arXiv subjects
Publications and source records attributed to Djoko Suprijanto.
In this work, we present novel approaches for constructing linear codes over $\ZZ_4$ from the known ones. We succeeded in obtaining new linear codes, many of which are optimal. In particular, we found all optimal codes for $k_1=2,~k_2=0$ and many optimal codes for $k_1=3,~k_2=0.$
This article focuses specifically on the study of self-dual double cyclic codes over a finite field $\mathbb{F}_q$. A self-dual double cyclic code is a double cyclic code that is equal to its dual. Structurally, a double cyclic code of length $(r,s)$ over $\mathbb{F}_q$ is a $\mathbb{F}_q[x]$-submodule of $\mathbb{F}_{q,r,s}:=\mathbb{F}_q[x]/\langle x^r-1\rangle\times\mathbb{F}_q[x]/\langle x^s-1\rangle$. Moreover, any double cyclic code of length $(r,s)$ over $\mathbb{F}_q$ is generated by two pairs of polynomials in $\mathbb{F}_{q,r,s}$. From the properties of the generating elements, we provide the necessary and sufficient conditions such that two pairs of polynomials in $\mathbb{F}_{q,r,s}$ generate a self-dual code. Furthermore, we examine the existence of self-dual double cyclic codes for some specific lengths: $(r,r)$; $(r,2r)$ and $(2r,r)$; and $(r,s)$, where $\gcd(r,s)=1$. For each case, we provide a construction method with some explicit examples over various finite fields. We also observe some connections between self-dual double cyclic codes and other classes of self-dual codes.
We obtain all possible parameters of Plotkin-optimal two-Lee weight projective codes over $\mathbb{Z}_4,$ together with their weight distributions. We show the existence of codes with these parameters as well as their weight distributions by constructing an infinite family of two-weight codes. Previously known codes constructed by Shi et al. (\emph{Des Codes Cryptogr.} {\bf 88}(3):1-13, 2020) can be derived as a special case of our results. We also prove that the Gray image of any Plotkin-optimal two-Lee weight projective codes over $\mathbb{Z}_4$ has the same parameters and weight distribution as some two-weight binary projective codes of type SU1 in the sense of Calderbank and Kantor (\emph{Bull. Lond. Math. Soc.} {\bf 18}:97-122, 1986).
In this work, we study a class of skew cyclic codes over the ring $R:=\mathbb{Z}_4+v\mathbb{Z}_4,$ where $v^2=v,$ with an automorphism $θ$ and a derivation $Δ_θ,$ namely codes as modules over a skew polynomial ring $R[x;θ,Δ_θ],$ whose multiplication is defined using an automorphism $θ$ and a derivation $Δ_θ.$ We investigate the structures of a skew polynomial ring $R[x;θ,Δ_θ].$ We define $Δ_θ$-cyclic codes as a generalization of the notion of cyclic codes. The properties of $Δ_θ$-cyclic codes as well as dual $Δ_θ$-cyclic codes are derived. As an application, some new linear codes over $\mathbb{Z}_4$ with good parameters are obtained by Plotkin sum construction, also via a Gray map as well as residue and torsion codes of these codes.
In this article, we investigate properties of cyclic codes over a finite non-chain ring $\mathbb{F}_q+v\mathbb{F}_q+v^2\mathbb{F}_q+v^3\mathbb{F}_q+v^4\mathbb{F}_q,$ where $q=p^r,$ $r$ is a positive integer, $p$ is an odd prime, $4 \mid (p-1),$ and $v^5=v.$ As an application, we construct several quantum error correcting codes over the finite field $\mathbb{F}_q.$
Alahmadi et al. ["Twisted centralizer codes", \emph{Linear Algebra and its Applications} {\bf 524} (2017) 235-249.] introduced the notion of twisted centralizer codes, $\mathcal{C}_{\mathbb{F}_q}(A,γ),$ defined as \[ \mathcal{C}_{\mathbb{F}_q}(A,γ)=\lbrace X \in \mathbb{F}_q^{n \times n}:~\ AX=γXA\rbrace, \] for $A \in \mathbb{F}_q^{n \times n},$ and $γ\in \mathbb{F}_q.$ Moreover, Alahmadi et al. ["On the dimension of twisted centralizer codes", \emph{Finite Fields and Their Applications} {\bf 48} (2017) 43-59.] also investigated the dimension of such codes and obtained upper and lower bounds for the dimension, and the exact value of the dimension only for cyclic or diagonalizable matrices $A.$ Generalizing and sharpening Alahmadi et al.'s results, in this paper, we determine the exact value of the dimension as well as provide an algorithm to construct an explicit basis of the codes for any given matrix $A.$
This paper studies an attitude control system design based on modified Rodrigues parameters feedback. It employs a linear continuous sliding mode controller. The sliding mode controller is able to bring the existence of the sliding motion asymptotically. Besides, the attitude control system equilibrium point is proved to have an asymptotic stability guarantee through further analysis. This stability analysis is conducted since the sliding mode existence on the designed sliding surface does not imply the stability guarantee of the system's equilibrium. This paper ends with some numerical examples that confirm the effectiveness of the designed attitude control system.
We investigate linear codes over the ring $\mathbb{Z}_4 + u\mathbb{Z}_4 + v\mathbb{Z}_4 + w\mathbb{Z}_4 + uv\mathbb{Z}_4 + uw\mathbb{Z}_4 + vw\mathbb{Z}_4 + uvw\mathbb{Z}_4$, with conditions $u^2=u$, $v^2=v$, $w^2=w$, $uv=vu$, $uw=wu$ and $vw=wv.$ We first analyze the structure of the ring and then define linear codes over this ring. Lee weight and Gray map for these codes are defined and MacWilliams relations for complete, symmetrized, and Lee weight enumerators are obtained. The Singleton bound as well as maximum distance separable codes are also considered. Furthermore, cyclic and quasi-cyclic codes are discussed, and some examples are also provided.
In this paper, we consider some structures of linear codes over the ring $\mathcal{R}_k=R[v_1,\dots,v_k],$ where $v_i^2=v_i$ forall $i=1,\dots,k),$ and $R$ is a finite commutative Frobenius ring.
We study the structure of linear codes over the ring $B_k$ which is defined by $\mathbb{F}_{p^r}[v_1,v_2,\ldots,v_k]/\langle v_i^2=v_i,~v_iv_j=v_jv_i \rangle_{i,j=1}^k.$ In order to study the codes, we begin with studying the structure of the ring $B_k$ via a Gray map which also induces a relation between codes over $B_k$ and codes over $\mathbb{F}_{p^r}.$ We consider Euclidean and Hermitian self-dual codes, MacWilliams relations, as well as Singleton-type bounds for these codes. Further, we characterize cyclic and quasi-cyclic codes using their images under the Gray map, and give the generators for these type of codes.
We study the non-rigidity of Euclidean $t$-designs, namely we study when Euclidean designs (in particular certain tight Euclidean designs) can be deformed keeping the property of being Euclidean $t$-designs. We show that certain tight Euclidean $t$-designs are non-rigid, and in fact satisfy a stronger form of non-rigidity which we call strong non-rigidity. This shows that there are plenty of non-isomorphic tight Euclidean $t$-designs for certain parameters, which seems to have been unnoticed before. We also include the complete classification of tight Euclidean $2$-designs.
We study $Θ_S-$cyclic codes over the family of rings $A_k.$ We characterize $Θ_S-$cyclic codes in terms of their binary images. A family of Hermitian inner-products is defined and we prove that if a code is $Θ_S-$cyclic then its Hermitian dual is also $Θ_S-$cyclic. Finally, we give constructions of $Θ_S-$cyclic codes.
In this paper we study the structure of $θ$-cyclic codes over the ring $B_k$ including its connection to quasi-$\tildeθ$-cyclic codes over finite field $\mathbb{F}_{p^r}$ and skew polynomial rings over $B_k.$ We also characterize Euclidean self-dual $θ$-cyclic codes over the rings. Finally, we give the generator polynomial for such codes and some examples of optimal Euclidean $θ$-cyclic codes.
In this paper we give the enumeration formulas for Euclidean self-dual skew-cyclic codes over finite fields when $(n,|θ|)=1$ and for some cases when $(n,|θ|)>1,$ where $n$ is the length of the code and $|θ|$ is the order of automorphism $θ.$
A finite set $X \seq \RR^n$ with a weight function $w : X \longrightarrow \RR_{>0}$ is called \emph{Euclidean $t$-design} in $\RR^n$ (supported by $p$ concentric spheres) if the following condition holds: \[ \sum_{i=1}^p \frac{w(X_i)}{|S_i|}\int_{S_i} f(\boldsymbol x)dσ_i(\boldsymbol x) =\sum_{\boldsymbol x \in X}w(\boldsymbol x) f(\boldsymbol x), \] for any polynomial $f(\boldsymbol x) \in \mbox{Pol}(\RR^n)$ of degree at most $t$. Here $S_i \seq \RR^n$ is a sphere of radius $r_i \geq 0,$ $X_i=X \cap S_i,$ and $σ_i(\boldsymbol x)$ is an $O(n)$-invariant measure on $S_i$ such that $|S_i|=r_i^{n-1}|S^{n-1}|$, with $|S_i|$ is the surface area of $S_i$ and $|S^{n-1}|$ is a surface area of the unit sphere in $\RR^n$. Recently, Bajnok (2006) constructed tight Euclidean $t$-designs in the plane ($n=2$) for arbitrary $t$ and $p.$ In this paper we show that for case $t=6$ and $p=2,$ tight Euclidean $6$-designs constructed by Bajnok is the unique configuration in $\RR^n$, for $2 \leq n \leq 8.$
Most methods of orbit determination are often difficult for numerical implementations since they are developed before the computer era. The recently developed mathematical technique of semi-definite programming (SDP) has been implemented for many problems in scientific fields including astrometry. This is a good opportunity to resolve orbits of binary systems located in the southern hemisphere since more than seventy years Bosscha Observatory had been continuously conducting observations of binary systems. Here we describe prospects of application of SDP for deriving orbital parameters of binary systems using data supplied by Bosscha Observatory that has been published in the Centre de Donnees astronomiques de Strasbourg. This study will support observers at Bosscha Observatory to appropriately select target stars belong to binary systems for their ongoing researches. Since SDP is a powerful scheme, free trial-and-error and human-independent judgment, we suggest that SDP may become a standard method for determining orbital parameters of binary systems.