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Adel Alahmadi

Publications and source records attributed to Adel Alahmadi.

At least 19 recordsLinked to original sources
arXiv2021

Cyclic and Quasi-Cyclic DNA Codes

In this paper, we discuss DNA codes that are cyclic or quasi-cyclic over $\Z_{4}+ω\Z_{4}$, where $ω^{2}=2+2ω$ along with methods to construct these with combinatorial constraints. We also generalize results obtained for the ring $\Z_{4}+ω\Z_{4}$, where $ω^{2}=2+2ω$, and some other rings to the sixteen rings $R_θ=\Z_{4}+ω\Z_{4}$, where $ω^{2}=θ\in \Z_{4}+ω\Z_{4}$, using the generalized Gau map and Gau distance in \cite{3}. We determine a relationship between the Gau distance and Hamming distance for linear codes over the sixteen rings $R_θ$ which enables us to attain an upper boundary for the Gau distance of free codes that are self-dual over the rings $R_θ$.

cs.IT↗
arXiv2021

DNA Codes over the Ring $\mathbb{Z}_4 + w\mathbb{Z}_4$

In this present work, we generalize the study of construction of DNA codes over the rings $\mathcal{R}_θ=\mathbb{Z}_4+w\mathbb{Z}_4$, $w^2 = θ$ for $θ\in \mathbb{Z}_4+w\mathbb{Z}_4$. Rigorous study along with characterization of the ring structures is presented. We extend the Gau map and Gau distance, defined in \cite{DKBG}, over all the $16$ rings $\mathcal{R}_θ$. Furthermore, an isometry between the codes over the rings $\mathcal{R}_θ$ and the analogous DNA codes is established in general. Brief study of dual and self dual codes over the rings is given including the construction of special class of self dual codes that satisfy reverse and reverse-complement constraints. The technical contributions of this paper are twofold. Considering the Generalized Gau distance, Sphere Packing-like bound, GV-like bound, Singleton like bound and Plotkin-like bound are established over the rings $\mathcal{R}_θ$. In addition to this, optimal class of codes are provided with respect to Singleton-like bound and Plotkin-like bound. Moreover, the construction of family of DNA codes is proposed that satisfies reverse and reverse-complement constraints using the Reed-Muller type codes over the rings $\mathcal{R}_θ$.

cs.IT↗
arXiv2019

Simplicity of Lie algebras of Poisson brackets

Let $A$ be an associative commutative algebra with $1$ over a field of zero characteristic, $\{,\} : A \times A \to A$ is a Poisson bracket, $Z = \{ a \in A \mid \{a, A\} = (0) \}.$ We prove that if $A$ is simple as a Poisson algebra then the Lie algebra $\frac{\{A,A\}}{\{A,A\}\cap Z}$ is simple.

math.RA↗
arXiv2018

Regular elements determined by generalized inverses

In a semiprime ring, von Neumann regular elements are determined by their inner inverses. In particular, for elements $a,b$ of a von Neumann regular ring $R$, $a=b$ if and only if $I(a)=I(b)$, where $I(x)$ denotes the set of inner inverses of $x\in R$. We also prove that, in a semiprime ring, the same is true for reflexive inverses.

math.RA↗
arXiv2017

Embeddings in Lie algebras of subexponential growth

We prove that an arbitrary countable dimensional Lie algebra over a field of characteristic $\neq 2$ that is locally of subexponential growth is embeddable in a finitely generated Lie algebra of subexponential growth.

math.RA↗
arXiv2017

On the dimension of twisted centralizer codes

Given a field $F$, a scalar $λ\in F$ and a matrix $A\in F^{n\times n}$, the twisted centralizer code $C_F(A,λ):=\{B\in F^{n\times n}\mid AB-λBA=0\}$ is a linear code of length $n^2$. When $A$ is cyclic and $λ\ne0$ we prove that $\dim C_F(A,λ)=\mathrm{deg}(\gcd(c_A(t),λ^n c_A(λ^{-1}t)))$ where $c_A(t)$ denotes the characteristic polynomial of $A$. We also show how $C_F(A,λ)$ decomposes, and we estimate the probability that $C_F(A,λ)$ is nonzero when $|F|$ is finite. Finally, we prove $\dim C_F(A,λ)\leqslant n^2/2$ for $λ\not\in\{0,1\}$ and `almost all' matrices $A$.

math.CO↗
arXiv2017

Matrix wreath products of algebras and embedding theorems

We introduce a new construction of matrix wreath products of algebras that is similar to wreath products of groups. We then use it to prove embedding theorems for Jacobson radical, nil, and primitive algebras. In §\ref{Section6}, we construct finitely generated nil algebras of arbitrary Gelfand-Kirillov dimension $\geq 8$ over a countable field which answers a question from \cite{8}.

math.RA↗
arXiv2017

Algebras and semigroups of locally subexponential growth

We prove that a countable dimensional associative algebra (resp. a countable semigroup) of locally subexponential growth is $M_\infty$-embeddable as a left ideal in a finitely generated algebra (resp. semigroup) of subexponential growth. Moreover, we provide bounds for the growth of the finitely generated algebra (resp. semigroup). The proof is based on a new construction of matrix wreath product of algebras.

math.RA↗
arXiv2017

Twisted Centralizer Codes

Given an $n\times n$ matrix $A$ over a field $F$ and a scalar $a\in F$, we consider the linear codes $C(A,a):=\{B\in F^{n\times n}\mid \,AB=aBA\}$ of length $n^2$. We call $C(A,a)$ a twisted centralizer code. We investigate properties of these codes including their dimensions, minimum distances, parity-check matrices, syndromes, and automorphism groups. The minimal distance of a centralizer code (when $a=1$) is at most $n$, however for $a\ne 0,1$ the minimal distance can be much larger, as large as $n^2$.

math.CO↗
arXiv2017

Long quasi-polycyclic $t-$CIS codes

We study complementary information set codes of length $tn$ and dimension $n$ of order $t$ called ($t-$CIS code for short). Quasi-cyclic and quasi-twisted $t$-CIS codes are enumerated by using their concatenated structure. Asymptotic existence results are derived for one-generator and have co-index $n$ by Artin's conjecture for quasi cyclic and special case for quasi twisted. This shows that there are infinite families of long QC and QT $t$-CIS codes with relative distance satisfying a modified Varshamov-Gilbert bound for rate $1/t$ codes. Similar results are defined for the new and more general class of quasi-polycyclic codes introduced recently by Berger and Amrani.

cs.IT↗
arXiv2017

On linear complementary-dual multinegacirculant codes

Linear codes with complementary-duals (LCD) are linear codes that intersect with their dual trivially. Multinegacirculant codes of index $2$ that are LCD are characterized algebraically and some good codes are found in this family. Exact enumeration is performed for indices 2 and 3, and for all indices $t$ for a special case of the co-index by using their concatenated structure. Asymptotic existence results are derived for the special class of such codes that are one-generator and have co-index a power of two by means of Dickson polynomials. This shows that there are infinite families of LCD multinegacirculant codes with relative distance satisfying a modified Varshamov-Gilbert bound.

cs.IT↗
arXiv2016

On self-dual double negacirculant codes

Double negacirculant (DN) codes are the analogues in odd characteristic of double circulant codes. Self-dual DN codes of odd dimension are shown to be consta-dihedral. Exact counting formulae are derived for DN codes. The special class of length a power of two is studied by means of Dickson polynomials, and is shown to contain families of codes with relative distances satisfying a modified Gilbert-Varshamov bound.

cs.IT↗
arXiv2016

On self-dual double circulant codes

Self-dual double circulant codes of odd dimension are shown to be dihedral in even characteristic and consta-dihedral in odd characteristic. Exact counting formulae are derived for them and used to show they contain families of codes with relative distance satisfying a modified Gilbert-Varshamov bound.

cs.IT↗
arXiv2015

The joint weight enumerator of an LCD code and its dual

A binary linear code is called {\em LCD} if it intersects its dual trivially. We show that the coefficients of the joint weight enumerator of such a code with its dual satisfy linear constraints, leading to a new linear programming bound on the size of an LCD code of given length and minimum distance. In addition, we show that this polynomial is, in general, an invariant of a matrix group of dimension $4$ and order $12$. Also, we sketch a Gleason formula for this weight enumerator.

math.MG↗