Quasi-twisted Codes Research Articles

A database of linear codes over F_13 with minimum distance bounds and new quasi-twisted codes from a heuristic search algorithm

Error control codes have been widely used in data communications and storage systems. One central problem in coding theory is to optimize the parameters of a linear code and construct codes with best possible parameters. There are tables of best-known linear codes over finite fields of sizes up to 9. Recently, there has been a growing interest in codes over $\mathbb{F}_{13}$ and other fields of size greater than 9. The main purpose of this work is to present a database of best-known linear codes over the field $\mathbb{F}_{13}$ together with upper bounds on the minimum distances. To find good linear codes to establish lower bounds on minimum distances, an iterative heuristic computer search algorithm is employed to construct quasi-twisted (QT) codes over the field $\mathbb{F}_{13}$ with high minimum distances. A large number of new linear codes have been found, improving previously best-known results. Tables of $[pm, m]$ QT codes over $\mathbb{F}_{13}$ with best-known minimum distances as well as a table of lower and upper bounds on the minimum distances for linear codes of length up to 150 and dimension up to 6 are presented.

Note on quasi-twisted codes and an application

Recently, Jia proposed the decompositions and trace representations of quasi-twisted (QT) codes over finite fields (Finite Fields Appl 18:237–257, 2012). The present paper can be viewed as a complementary part of Jia’s work. We investigate some other useful properties of \(\lambda \)-QT codes over finite fields, including the lower Hamming distance bounds, enumerations and searching algorithm for generators. As an interesting application of \(\lambda \)-QT codes over finite fields, we study \(\lambda \)-QT codes over the finite non-chain ring \(\mathbb {F}_q+v\mathbb {F}_q\) briefly.

A new iterative computer search algorithm for good quasi-twisted codes

As a generalization to cyclic and consta-cyclic codes, quasi-twisted (QT) codes contain many good linear codes. During the last twenty years, a lot of record-breaking codes have been found by computer search for good QT codes. But due to the time complexity, very few QT codes have been reported recently. In this paper, a new iterative, heuristic computer search algorithm is presented, and a lot of new QT codes have been obtained. With these results, a total of 45 entries in the code tables for the best-known codes have been improved. Also, as an example to show the effectiveness of the algorithm, 8 better binary quasi-cyclic codes with dimension 12 and $$m = 13$$m=13 than previously best-known results are constructed.

New linear codes from constacyclic codes

One of the main challenges of coding theory is to construct linear codes with the best possible parameters. Various algebraic and combinatorial methods along with computer searches are used to construct codes with better parameters. Given the computational complexity of determining the minimum distance of a code, exhaustive searches are not feasible for all but small parameter sets. Therefore, codes with certain algebraic structures are preferred for both theoretical and practical reasons. In this work we focus on the class of constacyclic codes to first generate all constacyclic codes exhaustively over small finite fields of order up to 9 to create a database of best constacyclic codes. We will then use this database as a building block for a search algorithm for new quasi-twisted codes. Our search on constacyclic codes has revealed 16 new codes, i.e. codes with better parameters than currently best-known linear codes. Given that constacyclic codes are well known, this is a surprising result. Moreover, using the standard constructions of puncturing, shortening or extending a given code, we also derived 55 additional new codes from these constacyclic codes. Hence, we achieved improvements on 71 entries in the database of best-known codes. We use a search strategy that is comprehensive, i.e. it computes every constacyclic code for a given length and shift constant, and it avoids redundantly examining constacyclic codes that are equivalent to either cyclic codes or other constacyclic codes.

On quasi-twisted codes over finite fields

In coding theory, quasi-twisted (QT) codes form an important class of codes which has been extensively studied. We decompose a QT code to a direct sum of component codes – linear codes over rings. Furthermore, given the decomposition of a QT code, we can describe the decomposition of its dual code. We also use the generalized discrete Fourier transform to give the inverse formula for both the nonrepeated-root and repeated-root cases. Then we produce a formula which can be used to construct a QT code given the component codes.

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

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.

An Explicit Construction of $2$-Generator Quasi-Twisted Codes

Quasi-twisted (QT) codes are a generalization of quasi-cyclic (QC) codes. Based on consta-cyclic simplex codes, a new explicit construction of a family of 2-generator quasi-twisted (QT) two-weight codes is presented. It is also shown that many codes in the family meet the Griesmer bound and therefore are length-optimal. New distance-optimal binary QC [195, 8, 96], [210, 8, 104], and [240, 8, 120] codes, and good ternary QC [208, 6, 135] and [221, 6, 144] codes are also obtained by the construction.

Constructing linear codes from some orbits of projectivities

It is proved that a linear code defined by some orbits of a projectivity is a degenerate quasi-twisted code. A lot of new linear codes over the field of q elements ( q = 3 , 4 , 5 , 8 , 9 ) improving the best known bounds (e.g., an optimal [ 90 , 9 , 54 ] 3 code) are found from such codes by some combinations of puncturing or extending.

Consta-Abelian Codes Over Galois Rings

We study n-length consta-Abelian codes (a generalization of the well-known Abelian codes and constacyclic codes) over Galois rings of characteristic p/sup a/, where n and p are coprime. A twisted discrete Fourier transform (DFT) is used to generalize transform domain results of Abelian and constacyclic codes, to consta-Abelian codes. Further, we characterize consta-Abelian codes invariant under two kinds of monomials, whose underlying permutations are effected by: i) multiplying the coordinates with a unit in the appropriate mixed-radix representation of the coordinate positions and ii) shifting the coordinates by t positions. All the codes studied here belong to the class of quasi-twisted codes which are known to contain some good codes. We show that the dual of a consta-Abelian code invariant under the two monomials is also a consta-Abelian code closed under both monomials.

New quasi-twisted degenerate ternary linear codes

Twenty six ternary linear quasi-twisted codes improving the best known lower bounds on minimum distance are constructed.

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.

Improved Bounds for Ternary Linear Codes of Dimension 8 Using Tabu Search

In this paper, new codes of dimension 8 are presented which give improved bounds on the maximum possible minimum distance of ternary linear codes. These codes belong to the class of quasi-twisted (QT) codes, and have been constructed using a stochastic optimization algorithm, tabu search. Twenty three codes are given which improve or establish the bounds for ternary codes. In addition, a table of upper and lower bounds for d3(n, 8) is presented for n ≤ 200.

New quasi-twisted quaternary linear codes

Let [n, k, d]/sub q/-codes be linear codes of length n, dimension k, and minimum Hamming distance d over GF(q). In this article, 31 new quaternary (q=4) codes are constructed which improve the respective lower bounds on minimum distance in Brouwer's tables.

Improved Bounds for Quaternary Linear Codes of Dimension 6

In this paper, twenty new codes of dimension 6 are presented which give improved bounds on the maximum possible minimum distance of quaternary linear codes. These codes belong to the class of quasi-twisted (QT) codes, and have been constructed using a stochastic optimization algorithm, tabu search. A table of upper and lower bounds for d4(n,6) is presented for n≤ 200.

Improvements to the bounds on optimal ternary linear codes of dimension 6

New ternary codes of dimension 6 are presented which improve the bounds on optimal linear codes. These codes belong to the class of quasi-twisted (QT) codes, and have been constructed using a greedy algorithm. This work extends previous results on QT codes of dimension 6. In particular, several new two-weight QT codes are presented. Numerous new optimal codes which meet the Griesmer bound are given, as well as others which establish lower bounds on the maximum minimum distance.

New optimal quaternary linear codes of dimension 5

In this correspondence, new optimal quaternary linear codes of dimension 5 are presented which extend previously known results. These codes belong to the class of quasi-twisted (QT) codes, and have been obtained using a greedy local search algorithm. Other codes are also given which provide a lower bound on the maximum possible minimum distance. The generator polynomials for these codes are tabulated, and the minimum distances of known QT codes are given.

New optimal ternary linear codes

The class of quasi-twisted (QT) codes is a generalization of the class of quasi-cyclic codes, similar to the way constacyclic codes are a generalization of cyclic codes. In this paper, rate 1/p QT codes over GF(3) are presented which have been constructed using integer linear programming and heuristic combinatorial optimization. Many of these attain the maximum possible minimum distance for any linear code with the given parameters, and several improve the maximum known minimum distances. Two of these new codes, namely (90, 6, 57) and (120, 6, 78), are optimal and so prove that d/sub 3/(90, 6)=57 and d/sub 3/(120, 6)=78.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>