Quasi-twisted Codes Research Articles

Self-orthogonal and quantum codes over chain rings

In this paper, we investigate the Gray images of codes over chain rings, leading to the derivation of infinite families of self-orthogonal linear codes over the residue field $\mathbb{F}_q$. We determine the parameters of optimal self-orthogonal and divisible linear codes. Additionally, we study the Gray images of quasi-twisted codes, resulting in some self-orthogonal Griesmer quasi-cyclic codes. Finally, we employ the CSS construction to derive some quantum codes based on self-orthogonal linear codes.

Gray Images of Constacyclic Codes over a Non-chain Extension of Z 4

Let [Formula: see text] be the ring of integers modulo 4. We study the [Formula: see text]-constacyclic and [Formula: see text]-cyclic codes over the non-chain ring [Formula: see text] for a unit [Formula: see text] in [Formula: see text]. We define several Gray maps and find that the respective Gray images of a quasi-cyclic code over [Formula: see text] are cyclic, quasi-cyclic or permutation equivalent to this code. For an odd positive integer [Formula: see text], we determine the generator polynomials of cyclic and [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text]. Further, we prove that a [Formula: see text]-cyclic code of length [Formula: see text] is a [Formula: see text]-constacyclic code if [Formula: see text] is odd, and a [Formula: see text]-quasi-twisted code if [Formula: see text] is even. A few examples are also incorporated, in which two parameters are new and one is best known to date.

Matrix product and quasi-twisted codes in one class

Many classical constructions, such as Plotkin’s and Turyn’s, were generalized by matrix product (MP) codes. Quasi-twisted (QT) codes, on the other hand, form an algebraically rich structure class that contains many codes with best-known parameters. We significantly extend the definition of MP codes to establish a broader class of generalized matrix product (GMP) codes that contains QT codes as well. We propose a generator matrix formula for any linear GMP code and provide a condition for determining the code size. We prove that any QT code has a GMP structure. Then we show how to build a generator polynomial matrix for a QT code from its GMP structure, and vice versa. Even though the class of QT codes contains many codes with best-known parameters, we present different examples of GMP codes with best-known parameters that are neither MP nor QT. Two different lower bounds on the minimum distance of GMP codes are presented; they generalize their counterparts in the MP codes literature. The second proposed lower bound replaces the non-singular by columns matrix with a less restrictive condition. Some examples are provided for comparing the two proposed bounds, as well as showing that these bounds are tight.

Quasi-twisted codes as contractions of quasi-cyclic codes

Quasi-twisted codes as contractions of quasi-cyclic codes

Ternary quantum codes constructed from a class of quasi-twisted codes

Ternary quantum codes constructed from a class of quasi-twisted codes

Lower Bound on the Minimum Distance of Single-Generator Quasi-Twisted Codes

We recall a classic lower bound on the minimum Hamming distance of constacyclic codes over finite fields, analogous to the well-known BCH bound for cyclic codes. This BCH-like bound serves as a foundation for proposing some minimum-distance lower bounds for single-generator quasi-twisted (QT) codes. Associating each QT code with a constacyclic code over an extension field, we obtain the first bound. This is the QT analogue to a result in the literature for quasi-cyclic codes. We point out some weaknesses in this bound and propose a novel bound that takes into account the Chinese remainder theorem approach to QT codes as well as the BCH bound of constacyclic codes. This proposed bound, in contrast to previous bounds in the literature, does not presuppose a specific form of code generator and does not require calculations in any extension field. We illustrate that our bound meets the one in the literature when the code generator adheres to the specific form assumed in that study. Various numerical examples enable us to compare and discuss these bounds.

New good quasi-cyclic codes and codes with improved minimum distance

One of the most important and challenging problems in coding theory is to construct codes with optimal parameters. As a generalization of cyclic codes, quasi-cyclic (QC) codes as well as quasi-twisted (QT) codes have been shown to contain record-breaking codes. In this paper, various computer algorithms have been used to search for good QC codes. A lot of good new QC codes have been found and they have been used to construct new linear codes. A total 11 new codes that improve the bound on the minimum distance are presented.

A note on constacyclic codes over the ring z_3[u, v

In this paper, we study λ-constacyclic codes over the ring R = Z 3 [u, v]/ for λ = (1 + u),(2 + 2u) and 2. We introduce a Gray map from R to Z 3 3 and show that the Gray image of a cyclic code is a quasi-cyclic code of index 3. It is proved that the Gray image of λ-constacyclic code over R is permutation equivalent to either quasi-cyclic or quasi-twisted code according to the value of λ. Moreover, we determine the structure of (1+u)-constacyclic codes for an odd length n over R and give some suitable examples.

New Binary Quantum Codes Derived from Quasi-Twisted Codes with Hermitian Inner Product

New Binary Quantum Codes Derived from Quasi-Twisted Codes with Hermitian Inner Product

A Comparison of Distance Bounds for Quasi-Twisted Codes

Spectral bounds on the minimum distance of quasi-twisted codes over finite fields are proposed, based on eigenvalues of polynomial matrices and the corresponding eigenspaces. They generalize the Semenov-Trifonov and Zeh-Ling bounds in a way similar to how the Roos and shift bounds extend the BCH and HT bounds for cyclic codes. The eigencodes of a quasi-twisted code in the spectral theory and the outer codes in its concatenated structure are related. A comparison based on this relation verifies that the Jensen bound always outperforms the spectral bound under special conditions, which yields a similar relation between the Lally and the spectral bounds. The performances of the Lally, Jensen and spectral bounds are presented in comparison with each other.

Several classes of asymptotically good quasi-twisted codes with a low index

Several classes of asymptotically good quasi-twisted codes with a low index

N-Dimension quasi-twisted codes of arbitrary length over finite fields

n-Dimension quasi-twisted (QT) codes are generalizations of 2-dimension QT codes and n-dimension quasi-cyclic codes. In this paper, we study some structural properties of n-dimension QT codes of arbitrary length over finite fields including the decomposition, the trace representation and the minimum Hamming distance bound.

Quantum Codes and Entanglement-Assisted Quantum Codes Derived from One-Generator Quasi-Twisted Codes

In this paper, we consider a special class of one-generator quasi-twisted (QT) codes of index 2 and explore their algebraic structure. By studying the form of Hermitian dual codes, a sufficient condition for self-orthogonality with Hermitian inner product is proposed. Using some one-generator QT codes, we construct some quantum codes on small finite fields. Moreover, a new method is provided to construct maximal-entanglement entanglement-assisted quantum codes via the class of QT codes. By comparison, our quantum codes and maximal-entanglement entanglement-assisted quantum codes have better parameters than any known codes.

A note on constacyclic codes over the ring z_3[u, v]/<u^2 −u, v^2, uv, vu>

In this paper, we study λ-constacyclic codes over the ring R = Z3[u, v]/<u2 − u, v2, uv, vu> for λ = (1 + u),(2 + 2u) and 2. We introduce a Gray map from R to Z33 and show that the Gray image of a cyclic code is a quasi-cyclic code of index 3. It is proved that the Gray image of λ-constacyclic code over R is permutation equivalent to either quasi-cyclic or quasi-twisted code according to the value of λ. Moreover, we determine the structure of (1+u)-constacyclic codes for an odd length n over R and give some suitable examples.

On normalized generating sets for generalized quasi-twisted codes over finite fields

In this short correspondence, we firstly determine the normalized generating set for generalized quasi-twisted codes over finite fields. Then we present an algorithm to construct it. As an application, some optimal or the best-known linear codes are derived from generalized quasi-twisted codes by applying our method.

Constructions of quasi-twisted quantum codes

In this work, our main objective is to construct quantum codes from quasi-twisted (QT) codes. At first, a necessary and sufficient condition for Hermitian self-orthogonality of QT codes is introduced by virtue of the Chinese remainder theorem. Then, we utilize these self-orthogonal QT codes to provide quantum codes via the famous Hermitian construction. Moreover, we present a new construction method of q-ary quantum codes, which can be viewed as an effective generalization of the Hermitian construction. General QT codes that are not self-orthogonal are also employed to construct quantum codes. As the computational results, some binary, ternary and quaternary quantum codes are constructed and their parameters are determined, which all cannot be deduced by the quantum Gilbert–Varshamov bound. In the binary case, a small number of quantum codes are derived with strictly improved parameters compared with the current records. In the ternary and quaternary cases, our codes fill some gaps or have better performances than the current results.

A bound on the minimum distance of generalized quasi-twisted codes

Generalized quasi-twisted (GQT) codes form a generalization of quasi-twisted (QT) codes and generalized quasi-cyclic (GQC) codes. By the Chinese remainder theorem, the GQT codes can be decomposed into a direct sum of some linear codes over Galois extension fields, which leads to the trace representation of the GQT codes. Using this trace representation, we first prove the minimum distance bound for GQT codes with two constituents. Then we generalize the result to GQT codes with s constituents. Finally, we present some examples to show that the bound is better than the well-known Esmaeili-Yari bound and sharp in many instances.

Construction of quasi-twisted codes and enumeration of defining polynomials

Let $d_{q}(n,k)$ be the maximum possible minimum Hamming distance of a linear [$n,k$] code over $\mathbb{F}_{q}$. Tables of best known linear codes exist for small fields and some results are known for larger fields. Quasi-twisted codes are constructed using $m \times m$ twistulant matrices and many of these are the best known codes. In this paper, the number of $m \times m$ twistulant matrices over $\mathbb{F}_q$ is enumerated and linear codes over $\mathbb{F}_{17}$ and $\mathbb{F}_{19}$ are constructed for $k$ up to $5$.

Some generalizations of the ASR search algorithm for quasitwisted codes

One of the most important and challenging problems in coding theory is explicit construction of linear codes with the best possible parameters. It is well known that the class of quasitwisted (QT) codes is asymptotically good and contains many linear codes with best known parameters (BKLCs). A search algorithm (ASR) on QT codes has been particularly effective to construct such codes. Recently, the ASR algorithm was generalized based on the notion of code equivalence. In this work, we introduce a new generalization of the ASR algorithm to include a broader scope of QT codes. As a result of implementing this algorithm, we have found eight new linear codes over the field 𝔽 5 . Furthermore, we have found seven additional new codes from the standard constructions of puncturing, shortening or Construction X. We also introduce a new search algorithm that can be viewed as a further generalization of ASR into the class multitwisted (MT) codes. Using this method, we have found many codes with best known parameters with more direct and desirable constructions than what is currently available in the database of BKLCs.

A modified Gilbert-Varshamov bound for self-dual quasi-twisted codes of index four

In this work, we study a family of self-dual four circulant codes and a family of self-dual four negacirculant codes, and the exact counting formula is derived for these families of codes. In addition, we prove that asymptotically good self-dual four circulant codes and negacirculant codes over finite fields exist, and both of them satisfy a modified Gilbert-Varshamov bound on the relative minimum distance with asymptotic rate 12.