Ghulam Mohammad

• Ph. D.
• Assistant Professor at Aligarh Muslim University

Let $\mathfrak{R}=\mathbb{Z}_{2}+u\mathbb{Z}_{2}$, where $u^2=0$, and $\textbf{S}=\mathbb{Z}_{2}+u\mathbb{Z}_{2}+v\mathbb{Z}_{2}+uv\mathbb{Z}_{2}$, where $u^{2}=v^{2}=0$, $uv=vu$. In this article, we study $\mathfrak{R}\textbf{S}$-additive cyclic, additive constacyclic, and additive dual codes. We find the structural properties of these codes. The...

Let $\mathbb{Z}_{2}=\{0,1\}$, $\mathfrak{R_{1}}=\mathbb{Z}_{2}+u\mathbb{Z}_{2}$, where $u^2=0$ and $\mathfrak{R_{u^{k}}}=\mathbb{Z}_{2}+u\mathbb{Z}_{2}+\cdots+u^{k-1 }\mathbb{Z}_{2}$, where $u^{k}=0$. In this article, we study $\mathbb{Z}_{2}\mathfrak{R_{1}}\mathfrak{R_{u^k}}$-additive cyclic, additive dual codes and their structural properties. Th...

Let $${\mathfrak {R}}= {\mathbb {Z}}_4[u,v]/\langle u^2-2,uv-2,v^2,2u,2v\rangle$$ R = Z 4 [ u , v ] / ⟨ u 2 - 2 , u v - 2 , v 2 , 2 u , 2 v ⟩ be a ring, where $${\mathbb {Z}}_{4}$$ Z 4 is a ring of integers modulo 4. This ring $${\mathfrak {R}}$$ R is a local non-chain ring of characteristic 4. The main objective of this article is to construct rev...

Let S = ℤ p [ u, v ] /〈u ² , v ² , uv − uv〉 be a semi-local ring, where p is a prime number. In the present article, we determine the generating sets of S and use them to construct the structures of ℤ p S -additive cyclic and constacyclic codes. The minimal polynomials and spanning sets of ℤ p S -additive cyclic and constacyclic codes are also dete...

Let $ \mathcal {R}_{k}= \mathbb {Z}_4[u_{1},u_{2},\ldots,u_{k}]/\langle u_{i}^2-u_{i},u_{i}u_{j}-u_{j}u_{i}\rangle $ be a non-chain ring of characteristic 4, where $ 1\leq i,j\leq k $ and $ k\geq 1 $ . In this article, we discuss reversible cyclic codes of odd lengths over the ring $ \mathcal {R}_{k} $ . We construct bijections between the elements...

Let $\mathfrak{A}=\mathbb Z_4+u\mathbb Z_4+v\mathbb Z_4$, where $u^2=u$, $v^2=v$ and $uv=vu=0,$ be a ring which is an extension of $\mathbb{Z}_{4}$. In this article, we study the structure of cyclic codes over $\mathfrak{A}$ and define a $\mathbb Z_2$-linear isometry $\Phi$ from $\mathfrak{A}^{n}$ to $\mathbb Z^{6n}_2.$ Based on the classical cycli...

Let $\mathbb{Z}_{2}=\{0,1\}$, $\mathfrak{R_{1}}=\mathbb{Z}_{2}+u\mathbb{Z}_{2}$, where $u^2=0$ and $\mathfrak{R_{2}}{=}\mathbb{Z}_{2}+u\mathbb{Z}_{2}+v\mathbb{Z}_{2}$, where $u^{2}{=}v^{2}=0{=}uv{=}vu$. In this article, we study $\mathbb{Z}_{2}\mathfrak{R_{1}}\mathfrak{R_{2}}$-additive cyclic, additive constacyclic and additive dual codes and find...

Suppose \(\mathbb {F}_{q}\) is a finite field with q elements and \(q=p^{t},\) where p is a prime and \(t\ge 1.\) Let \(\mathfrak {R}_{q}={\mathbb {F}}_q+u_{1}{\mathbb {F}}_q+u_{2}\mathbb F_q+u_{1}u_{2}{\mathbb {F}}_q\), where \(u_{1}^2=0\), \(u_{2}^2=0,\) \(u_{1}u_{2}=u_{2}u_{1}\) be a non-chain ring. A necessary and sufficient condition for a giv...

In this article, we inspect the structure of reversible cyclic codes over the ring S = F_2 + uF_2 + u^2 F_2 + · · · + u^{k−1}F_2 , where u^k = 0. We determine a unique set of generators for cyclic codes over S. We classify reversible cyclic codes with respect to their generators. We also give the condition for the dual of reversible cyclic codes of...

In this work, we study cyclic codes of length n over a finite commutative non-chain ring R = F _q[ u,v ]/⟨ u^2 −\gamma u, v^2 −\epsilon v , uv−vu ⟩ where \gamma, \epsilon∈ F_q * and we find new and better quantum error-correcting codes than previously known quantum error correcting codes. Then certain constraints are imposed on the generator polyno...

In this article, we consider a semi-local ring S=Fq+uFq, where u2=u, q=ps and p is a prime number. We define a multiplication yb=β(b)y+γ(b), where β is an automorphism and γ is a β-derivation on S so that S[y;β,γ] becomes a non-commutative ring which is known as skew polynomial ring. We give the characterization of S[y;β,γ] and obtain the most stri...

In this paper, we study v-skew constacyclic codes over the ring \(F_{q}+vF_{q}\), where \(v^2=1,~q=p^m\) and p is an odd prime. We obtain the structural properties of v-skew constacyclic codes over \(F_{q}+vF_{q}\) using decomposition method. The generator polynomials of v-skew constacyclic codes and their dual codes over R are obtained. Moreover,...

Let Rr = Fq + v1F Fq + ··· + vr Fq, where q is the power of prime, vi² = vi, vivj = vjvi = 0 for 1 ≤ i, j ≤ r and r ≥ 1. In this paper, the structure of λ-constacyclic codes over the ring Rr is studied and a Gray map ϕ from Rrn to Fq(r+1)n is given. The necessary and sufficient conditions for these codes to contain their Euclidean duals are determi...

Let m≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \ge 1$$\end{document} be a fixed integer and q be an odd prime such that q=pm\documentclass[12pt]{minimal} \use...

Let p be an odd prime, \(q=p^m\), \(R_1=\mathbb F_{q}+u{\mathbb {F}}_{q}+v{\mathbb {F}}_{q}+uv{\mathbb {F}}_{q}\) with \(u^2=1\), \(v^2=1\), \(uv=vu\) and \(R_2={\mathbb {F}}_{q}[u,v,w]/\langle u^2-1, v^2-1,w^2-1, uv-vu,vw-wv,wu-uw\rangle \) with \(u^2=1\), \(v^2=1\), \(w^2=1\), \(uv=vu\), \(vw=wv\), \(wu=uw\). In this paper, \({\mathbb {F}}_q R_1R...

Let W be a fixed k-dimensional subspace of an n-dimensional vector space V such that n − k ≥ 1. In this paper, we introduce a graph structure, called the subspace based subspace inclusion graph I W n (V), where the vertex set V(I W n (V)) is the collection of all subspaces U of V such that U+W = V and U W, i.e., V(I W n (V)) = {U ⊆ V | U+W = V, U W...

Constacyclic codes over the ring F p [u, v]/ u 2 − 1, v 3 − v, uv − vu and their applications Abstract The objective of this paper is to investigate the structural properties of (λ 1 +uλ 2 + vλ 3 + v 2 λ 4 + uvλ 5 + uv 2 λ 6)-constacyclic codes over the ring F p [u, v]/ u 2 − 1, v 3 − v, uv − vu for odd prime p. Precisely, we prove that the Gray im...

In this paper, we study cyclic codes over a finite non-chain ring \(\mathbb {F}_{p^{m}}+v\mathbb {F}_{p^{m}}\) with v2 = 1 and find new and better quantum error-correcting codes than previously known quantum error correcting codes over \(\mathbb {F}_{p^{m}}\). Therewith, we characterize the LCD codes and obtain many new LCD codes. Also, we prove th...

In this paper, we study the structural properties of (α+u1β+u2γ+u1u2δ)-constacyclic codes over R=Fq[u1,u2]/〈u12−u1,u22−u2,u1u2−u2u1〉 where q=pm for odd prime p and m≥1. We derive the generators of constacyclic and dual constacyclic codes. We have shown that Gray image of a constacyclic code of length n is a quasi constacyclic code of length 4n. Als...

Let p be an odd prime, and k be an integer such that gcd(k,p) = 1. Using pairwise orthogonal idempotents γ1,γ2,γ3 of the ring ℛ = 𝔽p[u]/〈uk+1 − u〉, with γ1 + γ2 + γ3 = 1, ℛ is decomposed as ℛ = γ1ℛ⊕ γ2ℛ⊕ γ3ℛ, which contains the ring R = γ1𝔽p ⊕ γ2𝔽p ⊕ γ3𝔽p as a subring. It is shown that, for λ0,λk ∈ 𝔽p, λ0 + ukλ k ∈ R, and it is invertible if and on...

Let p be an odd prime, and k be an integer such that gcd(k, p) = 1. Using pairwise orthogonal idempotents γ 1 , γ 2 , γ 3 of the ring R = F p [u]/u k+1 − u, with γ 1 + γ 2 + γ 3 = 1, R is decomposed as R = γ 1 R ⊕ γ 2 R ⊕ γ 3 R, which contains the ring R = γ 1 F p ⊕ γ 2 F p ⊕ γ 3 F p as a subring. It is shown that, for λ 0 , λ k ∈ F p , λ 0 + u k λ...

In this paper, we study a class of skew constacyclic codes over the ring \(R=F_q+u_1F_{q}+\cdots +u_{2m}F_{q}\), where \(u_i^2=u_i\), \(u_iu_j=u_ju_i=0\), for \(i,j=1,2,\ldots ,2m ~,~ i \ne j\) and \(q=p^s\), and derive the generator polynomials of this class of codes over R. Also, by using Calderbank–Shor–Steane construction, some new non-binary q...

In this paper, quantum codes over Fp from cyclic codes over the ring Fp[u, v]/〈u2 − 1, v3 − v, uv − vu〉, where u2 = 1, v3 = v, uv = vu and p is an odd prime have been studied. We give the structure of cyclic codes over the ring Fp[u, v]/〈u2 − 1, v3 − v, uv − vu〉 and obtain quantum codes over Fp using self-orthogonal property of these classes of cod...

In the present paper, we study (1 − 2v²)-skew constacyclic codes over the ring Fq + vFq + v²Fq, where v³ = v, q = pm and p is an odd prime.We investigate the structural properties of skew cyclic codes over Fq + vFq + v²Fq using decomposition method. By defining a Graymap from Fq +vFq +v²Fq to F³q , it has been proved that the Gray image of a (1 − 2...

In this paper, we study nonbinary quantum codes from cyclic codes over the ring Fp[u]/(u3 - u) for odd prime p. We give the constructional properties of linear and cyclic codes over the ring Fp[u]/(u3 - u). As an applications of these classes of codes, we obtain quantum codes over Fp using self-orthogonal property. Moreover, we find some new nonbin...

In this paper, we study quantum codes over \(F_q\) from cyclic codes over \(F_q+uF_q+vF_q+uvF_q,\) where \(u^2=u,~v^2=v,~uv=vu,~q=p^m\), and p is an odd prime. We give the structure of cyclic codes over \(F_q+uF_q+vF_q+uvF_q\) and obtain self-orthogonal codes over \(F_q\) as Gray images of linear and cyclic codes over \(F_q+uF_q+vF_q+uvF_q\). In pa...

In the present paper, we study skew cyclic codes over the finite semi-local ring [Formula: see text], where [Formula: see text] and [Formula: see text] is an odd prime. We define a Gray map from [Formula: see text] to [Formula: see text] and investigate the structural properties of skew cyclic codes over [Formula: see text] using decomposition meth...

In the present paper, we study skew cyclic codes over the ring $F_{q}+vF_{q}+v^2F_{q}$, where $v^3=v,~q=p^m$ and $p$ is an odd prime. We investigate the structural properties of skew cyclic codes over $F_{q}+vF_{q}+v^2F_{q}$ using decomposition method. By defining a Gray map from $F_{q}+vF_{q}+v^2F_{q}$ to $F_{q}^3$, it has been proved that the Gra...

Let $R=\mathbb{Z}_4+u\mathbb{Z}_4,$ where $\mathbb{Z}_4$ denotes the ring of integers modulo $4$ and $u^2=0$. In the present paper, we introduce a new Gray map from $R^n$ to $\mathbb{Z}_{4}^{2n}.$ We study $(1+2u)$-constacyclic codes over $R$ of odd lengths with the help of cyclic codes over $R$. It is proved that the Gray image of $(1+2u)$-constac...

Let $R$ be a finite chain ring with maximal ideal $m=<\gamma>$ and residue field $K$, and let $t$ be the nilpotency index of $\gamma$. For a given unit $\lambda\in R$, a linear code $C$ over $R$ is said to be $\lambda$-constacyclic, if $(\lambda c_{n-1}, c_0, c_1, ..., c_{n-2})\in C$ whenever $(c_0, c_1, ..., c_{n-1})\in C$. In the present paper, w...

In this paper, we study quantum error-correcting codes from cyclic codes over the ring R, where R is a non-chain extension of Fp. We give a method to obtain self-orthogonal codes over Fp as Gray images of linear and cyclic codes over the ring R. Finally, we derive the parameters of associated quantum error-correcting codes.

Let $R=F_3+vF_3=\{0, 1, 2, v, 2v, v+1, v+2, 2v+1, 2v+2\}$ be a ring, where $v^2=1$. In this paper, we study skew cyclic codes over $R$ and investigate the structural properties of skew polynomial ring $R[x, \theta]$ and the set $R[x, \theta]/<x^n-1>$, where $\theta$ is a non trivial automorphism of $R$. Further, we prove that skew cyclic codes over...

Let $R=F_3+vF_3$ be a finite commutative ring, where $v^2=1$. It is a finite semi-local ring, not a chain ring. In this paper, we give a construction for quantum codes from cyclic codes over $R$. We derive self-orthogonal codes over $F_3$ as Gray images of linear and cyclic codes over $R$. In particular, we use two codes associated with a cyclic co...

Let $R=F_p+vF_p$, where $v^2=1$. Then $R$ is a finite commutative semi-local ring but not a chain ring for $p>2$. In this paper, we define a Gray map from $R$ to $F_{p}^{2}$ and study $-v$-constacyclic codes over $R$. It is shown that the image of a $-v$-constacyclic code of length $n$ over $R$ under the Gray map is a distance-invariant linear cycl...

In this paper, we study skew cyclic codes over the finite non-chain ring R. We investigate the structural properties of skew polynomial ring of the ring R with respect to a non-trivial automorphism. Further, we prove that skew cyclic codes over R are equivalent to either cyclic codes or quasi-cyclic codes.

Let R ¼ F 3 þ vF 3 be a ¯nite commutative ring, where v 2 ¼ 1. It is a ¯nite semi-local ring, not a chain ring. In this paper, we give a construction for quantum codes from cyclic codes over R. We derive self-orthogonal codes over F 3 as Gray images of linear and cyclic codes over R. In particular , we use two codes associated with a cyclic code ov...