Habibul Islam
@iitp.ac.in
Department of Mathematics
Indian Institute of Technology Patna
EDUCATION
I have done B.Sc in Mathematics(Hons) from West Bengal State University, India in 2013. I have received M.Sc in Pure mathematics degree from University of Calcutta, India in 2016. Now, I am pursing PhD in mathematics, specialized in algebraic coding theory in the department of mathematics, Indian Institute of Technology Patna
RESEARCH INTERESTS
Algebraic Coding Theory, Codes over rings, Quantum error-correcting codes, LCD codes, Weight distribution, Cyclic codes over Matrix rings, Additive cyclic and constacyclic codes.
Scopus Publications
Scholar Citations
Scholar h-index
Scholar i10-index
Scopus Publications
Habibul Islam, Om Prakash, and Dipak Kumar Bhunia
Taru Publications
Let Fp be the finite field of order p and M3(Fp) the ring of 3 × 3 matrices over Fp, where p is a prime. For certain prime p, we determine the complete algebraic properties of cyclic codes of length N (p | N) over M3(Fp). We define an isometry from M3(Fp) to Fp3 + eFp3 + e2Fp3, where e3 = 1. As an outcome, we derive numerous optimal and good linear F8 codes induced from F8 -images of cyclic codes over M3(F2).
Ram Krishna Verma, , Om Prakash, Ashutosh Singh, and Habibul Islam
American Institute of Mathematical Sciences (AIMS)
<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} $\\end{document}</tex-math></inline-formula>. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.</p>
Om Prakash, , Shikha Yadav, Habibul Islam, Patrick Solé, and
American Institute of Mathematical Sciences (AIMS)
<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M2">\\begin{document}$ \\mathbb{Z}_4 $\\end{document}</tex-math></inline-formula> be the ring of integers modulo <inline-formula><tex-math id="M3">\\begin{document}$ 4 $\\end{document}</tex-math></inline-formula>. This paper studies mixed alphabets <inline-formula><tex-math id="M4">\\begin{document}$ \\mathbb{Z}_4\\mathbb{Z}_4[u^3] $\\end{document}</tex-math></inline-formula>-additive cyclic and <inline-formula><tex-math id="M5">\\begin{document}$ \\lambda $\\end{document}</tex-math></inline-formula>-constacyclic codes for units <inline-formula><tex-math id="M6">\\begin{document}$ \\lambda = 1+2u^2,3+2u^2 $\\end{document}</tex-math></inline-formula>. First, we obtain the generator polynomials and minimal generating set of additive cyclic codes. Then we extend our study to determine the structure of additive constacyclic codes. Further, we define some Gray maps and obtain <inline-formula><tex-math id="M7">\\begin{document}$ \\mathbb{Z}_4 $\\end{document}</tex-math></inline-formula>-images of such codes. Finally, we present numerical examples that include six new and two best-known quaternary linear codes.</p>
Indibar Debnath, Om Prakash, and Habibul Islam
Springer Science and Business Media LLC
Indibar Debnath, Om Prakash, and Habibul Islam
Springer Science and Business Media LLC
Habibul Islam and Dipak Kumar Bhunia
Springer Science and Business Media LLC
AbstractAs a tool towards quantum error correction, additive conjucyclic codes have gained great attention. But, their algebraic structure is completely unknown over finite fields (except $${\\mathbb {F}}_{q^2}$$ F q 2 ) as well as rings. In this article, we investigate the structure of additive conjucyclic codes over Galois rings $$GR(2^r,2)$$ G R ( 2 r , 2 ) , where $$r\\ge 2$$ r ≥ 2 is an integer. We develop a one-to-one correspondence between the family of additive conjucyclic codes of length n over $$GR(2^r,2)$$ G R ( 2 r , 2 ) and the family of linear cyclic codes of length 2n over $${\\mathbb {Z}}_{2^r}$$ Z 2 r . This correspondence helps to obtain additive conjucyclic codes over $$GR(2^r,2)$$ G R ( 2 r , 2 ) via known linear cyclic codes over $${\\mathbb {Z}}_{2^r}$$ Z 2 r . We prove that the trace dual $${\\mathscr {C}}^{Tr}$$ C Tr of an additive conjucyclic code $${\\mathscr {C}}$$ C is also an additive conjucyclic code. Moreover, we derive a necessary and sufficient condition of additive conjucyclic codes to be self-dual. We further propose a technique for constructing linear cyclic codes over $${\\mathbb {Z}}_{2^r}$$ Z 2 r contained in additive conjucyclic codes over $$GR(2^r,2)$$ G R ( 2 r , 2 ) . Last but not least, we explicitly derive the generator matrices for these codes.
Om Prakash, Habibul Islam, and Ram Krishna Verma
Springer International Publishing
Habibul Islam and Anna-Lena Horlemann
IEEE
For a prime power q, an integer m and 0 ≤ e ≤ m − 1 we study the e-Galois hull dimension of Gabidulin codes Gk(α) of length m and dimension k over ${\\mathbb{F}_{{q^m}}}$. Using a self-dual basis α of ${\\mathbb{F}_{{q^m}}}$ over ${\\mathbb{F}_q}$, we first explicitly compute the hull dimension of Gk(α). Then a necessary and sufficient condition of Gk(α) to be linear complementary dual (LCD), self-orthogonal and self-dual will be provided. We prove the existence of e-Galois (where $e = \\frac{m}{2}$) self-dual Gabidulin codes of length m for even q, which is in contrast to the known fact that Euclidean self-dual Gabidulin codes do not exist for even q. As an application, we construct two classes of MDS entangled-assisted quantum error-correcting codes (MDS EAQECCs) whose parameters have more flexibility compared to known codes in this context.
Habibul Islam, Om Prakash, and Dipak Kumar Bhunia
Springer Science and Business Media LLC
Om Prakash, Shikha Yadav, Habibul Islam, and Patrick Solé
Springer Science and Business Media LLC
Shikha Patel, Om Prakash, and Habibul Islam
Springer Science and Business Media LLC
Shikha Patel, Om Prakash, and Habibul Islam
Springer Science and Business Media LLC
Habibul Islam, Shikha Patel, Om Prakash, and Patrick Solé
Springer Science and Business Media LLC
Habibul Islam, Om Prakash, and Dipak Kumar Bhunia
Springer Science and Business Media LLC
In this article, for a prime p such that $$p\\equiv 2$$ p ≡ 2 or $$3 \\pmod {5}$$ 3 ( mod 5 ) , we identify cyclic codes of length N over $$R=M_{2}({\\mathbb {F}}_{p}+u{\\mathbb {F}}_{p})$$ R = M 2 ( F p + u F p ) , $$u^2=0$$ u 2 = 0 as right R -submodules of $$R/\\langle x^N-1\\rangle $$ R / ⟨ x N - 1 ⟩ . Also, we define an isometry from $$M_{2}({\\mathbb {F}}_{p}+u{\\mathbb {F}}_{p})$$ M 2 ( F p + u F p ) to $${\\mathbb {F}}_{p^2}+u{\\mathbb {F}}_{p^2}+v{\\mathbb {F}}_{p^2}+uv{\\mathbb {F}}_{p^2}$$ F p 2 + u F p 2 + v F p 2 + u v F p 2 , where $$u^2=v^2=0,uv=vu$$ u 2 = v 2 = 0 , u v = v u and determine the structure of cyclic codes, in particular self-dual cyclic codes of length N where $$\\gcd (N,p)=1$$ gcd ( N , p ) = 1 . Moreover, several optimal and near to optimal codes are obtained as the Gray images of these codes over R .
Shikha Patel, Habibul Islam, and Om Prakash
Springer Science and Business Media LLC
Ram Krishna Verma, Om Prakash, Habibul Islam, and Ashutosh Singh
Springer Science and Business Media LLC
Habibul Islam, Om Prakash, and Ram Krishna Verma
American Institute of Mathematical Sciences (AIMS)
<p style='text-indent:20px;'>For any odd prime <inline-formula><tex-math id="M2">\\begin{document}$ p $\\end{document}</tex-math></inline-formula>, we study constacyclic codes of length <inline-formula><tex-math id="M3">\\begin{document}$ n $\\end{document}</tex-math></inline-formula> over the finite commutative non-chain ring <inline-formula><tex-math id="M4">\\begin{document}$ R_{k,m} = \\mathbb{F}_{p^m}[u_1,u_2,\\dots,u_k]/\\langle u^2_i-1,u_iu_j-u_ju_i\\rangle_{i\\neq j = 1,2,\\dots,k} $\\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\\begin{document}$ m,k\\geq 1 $\\end{document}</tex-math></inline-formula> are integers. We determine the necessary and sufficient condition for these codes to contain their Euclidean duals. As an application, from the dual containing constacyclic codes, several MDS, new and better quantum codes compare to the best known codes in the literature are obtained.</p>
Om Prakash, Habibul Islam, and Arindam Ghosh
Springer Nature Singapore
Habibul Islam and Om Prakash
Informa UK Limited
Habibul Islam and Om Prakash
Informa UK Limited
Abstract For a prime p and integer k > 1, we find the unique set of generators for cyclic codes over . Besides, the necessary and sufficient conditions for cyclic codes to be reversible are obtained. Finally, as an application, some computational examples are given in support of our results.
Habibul Islam, Om Prakash, and Patrick Solé
American Institute of Mathematical Sciences (AIMS)
<p style='text-indent:20px;'>We study mixed alphabet cyclic and constacyclic codes over the two alphabets <inline-formula><tex-math id="M2">\\begin{document}$ \\mathbb{Z}_{4}, $\\end{document}</tex-math></inline-formula> the ring of integers modulo <inline-formula><tex-math id="M3">\\begin{document}$ 4 $\\end{document}</tex-math></inline-formula>, and its quadratic extension <inline-formula><tex-math id="M4">\\begin{document}$ \\mathbb{Z}_{4}[u] = \\mathbb{Z}_{4}+u\\mathbb{Z}_{4}, u^{2} = 0. $\\end{document}</tex-math></inline-formula> Their generator polynomials and minimal spanning sets are obtained. Further, under new Gray maps, we find cyclic, quasi-cyclic codes over <inline-formula><tex-math id="M5">\\begin{document}$ \\mathbb{Z}_{4} $\\end{document}</tex-math></inline-formula> as the Gray images of both <inline-formula><tex-math id="M6">\\begin{document}$ \\lambda $\\end{document}</tex-math></inline-formula>-constacyclic and skew <inline-formula><tex-math id="M7">\\begin{document}$ \\lambda $\\end{document}</tex-math></inline-formula>-constacyclic codes over <inline-formula><tex-math id="M8">\\begin{document}$ \\mathbb{Z}_{4}[u] $\\end{document}</tex-math></inline-formula>. Moreover, it is proved that the Gray images of <inline-formula><tex-math id="M9">\\begin{document}$ \\mathbb{Z}_{4}\\mathbb{Z}_{4}[u] $\\end{document}</tex-math></inline-formula>-additive constacyclic and skew <inline-formula><tex-math id="M10">\\begin{document}$ \\mathbb{Z}_{4}\\mathbb{Z}_{4}[u] $\\end{document}</tex-math></inline-formula>-additive constacyclic codes are generalized quasi-cyclic codes over <inline-formula><tex-math id="M11">\\begin{document}$ \\mathbb{Z}_{4} $\\end{document}</tex-math></inline-formula>. Finally, several new quaternary linear codes are obtained from these cyclic and constacyclic codes.</p>
Habibul Islam, Edgar Martínez-Moro, and Om Prakash
Elsevier BV
Shikha Yadav, Habibul Islam, Om Prakash, and Patrick Solé
Springer Science and Business Media LLC