Taher Abualrub’s research while affiliated with American University of Sharjah and other places

Let p be a prime integer and Fp the finite field of order p. Moreover, let q1 and q2 be prime integers such that p is a quadratic residue modulo q1 and q2. In this paper, we will introduce the class of double quadratic residue (QR) codes of length n=q1+q2, over the ring Fp×Fp. To study these codes and their properties, we will focus on two cases: In case I, we will assume that q1 and q2 are equal and then study double QR codes over the ring Fp×Fp of length n=2q, where q is a prime integer and p is a quadratic residue modulo p. Then in case II, we will study QR codes of length n=q1+q2, where q1 and q2 are distinct. We will stud the duals of double QR codes over the ring Fp×Fp and classify double QR codes thar are linear complementary dual codes (LCD). We will show that LCD double QR codes can be applied in secret sharing scheme.

In this paper we study the structure and properties of additive right and left polycyclic codes induced by a nonbinary vector $\textbf{a}\in \mathbb {F} _{4}^{n}$, where $\mathbb {F}_{4}$ is the finite field of order 4. We show that additive right and left polycyclic codes are $\mathbb {F}_{2}[x]$ -submodules of the rings $R_{n}=\mathbb {F}_{4}\left[ x\right] /\left\langle x^{n}-a\left( x\right) \right\rangle$ and $S_{n}=\mathbb {F}_{4}\left[ x \right] /\left\langle x^{n}-a_{r}\left( x\right) \right\rangle$ respectively. We also show that these codes are invariant under multiplication by a certain matrix D, and construct their generator polynomials. Moreover, we study the relationship between additive polycyclic codes and linear polycyclic codes. We identify cases in which additive right polycyclic codes are linear right polycyclic codes and other cases in which additive right polycyclic codes are not linear right polycyclic codes. Finally, we give some applications of these codes by constructing examples of codes with good parameters.

In this paper, we study the class of [Formula: see text]-double cyclic codes of length [Formula: see text] We give a closed formula for the number of [Formula: see text] -double cyclic codes of length [Formula: see text] for any integers [Formula: see text] and [Formula: see text] that are relatively prime to [Formula: see text] Moreover, we give a closed formula for the number of quasi-cyclic (QC) codes of length [Formula: see text] and index [Formula: see text] We also provide formulas for the number of separable and non-separable [Formula: see text]-double cyclic codes of length [Formula: see text] In order to illustrate the results, we calculate the number of some codes with different [Formula: see text] and [Formula: see text]. Moreover, we list optimal parameter [Formula: see text]-double cyclic codes for specific values of [Formula: see text] and [Formula: see text].

This article studies skew generalized quasi-cyclic codes (SGQC-codes) over finite fields for any length n. We derive generator polynomials and cardinality of SGQC codes. Moreover, we show that the dual of any length SGQC-code is also an SGQC-code. Our search results lead to the construction of fifteen new 2-generator SGQC codes over the finite field F4 with minimum distances exceeding the minimum distances of the previously best known F4-linear codes with comparable parameters.

In this paper, we use a novel approach to describe generator polynomials of quasi-cyclic (QC) and generalized QC (GQC) codes over finite fields. Our study of QC- and GQC-codes will be general and not only restricted to one-generator codes. We prove that generator polynomials of QC-codes and GQC-codes are unique. Further, we use our results to obtain an expression for the dimensions of QC-codes and GQC-codes. As an application of our construction of these codes, we obtain many optimal linear codes over finite fields [Formula: see text] and [Formula: see text].

In this paper, we will study the structure of [Formula: see text]-additive codes where [Formula: see text] is the well-known ring of 4 elements and [Formula: see text] is the ring given by [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text]. We will classify all maximum distance separable codes with respect to the Singleton bound (MDSS) over [Formula: see text] Then we will focus on [Formula: see text]-additive cyclic and constacyclic codes. We will find a unique set of generator polynomials for these codes and determine minimum spanning sets for them. We will also find the generator polynomials for the dual of any [Formula: see text]-additive cyclic or constacyclic code.

In this paper, we introduce double Quadratic Residue Codes (QRC) of length n=p+q for prime numbers p and q in the ambient space ${{\mathbb {F}}} _{2}^{p}\times {{\mathbb {F}}}_{2}^{q}.$ We give the structure of separable and non-separable double QRC over this alphabet and we show that interesting double QR codes in this space exist only in the case when p=q. We give the main properties for these codes such as their idempotent generators and their duals. We relate these codes to codes over rings and show how they can be used to construct interesting lattices. As an applications of these codes, we provide examples of self-dual, formally self-dual and optimal double QRC. We also provide examples of best known quantum codes that are derived from double-QRC in this setting.

In this paper, we introduce ${\mathbb {F}}_{q^{2}}$-double cyclic codes of length n=r+s, where ${\mathbb {F}}_{q^{2}}$ is the Galois field of $q^{2}$ elements, q is a power of a prime integer p and r, s are positive integers. We determine the generator polynomials for any ${{\mathbb {F}}_{q^{2}} }$-double cyclic code. For any ${\mathbb {F}}_{q^{2}}$-double cyclic code ${\mathcal {C}},$ we will define the Euclidean dual code ${\mathcal {C}}^{\perp }$ based on the Euclidean inner product and the Hermitian dual code ${\mathcal {C}} ^{\perp _{H}}$ based on the Hermitian inner product. We will construct a relationship between ${\mathcal {C}}^{\perp }$ and ${\mathcal {C}}^{\perp _{H}}$ and then find the generator polynomials for the Hermitian dual code ${\mathcal {C}}^{\perp _{H}}.$ As an application of our work, we will present examples of optimal parameter linear codes over the finite field ${\mathbb {F}} _{4}$ and also examples of optimal quantum codes that were derived from ${\mathbb {F}}_{4}$-double cyclic codes using the Hermitian inner product.