Introduction to coding theory (2nd edition), by J.H. van Lint. Pp.183 DM86. 1992. ISBN 0-387-54894-7 (Springer)

Cyclic codes and their various generalizations, such as quasi-twisted (QT) codes, have a special place in algebraic coding theory. Among other things, many of the best-known or optimal codes have been obtained from these classes. In this work we introduce a new generalization of QT codes that we call multi-twisted (MT) codes and study some of their basic properties. Presenting several methods of constructing codes in this class and obtaining bounds on the minimum distances, we show that there exist codes with good parameters in this class that cannot be obtained as QT or constacyclic codes. This suggests that considering this larger class in computer searches is promising for constructing codes with better parameters than currently best-known linear codes. Working with this new class of codes motivated us to consider a problem about binomials over finite fields and to discover a result that is interesting in its own right.

A Generalization of Quasi-twisted Codes: Multi-twisted codes

A generalization of quasi-twisted codes: Multi-twisted codes

In this paper, we introduce two classes of MacDonald codes over the finite non-chain ring \begin{document}$ \mathbb{F}_p+v\mathbb{F}_p+v^2\mathbb{F}_p $\end{document} and their torsion codes which are linear codes over \begin{document}$ \mathbb{F}_p $\end{document} , where \begin{document}$ p $\end{document} is an odd prime and \begin{document}$ v^3 = v $\end{document} . We give the complete weight enumerator of two classes of torsion codes. As an application, systematic authentication codes are obtained by these torsion codes.

https://www.aimsciences.org/article/exportPdf?id=65ff648c-d548-4365-b618-04935b0964ae

Complete weight enumerator of torsion codes

Let p be an odd prime, q = p
 m
, R = F
 q
 + uF
 q
 with u
 2
 = 1, and S = F
 q
 + uF
 q
 + vF
 q
 + uvF
 q
 with u
 2
 = 1, v
 2
 = 1, uv = vu. In this paper, F
 q
RS-cyclic codes over F
 q
RS are studied. As an application, we present a construction of quantum error-correcting codes (QECCs) from the F
 q
RS-cyclic codes over F
 q
RS, which provides new QECCs. We also consider linear complementary dual (LCD) codes from the F
 q
RS-cyclic codes over F
 q
RS. Among others, we construct a Gray map over FqRS and discuss the Gray images of F
 q
RS-cyclic codes over F
 q
.

/pdf/on-the-structure-of-cyclic-codes-over-mathbb-f-q-rs-and-273o4gahx8.pdf

On the Structure of Cyclic Codes Over $¥mathbb{F}_{q}{RS}$ and Applications in Quantum and LCD Codes Constructions

On double cyclic codes over Z 4

Let $R=\mathbb{Z}_4$ be the integer ring mod $4$. A double cyclic code of length $(r,s)$ over $R$ is a set that can be partitioned into two parts that any cyclic shift of the coordinates of both parts leaves invariant the code. These codes can be viewed as $R[x]$-submodules of $R[x]/(x^r-1)\times R[x]/(x^s-1)$. In this paper, we determine the generator polynomials of this family of codes as $R[x]$-submodules of $R[x]/(x^r-1)\times R[x]/(x^s-1)$. Further, we also give the minimal generating sets of this family of codes as $R$-submodules of $R[x]/(x^r-1)\times R[x]/(x^s-1)$. Some optimal or suboptimal nonlinear binary codes are obtained from this family of codes. Finally, we determine the relationship of generators between the double cyclic code and its dual.

On double cyclic codes over Z_4

This paper is concerned with \begin{document} ${{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$ \end{document} -additive cyclic codes. These codes can be identified as submodules of the ring \begin{document} ${\mathbb{Z}}_{2}[x]/\langle x^r-1\rangle × {\mathbb{Z}}_{2}[x]/\langle x^s-1\rangle × {\mathbb{Z}}_{4}[x]/\langle x^t-1\rangle$ \end{document} . There are two major ingredients. First, we determine the generator polynomials and minimum generating sets of this kind of codes. Furthermore, we investigate their dual codes. We determine the structure of the dual of separable \begin{document} ${{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$ \end{document} -additive cyclic codes completely. For the dual of non-separable \begin{document} ${{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$ \end{document} -additive cyclic codes, we give their structural properties in a few special cases.

https://www.aimsciences.org/article/exportPdf?id=7b29f864-37a2-4d3f-946d-0b4fe82ec0d3

\begin{document}$ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$\end{document} -additive cyclic codes

In this short paper, we determine the minimal generating set of 1-generator generalized quasi-cyclic codes over ź4$\mathbb {Z}_{4}$. We also determine their rank and introduce a lower bound for the minimum distance of free 1-generator generalized quasi-cyclic codes. Further, we construct some new ź4$\mathbb {Z}_{4}$-linear codes and we obtain some good binary nonlinear codes using the usual Gray map.

1-generator generalized quasi-cyclic codes over ź4$\mathbb {Z}_{4}$

In this paper, we study 1-generator quasi-cyclic and generalized quasi-cyclic codes over the ring $R=\frac{{{\mathbb {Z}_4}[u]}}{{\left\langle {{u^2} - 1} \right\rangle }}$. We determine the structure of the generators and the minimal generating sets of 1-generator QC and GQC codes. We also give a lower bound for the minimum distance of free 1-generator quasi-cyclic and generalized quasi-cyclic codes over this ring, respectively. Furthermore, some new $\mathbb {Z}_4$-linear codes via the Gray map which have better parameters than the best known $\mathbb {Z}_4$-linear codes are presented.

Tingting Wu

Papers

On double cyclic codes over Z 4

On double cyclic codes over Z_4

\begin{document}$ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$\end{document} -additive cyclic codes

1-generator generalized quasi-cyclic codes over ź4$\mathbb {Z}_{4}$

1-Generator quasi-cyclic and generalized quasi-cyclic codes over the ring \(\frac{{{\mathbb {Z}_4}[u]}}{{\left\langle {{u^2} - 1}\right\rangle }}\)