyclic codes over Z/sub 4/ and some new binary codes

Previously, (linear) codes over Z/sub 4/ and quasi-cyclic (QC) codes (over fields) have been shown to yield useful results in coding theory. Combining these two ideas we study Z/sub 4/-QC codes and obtain new binary codes using the usual Gray map. Among the new codes, the lift of the famous Golay code to Z/sub 4/ produces a new binary code, a (92, 2/sup 24/, 28)-code, which is the best among all binary codes (linear or nonlinear). Moreover, we characterize cyclic codes corresponding to free modules in terms of their generator polynomials.

Skew cyclic codes of arbitrary length

In this paper, we study a special type of linear codes, called skew cyclic codes, in the most general case. This set of codes is a generalisation of cyclic codes but constructed using a non-commutative ring called the skew polynomial ring. In previous works, these codes have been studied with certain restrictions on their length. This work examines their structure for an arbitrary length without any restriction. Our results show that these codes are equivalent to either cyclic codes or quasi-cyclic codes, hence establish strong connections with well-known classes of codes.

On the Construction of Skew Quasi-Cyclic Codes

In this paper, we study a special type of quasi-cyclic (QC) codes called skew QC codes. This set of codes is constructed using a noncommutative ring called the skew polynomial ring F[x;¿]. After a brief description of the skew polynomial ring F[x;¿], it is shown that skew QC codes are left submodules of the ring Rsl=( F[x;¿]/(xs-1) )l. The notions of generator and parity-check polynomials are given. We also introduce the notion of similar polynomials in the ring F[x;¿] and show that parity-check polynomials for skew QC codes are unique up to similarity. Our search results lead to the construction of several new codes with Hamming distances exceeding the Hamming distances of the previously best known linear codes with comparable parameters.

On cyclic codes over Z4 + uZ4 and their Z4-images

Linear codes over the ring ℤ4 + uℤ4 have been introduced recently. In this paper, we study cyclic codes over this ring. We determine algebraic structures of cyclic codes over ℤ4 + uℤ4 and obtain basic facts about their generators. Making use of their algebraic structure, we conducted a computer search for cyclic codes over ℤ4 + uℤ4 and obtained many new linear codes over ℤ4. These codes have been added to the database of ℤ4 codes.

New quinary linear codes from quasi-twisted codes and their duals

Abstract One of the central problems in algebraic coding theory is construction of linear codes with best possible parameters. Quasi-twisted (QT) codes have been promising to solve this problem. Despite extensive search in this class and discovery of a large number of new codes, we have been able to find still more new codes that are QT over the alphabet F 5 using a more comprehensive search strategy.

Cyclic and some constacyclic codes over the ring Z 4 u { u 2 - 1 }

In this paper, we study cyclic codes and constacyclic codes with shift constant ( 2 + u ) over R = Z 4 + u Z 4 , where u 2 = 1 . We determine the form of the generators of the cyclic codes over this ring and their spanning sets. Considering their Z 4 images, we prove that the Z 4 -image of a ( 2 + u ) -constacyclic code of odd length is a cyclic code over Z 4 . We also present many examples of cyclic codes over R whose Z 4 -images have better parameters than previously best-known Z 4 -linear codes.

Some Open Problems on Quasi-Twisted and Related Code Constructions and Good Quaternary Codes

One of the most important and challenging problems in coding theory is to construct codes with the best possible parameters. Quasi-cyclic (QC) and the larger class of quasi- twisted (QT) codes have been proven to contain many good codes (with best-known parameters). In this paper, we review some open problems concerning these codes, introduce generalizations of QT codes, and suggest some constructions involving QT codes. We also present some new and good quaternary codes.

New quasi-cyclic codes over F5

Abstract One of the most important problems of coding theory is to construct codes with best possible minimum distances. In this paper, we use the algebraic structure of quasi-cyclic codes and the BCH type bound on the minimum distance to search for quasicyclic codes over F 5 , the finite field with five elements, which improve the minimum distances of best-known linear codes. We construct 15 new linear codes of this type.

LDPC Codes of Arbitrary Girth

For regular, degree two LDPC (low density parity-check) codes, there is a strong relationship between high girth and performance. This article presents a greedy algorithm, called successive level growth (SLG), for the construction of LDPC codes with arbitrarily specified girth. The simulation results show that our codes exhibit significant coding gains over randomly constructed LDPC codes and in some cases outperform PEG codes in the additive white Gaussian noise channel.