Constacyclic codes over the ring Fq[u,v,w]/〈u2 − 1, v2 − 1,w3 − w, uv − vu,vw − wv,wu − uw〉

Let [Formula: see text] be the ring [Formula: see text], where [Formula: see text] for any odd prime [Formula: see text] and positive integer [Formula: see text]. In this paper, we study constacyclic codes over the ring [Formula: see text]. We define a Gray map by a matrix and decompose a constacyclic code over the ring [Formula: see text] as the direct sum of constacyclic codes over [Formula: see text], we also characterize self-dual constacyclic codes over the ring [Formula: see text] and give necessary and sufficient conditions for constacyclic codes to be dual-containing. As an application, we give a method to construct quantum codes from dual-containing constacyclic codes over the ring [Formula: see text].

New Quantum and LCD Codes over Finite Fields of Even Characteristic

For an integer m ≥ 1, we study cyclic codes of length l over a commutative non-chain ring F2m + uF2m , where u2 = u . With a new Gray map and Euclidean dual-containing cyclic codes, we provide many new and superior codes to the best-known quantum error-correcting codes. Also, we characterise LCD codes of length l with respect to their generator polynomials and prove that F2m − image of an LCD code of length l is an LCD code of length 2l . Finally, we provide several optimal LCD codes from the Gray images of LCD codes over F2m + uF2m .

Quantum Codes Obtained through Constacyclic Codes over Z3 + νZ3 + ωZ3 + νωZ3

This paper is concerned with, structural properties and construction of quantum codes over Z3 by using constacyclic codes over the finite commutative non-chain ring R = Z3 + νZ3 + ωZ3 + νωZ3 where ν2 = 1, ω2 = ω, νω = νω, and Z3 is field having 3 elements with characteristic 3. A Gray map is defined between R and Z43. The parameters of quantum codes over Z3 are obtained by decomposing constacyclic codes into cyclic and negacyclic codes over Z3. As an application, some examples of quantum codes of arbitrary length, are also obtained.

G-codes, self-dual G-codes and reversible G-codes over the ring ${\mathscr{B}}_{j,k}$

In this work, we study a new family of rings, ${\mathscr{B}}_{j,k}$ B j , k , whose base field is the finite field ${\mathbb {F}}_{p^{r}}$ F p r . We study the structure of this family of rings and show that each member of the family is a commutative Frobenius ring. We define a Gray map for the new family of rings, study G-codes, self-dual G-codes, and reversible G-codes over this family. In particular, we show that the projection of a G-code over ${\mathscr{B}}_{j,k}$ B j , k to a code over ${\mathscr{B}}_{l,m}$ B l , m is also a G-code and the image under the Gray map of a self-dual G-code is also a self-dual G-code when the characteristic of the base field is 2. Moreover, we show that the image of a reversible G-code under the Gray map is also a reversible $G^{2^{j+k}}$ G 2 j + k -code. The Gray images of these codes are shown to have a rich automorphism group which arises from the algebraic structure of the rings and the groups. Finally, we show that quasi-G codes, which are the images of G-codes under the Gray map, are also Gs-codes for some s.

Three Authentication Schemes without Secrecy over Finite Fields and Galois Rings

We provide three new authentication schemes without secrecy. The first two on finite fields and Galois rings, using Gray map for this link. The third construction is based on Galois rings. The main achievement in this work is to obtain optimal impersonation and substitution probabilities in the schemes. Additionally, in the first and second scheme, we simplify the source space and obtain a better relationship between the size of the message space and the key space than the one given in a recent paper. Finally, we provide a third scheme on Galois rings.

Three authentication schemes without secrecy over finite fields and Galois rings

We give three new authentication schemes without secrecy. The first two on finite fields and Galois rings, using Gray map for this link. The third construction is given on Galois rings. The main achievement in this work is to obtain optimal impersonation and substitution probabilities in the schemes. Additionally, in the first and second scheme, we simplify the source space and bring a better relationship between the size of the message space and the key space than the given in [8]. Finally, we provide a third scheme on Galois rings, which generalizes the scheme over finite fields constructed in [9].

New quantum codes from skew constacyclic codes

<p style='text-indent:20px;'>For an odd prime <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula> and positive integers <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \ell $\end{document}</tex-math></inline-formula>, let <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{F}_{p^m} $\end{document}</tex-math></inline-formula> be the finite field with <inline-formula><tex-math id="M5">\begin{document}$ p^{m} $\end{document}</tex-math></inline-formula> elements and <inline-formula><tex-math id="M6">\begin{document}$ R_{\ell,m} = \mathbb{F}_{p^m}[v_1,v_2,\dots,v_{\ell}]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle_{1\leq i, j\leq \ell} $\end{document}</tex-math></inline-formula>. Thus <inline-formula><tex-math id="M7">\begin{document}$ R_{\ell,m} $\end{document}</tex-math></inline-formula> is a finite commutative non-chain ring of order <inline-formula><tex-math id="M8">\begin{document}$ p^{2^{\ell} m} $\end{document}</tex-math></inline-formula> with characteristic <inline-formula><tex-math id="M9">\begin{document}$ p $\end{document}</tex-math></inline-formula>. In this paper, we aim to construct quantum codes from skew constacyclic codes over <inline-formula><tex-math id="M10">\begin{document}$ R_{\ell,m} $\end{document}</tex-math></inline-formula>. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.</p>

On one-lee weight and two-lee weight $ \mathbb{Z}_2\mathbb{Z}_4[u] $ additive codes and their constructions

<p style='text-indent:20px;'>This paper mainly study <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document}</tex-math></inline-formula>-additive codes. A Gray map from <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{Z}_{2}^{\alpha}\times\mathbb{Z}_{4}^{\beta}[u] $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{Z}_{4}^{\alpha+2\beta} $\end{document}</tex-math></inline-formula> is defined, and we prove that is a weight preserving and distance preserving map. A MacWilliams-type identity between the Lee weight enumerator of a <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document}</tex-math></inline-formula>-additive code and its dual is proved. Some properties of one-weight <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document}</tex-math></inline-formula>-additive codes and two-weight projective <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document}</tex-math></inline-formula>-additive codes are discussed. As main results, some construction methods for one-weight and two-weight <inline-formula><tex-math id="M8">\begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document}</tex-math></inline-formula>-additive codes are studied, meanwhile several examples are presented to illustrate the methods.</p>

Skew cyclic codes over 𝔽4R

This paper considers a new alphabet set, which is a ring that we call [Formula: see text], to construct linear error-control codes. Skew cyclic codes over this ring are then investigated in details. We define a nondegenerate inner product and provide a criteria to test for self-orthogonality. Results on the algebraic structures lead us to characterize [Formula: see text]-skew cyclic codes. Interesting connections between the image of such codes under the Gray map to linear cyclic and skew-cyclic codes over [Formula: see text] are shown. These allow us to learn about the relative dimension and distance profile of the resulting codes. Our setup provides a natural connection to DNA codes where additional biomolecular constraints must be incorporated into the design. We present a characterization of [Formula: see text]-skew cyclic codes which are reversible complement.

Constacyclic and Linear Complementary Dual Codes Over Fq + uFq

This article discusses linear complementary dual (LCD) codes over ℜ = Fq+uFq(u2=1) where q is a power of an odd prime p. Authors come up with a new Gray map from ℜn to F2nq and define a new class of codes obtained as the gray image of constacyclic codes over .ℜ Further, we extend the study over Euclidean and Hermitian LCD codes and establish a relation between reversible cyclic codes and Euclidean LCD cyclic codes over ℜ. Finally, an application of LCD codes in multisecret sharing scheme is given.