arXiv subjects
Publications and source records attributed to Kenza Guenda.
In this paper we give the generalization of lifted codes over any finite chain ring. This has been done by using the construction of finite chain rings from $p$-adic fields. Further we propose a lattice construction from linear codes over finite chain rings using lifted codes.
Given a commutative ring $R$ with identity, a matrix $A\in M_{s\times l}(R)$, and $R$-linear codes $\mathcal{C}_1, \dots, \mathcal{C}_s$ of the same length, this article considers the hull of the matrix-product codes $[\mathcal{C}_1 \dots \mathcal{C}_s]\,A$. Consequently, it introduces various sufficient conditions under which $[\mathcal{C}_1 \dots \mathcal{C}_s]\,A$ is a linear complementary dual (LCD) code. As an application, LCD matrix-product codes arising from torsion codes over finite chain rings are considered. Highlighting examples are also given.
In this paper, a linear $\ell$-intersection pair of codes is introduced as a generalization of linear complementary pairs of codes. Two linear codes are said to be a linear $\ell$-intersection pair if their intersection has dimension $\ell$. Characterizations and constructions of such pairs of codes are given in terms of the corresponding generator and parity-check matrices. Linear $\ell$-intersection pairs of MDS codes over $\mathbb{F}_q$ of length up to $q+1$ are given for all possible parameters. As an application, linear $\ell$-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.
This paper considers the construction of isodual quasi-cyclic codes. First we prove that two quasi-cyclic codes are permutation equivalent if and only if their constituent codes are equivalent. This gives conditions on the existence of isodual quasi-cyclic codes. Then these conditions are used to obtain isodual quasi-cyclic codes. We also provide a construction for isodual quasi-cyclic codes as the matrix product of isodual codes.
In this paper, we give conditions for the existence of Hermitian self-dual $Θ-$cyclic and $Θ-$negacyclic codes over the finite chain ring $\mathbb{F}_q+u\mathbb{F}_q$. By defining a Gray map from $R=\mathbb{F}_q+u\mathbb{F}_q$ to $\mathbb{F}_{q}^{2}$, 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 $\mathbb{F}_q$ of index $2$. We also extend an algorithm of Boucher and Ulmer \cite{BF3} to construct self-dual skew cyclic codes based on the least common left multiples of non-commutative polynomials over $\mathbb{F}_q+u\mathbb{F}_q$.
In this paper, the investigation on the algebraic structure of the ring $\frac{\mathbb{F}_q[v]}{\langle\,v^q-v\,\rangle}$ and the description of its automorphism group, enable to study the algebraic structure of codes and their dual over this ring. We explore the algebraic structure of skew-constacyclic codes, by using a linear Gray map and we determine their generator polynomials. Necessary and sufficient conditions for the existence of self-dual skew cyclic and self-dual skew negacyclic codes over $\frac{\mathbb{F}_q[v]}{\langle\,v^q-v\,\rangle}$ are given.
In this paper, we study skew constacyclic codes over the ring $\mathbb{Z}_{q}R$ where $R=\mathbb{Z}_{q}+u\mathbb{Z}_{q}$, $q=p^{s}$ for a prime $p$ and $u^{2}=0$. We give the definition of these codes as subsets of the ring $\mathbb{Z}_{q}^αR^β$. Some structural properties of the skew polynomial ring $ R[x,θ]$ are discussed, where $ θ$ is an automorphism of $R$. We describe the generator polynomials of skew constacyclic codes over $ R $ and $\mathbb{Z}_{q}R$. Using Gray images of skew constacyclic codes over $\mathbb{Z}_{q}R$ we obtained some new linear codes over $\mathbb{Z}_4$. Further, we have generalized these codes to double skew constacyclic codes over $\mathbb{Z}_{q}R$.
In this paper we study the relationship between vectorial (Boolean) functions and cyclic codes in the context of algebraic attacks. We first derive a direct link between the annihilators of a vectorial function (in univariate form) and certain $2^{n}$-ary cyclic codes (which we prove that they are LCD codes) extending results due to Rønjom and Helleseth. The knowledge of the minimum distance of those codes gives rise to a lower bound on the algebraic immunity of the associated vectorial function. Furthermore, we solve an open question raised by Mesnager and Cohen. We also present some properties of those cyclic codes (whose generator polynomials determined by vectorial functions) as well as their weight enumerator. In addition we generalize the so-called algebraic complement and study its properties.
Authenticated Key Exchange (AKE) protocols represent an important cryptographic mechanism that enables several parties to communicate securely over an open network. Elashry, Mu and Susilo proposed in 2015 an Identity Based Authenticated Key Exchange (IBAKE) protocol where different parties establish secure communication by means of their public identities. The authors also introduced a new security notion for IBAKE protocols called resiliency, that is, if a shared secret between a group of parties is compromised or leaked, they can generate another completely new shared secret without the need to set up a new key exchange session. They then proved that their IBAKE protocol satisfies this security notion. We analyze the security of their protocol and prove that it has a major security flaw which renders it insecure against an impersonation attack. We also disprove the resiliency property of their scheme by proposing an attack where an adversary can compute any share secret key if just one secret bit is leaked.
In this paper, we construct a new class of complete permutation monomials and several classes of permutation polynomials. Further, by giving another characterization of o-polynomials, we obtain a class of permutation polynomials of the form $G(x)+ γTr(H(x))$, where G(X) is neither a permutation nor a linearized polynomial. This is an answer to the open problem 1 of Charpin and Kyureghyan in [P. Charpin and G. Kyureghyan, When does $G(x)+ γTr(H(x))$ permute $\mathbb{F}_{p^n}$?, Finite Fields and Their Applications 15 (2009) 615--632].
This paper present the construction cyclic isodual codes over finite fields and finite chain rings. These codes are monomially equivalent to their dual. Conditions are given for the existence of cyclic isodual codes. In addition, the concept of duadic codes over finite fields is extended to finite chain rings. Several constructions of isodual cyclic codes and self-dual codes are given.
This paper considers the equivalence problem for quasi-cyclic codes over finite fields. The results obtained are used to construct isodual quasi-cyclic codes.
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.
In this paper, we construct linear codes over $\mathbb{Z}_4$ with bounded $GC$-content. The codes are obtained using a greedy algorithm over $\mathbb{Z}_4$. 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 $d$.
This paper is dealing with DNA cyclic codes which play an important role in DNA computing and have attracted a particular attention in the literature. Firstly, we introduce a new family of DNA cyclic codes over the ring $R=\mathbb{F}_2[u]/(u^6)$. Such codes have theoretical advantages as well as several applications in DNA computing. A direct link between the elements of such a ring and the $64$ codons used in the amino acids of the living organisms is established. Such a correspondence allows us to extend the notion of the edit distance to the ring $R$ which is useful for the correction of the insertion, deletion and substitution errors. Next, we define the Lee weight, the Gray map over the ring $R$ as well as the binary image of the cyclic DNA codes allowing the transfer of studying DNA codes into studying binary codes. Secondly, we introduce another new family of DNA skew cyclic codes constructed over the ring $\tilde {R}=\mathbb{F}_2+v\mathbb{F}_2=\{0,1,v,v+1\}$ where $v^2=v$ and study their property of being reverse-complement. We show that the obtained code is derived from the cyclic reverse-complement code over the ring $\tilde {R}$. We shall provide the binary images and present some explicit examples of such codes.
In this paper, we give an important isomorphism between contacyclic codes and cyclic codes over finite principal ideal rings. Necessary and sufficient conditions for the existence of non-trivial cyclic self-dual codes over finite principal ideal rings are given.
In this paper we investigate the structure of repeated root constacyclic codes of length $2^amp^r$ over $\mathbb{F}_{p^s}$ with $a\geq1$ and $(m,p)=1$. We characterize the codes in terms of their generator polynomials. This provides simple conditions on the existence of self-dual negacyclic codes. Further, we gave cases where the constacyclic codes are equivalent to cyclic codes.
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.