In this paper, a linear ℓ-intersection pair of codes is introduced as a generalization of linear ... more In this paper, a linear ℓ-intersection pair of codes is introduced as a generalization of linear complementary pairs of codes. Two linear codes are said to be a linear ℓ-intersection pair if their intersection has dimension ℓ. Characterizations and constructions of such pairs of codes are given in terms of the corresponding generator and parity-check matrices. Linear ℓ-intersection pairs of MDS codes over F q of length up to q + 1 are given for all possible parameters. As an application, linear ℓ-intersection pairs of codes are used to construct entanglement-assisted quantum error correcting codes. This provides a large number of new MDS entanglement-assisted quantum error correcting codes.
For λ an n-th power of a unit in a finite chain ring we prove that λ-constacyclic repeated-root c... more For λ an n-th power of a unit in a finite chain ring we prove that λ-constacyclic repeated-root codes over some finite chain rings are equivalent to cyclic codes. This allows us to simplify the structure of some constacylic codes. We also study the α+pβconstacyclic codes of length p s over the Galois ring GR(p e , r).
In this paper, we construct linear codes over Z4 with bounded GCcontent. The codes are obtained u... more In this paper, we construct linear codes over Z4 with bounded GCcontent. The codes are obtained using a greedy algorithm over Z4. Further, upper and lower bounds are derived for the maximum size of DNA codes of length n with constant GC-content w and edit distance d.
Entanglement-assisted quantum error correcting codes (EAQECCs) are a simple and fundamental class... more Entanglement-assisted quantum error correcting codes (EAQECCs) are a simple and fundamental class of codes. They allow for the construction of quantum codes from classical codes by relaxing the duality condition and using pre-shared entanglement between the sender and receiver. However, in general it is not easy to determine the number of shared pairs required to construct an EAQECC. In this paper, we show that this number is related to the hull of the classical code. Using this fact, we give methods to construct EAQECCs requiring desirable amount of entanglement. This leads to design families of EAQECCs with good error performance. Moreover, we construct maximal entanglement EAQECCs from LCD codes. Finally, we prove the existence of asymptotically good EAQECCs in the odd characteristic case.
We construct codes over the ring F 2 +uF 2 with u 2 = 0. These code are designed for use in DNA c... more We construct codes over the ring F 2 +uF 2 with u 2 = 0. These code are designed for use in DNA computing applications. The codes obtained satisfy the reverse complement constraint, the GC content constraint and avoid the secondary structure. they are derived from the cyclic complement reversible codes over the ring F 2 + uF 2. We also construct an infinite family of BCH DNA codes.
In this paper, we give conditions for the existence of Hermitian selfdual Θ−cyclic and Θ−negacycl... more In this paper, we give conditions for the existence of Hermitian selfdual Θ−cyclic and Θ−negacyclic codes over the finite chain ring F q + uF q. By defining a Gray map from R = F q + uF q to F 2 q , we prove that the Gray images of skew cyclic codes of odd length n over R with even characteristic are equivalent to skew quasi-twisted codes of length 2n over F q of index 2. We also extend an algorithm of Boucher and Ulmer [9] to construct self-dual skew cyclic codes based on the least common left multiples of non-commutative polynomials over F q + uF q .
In this paper, we investigate the structure and properties of skew negacyclic codes and skew quas... more In this paper, we investigate the structure and properties of skew negacyclic codes and skew quasi-negacyclic codes over the ring [Formula: see text] Some structural properties of [Formula: see text] are discussed, where [Formula: see text] is an automorphism of [Formula: see text] A skew quasi-negacyclic code of length [Formula: see text] with index [Formula: see text] over [Formula: see text] is viewed both as in the conventional row circulant form and also as an [Formula: see text]-submodule of [Formula: see text], where [Formula: see text] is the Galois extension ring of degree [Formula: see text] over [Formula: see text] and [Formula: see text] is an automorphism of [Formula: see text] A sufficient condition for one generator skew quasi-negacyclic codes to be free is determined. Some distance bounds for free one generator skew quasi-negacyclic codes are discussed. Furthermore, given the decomposition of a skew quasi-negacyclic code, we provide the decomposition of its dual code...
In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-d... more In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring R = F q + vF q + v 2 F q , where v 3 = v, for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over F q and extend these to codes over R.
In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-d... more In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring $R=\F_{q}+v\F_{q}+v^{2}\F_{q}$, where $v^{3}=v$, for $q$ odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over $R$. Further, we give bounds on the minimum distance of LCD codes over $\F_q$ and extend these to codes over $R$.
Journal of Applied Mathematics and Computing, 2017
In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-d... more In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring R = F q + vF q + v 2 F q , where v 3 = v, for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over F q and extend these to codes over R.
In this paper we define $\mathbb{Z}_{2}\mathbb{Z}_{4}-$Simplex and MacDonald Codes of type $\alph... more In this paper we define $\mathbb{Z}_{2}\mathbb{Z}_{4}-$Simplex and MacDonald Codes of type $\alpha $ and $\beta $ and we give the covering radius of these codes.
In this paper we extend the work of Lisonek and Singh on construction X for quantum error-correct... more In this paper we extend the work of Lisonek and Singh on construction X for quantum error-correcting codes to finite fields of order p 2 where p is prime. The results obtained are applied to the dual of Hermitian repeated root cyclic codes to generate new quantum error-correcting codes.
In this paper, we study the linear codes over the commutative ring R = F q + vF q + v 2 F q and t... more In this paper, we study the linear codes over the commutative ring R = F q + vF q + v 2 F q and their Gray images, where v 3 = v. We define the Lee weight of the elements of R, we give a Gray map from R n to F 3n q and we give the relation between the dual and the Gray image of a code. This allows us to investigate the structure and properties of self-dual cyclic, formally self-dual and the Gray image of formally self-dual codes over R. Further, we give several constructions of formally self-dual codes over R.
In this paper, we study the linear codes over the commutative ring R = F q + vF q + v 2 F q and t... more In this paper, we study the linear codes over the commutative ring R = F q + vF q + v 2 F q and their Gray images, where v 3 = v. We define the Lee weight of the elements of R, we give a Gray map from R n to F 3n q and we give the relation between the dual and the Gray image of a code. This allows us to investigate the structure and properties of self-dual cyclic, formally self-dual and the Gray image of formally self-dual codes over R. Further, we give several constructions of formally self-dual codes over R.
RAIRO - Theoretical Informatics and Applications, 2021
In this paper, we present a new variant of the Niederreiter Public Key Encryption (PKE) scheme wh... more In this paper, we present a new variant of the Niederreiter Public Key Encryption (PKE) scheme which is resistant against recent attacks. The security is based on the hardness of the Rank Syndrome Decoding (RSD) problem and it presents a (u|u + υ)-construction code using two different types of codes: Ideal Low Rank Parity Check (ILRPC) codes and λ-Gabidulin codes. The proposed encryption scheme benefits are a larger minimum distance, a new efficient decoding algorithm and a smaller ciphertext and public key size compared to the Loidreau’s variants and to its IND-CCA secure version.
In this paper we propose a new rank code based signature scheme that used a concatenation of the ... more In this paper we propose a new rank code based signature scheme that used a concatenation of the LRPC and the λ-Gabidulin codes. Our construction benefits from the decoding algorithm of both of codes a considerable security levels with a moderate public key size
Notes on Number Theory and Discrete Mathematics, 2020
For x large enough, there exists a primitive sequence \mathcal{A}, such that \begin{equation*} \s... more For x large enough, there exists a primitive sequence \mathcal{A}, such that \begin{equation*} \sum\limits_{a\in \mathcal{A}}\frac{1}{a(\log a+x)}\gg \sum\limits_{p\in \mathcal{P}}\frac{1}{p(\log p+x)}\text{,} \end{equation*} where \mathcal{P} denotes the set of prime numbers.
In this paper, we propose a new public key cryptosystem in a non-commutative group over group rin... more In this paper, we propose a new public key cryptosystem in a non-commutative group over group ring, using a hard problem, Factorization with Discrete Logarithm Problem (FDLP). The security analysis of the proposed scheme is discussed and it is shown that the system is secure.
In this paper, a linear ℓ-intersection pair of codes is introduced as a generalization of linear ... more In this paper, a linear ℓ-intersection pair of codes is introduced as a generalization of linear complementary pairs of codes. Two linear codes are said to be a linear ℓ-intersection pair if their intersection has dimension ℓ. Characterizations and constructions of such pairs of codes are given in terms of the corresponding generator and parity-check matrices. Linear ℓ-intersection pairs of MDS codes over F q of length up to q + 1 are given for all possible parameters. As an application, linear ℓ-intersection pairs of codes are used to construct entanglement-assisted quantum error correcting codes. This provides a large number of new MDS entanglement-assisted quantum error correcting codes.
For λ an n-th power of a unit in a finite chain ring we prove that λ-constacyclic repeated-root c... more For λ an n-th power of a unit in a finite chain ring we prove that λ-constacyclic repeated-root codes over some finite chain rings are equivalent to cyclic codes. This allows us to simplify the structure of some constacylic codes. We also study the α+pβconstacyclic codes of length p s over the Galois ring GR(p e , r).
In this paper, we construct linear codes over Z4 with bounded GCcontent. The codes are obtained u... more In this paper, we construct linear codes over Z4 with bounded GCcontent. The codes are obtained using a greedy algorithm over Z4. Further, upper and lower bounds are derived for the maximum size of DNA codes of length n with constant GC-content w and edit distance d.
Entanglement-assisted quantum error correcting codes (EAQECCs) are a simple and fundamental class... more Entanglement-assisted quantum error correcting codes (EAQECCs) are a simple and fundamental class of codes. They allow for the construction of quantum codes from classical codes by relaxing the duality condition and using pre-shared entanglement between the sender and receiver. However, in general it is not easy to determine the number of shared pairs required to construct an EAQECC. In this paper, we show that this number is related to the hull of the classical code. Using this fact, we give methods to construct EAQECCs requiring desirable amount of entanglement. This leads to design families of EAQECCs with good error performance. Moreover, we construct maximal entanglement EAQECCs from LCD codes. Finally, we prove the existence of asymptotically good EAQECCs in the odd characteristic case.
We construct codes over the ring F 2 +uF 2 with u 2 = 0. These code are designed for use in DNA c... more We construct codes over the ring F 2 +uF 2 with u 2 = 0. These code are designed for use in DNA computing applications. The codes obtained satisfy the reverse complement constraint, the GC content constraint and avoid the secondary structure. they are derived from the cyclic complement reversible codes over the ring F 2 + uF 2. We also construct an infinite family of BCH DNA codes.
In this paper, we give conditions for the existence of Hermitian selfdual Θ−cyclic and Θ−negacycl... more In this paper, we give conditions for the existence of Hermitian selfdual Θ−cyclic and Θ−negacyclic codes over the finite chain ring F q + uF q. By defining a Gray map from R = F q + uF q to F 2 q , we prove that the Gray images of skew cyclic codes of odd length n over R with even characteristic are equivalent to skew quasi-twisted codes of length 2n over F q of index 2. We also extend an algorithm of Boucher and Ulmer [9] to construct self-dual skew cyclic codes based on the least common left multiples of non-commutative polynomials over F q + uF q .
In this paper, we investigate the structure and properties of skew negacyclic codes and skew quas... more In this paper, we investigate the structure and properties of skew negacyclic codes and skew quasi-negacyclic codes over the ring [Formula: see text] Some structural properties of [Formula: see text] are discussed, where [Formula: see text] is an automorphism of [Formula: see text] A skew quasi-negacyclic code of length [Formula: see text] with index [Formula: see text] over [Formula: see text] is viewed both as in the conventional row circulant form and also as an [Formula: see text]-submodule of [Formula: see text], where [Formula: see text] is the Galois extension ring of degree [Formula: see text] over [Formula: see text] and [Formula: see text] is an automorphism of [Formula: see text] A sufficient condition for one generator skew quasi-negacyclic codes to be free is determined. Some distance bounds for free one generator skew quasi-negacyclic codes are discussed. Furthermore, given the decomposition of a skew quasi-negacyclic code, we provide the decomposition of its dual code...
In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-d... more In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring R = F q + vF q + v 2 F q , where v 3 = v, for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over F q and extend these to codes over R.
In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-d... more In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring $R=\F_{q}+v\F_{q}+v^{2}\F_{q}$, where $v^{3}=v$, for $q$ odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over $R$. Further, we give bounds on the minimum distance of LCD codes over $\F_q$ and extend these to codes over $R$.
Journal of Applied Mathematics and Computing, 2017
In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-d... more In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring R = F q + vF q + v 2 F q , where v 3 = v, for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over F q and extend these to codes over R.
In this paper we define $\mathbb{Z}_{2}\mathbb{Z}_{4}-$Simplex and MacDonald Codes of type $\alph... more In this paper we define $\mathbb{Z}_{2}\mathbb{Z}_{4}-$Simplex and MacDonald Codes of type $\alpha $ and $\beta $ and we give the covering radius of these codes.
In this paper we extend the work of Lisonek and Singh on construction X for quantum error-correct... more In this paper we extend the work of Lisonek and Singh on construction X for quantum error-correcting codes to finite fields of order p 2 where p is prime. The results obtained are applied to the dual of Hermitian repeated root cyclic codes to generate new quantum error-correcting codes.
In this paper, we study the linear codes over the commutative ring R = F q + vF q + v 2 F q and t... more In this paper, we study the linear codes over the commutative ring R = F q + vF q + v 2 F q and their Gray images, where v 3 = v. We define the Lee weight of the elements of R, we give a Gray map from R n to F 3n q and we give the relation between the dual and the Gray image of a code. This allows us to investigate the structure and properties of self-dual cyclic, formally self-dual and the Gray image of formally self-dual codes over R. Further, we give several constructions of formally self-dual codes over R.
In this paper, we study the linear codes over the commutative ring R = F q + vF q + v 2 F q and t... more In this paper, we study the linear codes over the commutative ring R = F q + vF q + v 2 F q and their Gray images, where v 3 = v. We define the Lee weight of the elements of R, we give a Gray map from R n to F 3n q and we give the relation between the dual and the Gray image of a code. This allows us to investigate the structure and properties of self-dual cyclic, formally self-dual and the Gray image of formally self-dual codes over R. Further, we give several constructions of formally self-dual codes over R.
RAIRO - Theoretical Informatics and Applications, 2021
In this paper, we present a new variant of the Niederreiter Public Key Encryption (PKE) scheme wh... more In this paper, we present a new variant of the Niederreiter Public Key Encryption (PKE) scheme which is resistant against recent attacks. The security is based on the hardness of the Rank Syndrome Decoding (RSD) problem and it presents a (u|u + υ)-construction code using two different types of codes: Ideal Low Rank Parity Check (ILRPC) codes and λ-Gabidulin codes. The proposed encryption scheme benefits are a larger minimum distance, a new efficient decoding algorithm and a smaller ciphertext and public key size compared to the Loidreau’s variants and to its IND-CCA secure version.
In this paper we propose a new rank code based signature scheme that used a concatenation of the ... more In this paper we propose a new rank code based signature scheme that used a concatenation of the LRPC and the λ-Gabidulin codes. Our construction benefits from the decoding algorithm of both of codes a considerable security levels with a moderate public key size
Notes on Number Theory and Discrete Mathematics, 2020
For x large enough, there exists a primitive sequence \mathcal{A}, such that \begin{equation*} \s... more For x large enough, there exists a primitive sequence \mathcal{A}, such that \begin{equation*} \sum\limits_{a\in \mathcal{A}}\frac{1}{a(\log a+x)}\gg \sum\limits_{p\in \mathcal{P}}\frac{1}{p(\log p+x)}\text{,} \end{equation*} where \mathcal{P} denotes the set of prime numbers.
In this paper, we propose a new public key cryptosystem in a non-commutative group over group rin... more In this paper, we propose a new public key cryptosystem in a non-commutative group over group ring, using a hard problem, Factorization with Discrete Logarithm Problem (FDLP). The security analysis of the proposed scheme is discussed and it is shown that the system is secure.
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