Pub Date : 2021-09-15DOI: 10.13069/jacodesmath.1000837
Pankaj Kumar, P. Devi
{"title":"Minimum distance and idempotent generators of minimal cyclic codes of length ${p_1}^{alpha_1}{p_2}^{alpha_2}{p_3}^{alpha_3}$","authors":"Pankaj Kumar, P. Devi","doi":"10.13069/jacodesmath.1000837","DOIUrl":"https://doi.org/10.13069/jacodesmath.1000837","url":null,"abstract":"","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48953757","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-31DOI: 10.13069/JACODESMATH.935938
Gaurav Mittal, R. Sharma
In this paper, we characterize the unit groups of semisimple group algebras $mathbb{F}_qG$ of non-metabelian groups of order $108$, where $F_q$ is a field with $q=p^k$ elements for some prime $p > 3$ and positive integer $k$. Up to isomorphism, there are $45$ groups of order $108$ but only $4$ of them are non-metabelian. We consider all the non-metabelian groups of order $108$ and find the Wedderburn decomposition of their semisimple group algebras. And as a by-product obtain the unit groups.
{"title":"On unit group of finite semisimple group algebras of non-metabelian groups of order 108","authors":"Gaurav Mittal, R. Sharma","doi":"10.13069/JACODESMATH.935938","DOIUrl":"https://doi.org/10.13069/JACODESMATH.935938","url":null,"abstract":"In this paper, we characterize the unit groups of semisimple group algebras $mathbb{F}_qG$ of non-metabelian groups of order $108$, where $F_q$ is a field with $q=p^k$ elements for some prime $p > 3$ and positive integer $k$. Up to isomorphism, there are $45$ groups of order $108$ but only $4$ of them are non-metabelian. We consider all the non-metabelian groups of order $108$ and find the Wedderburn decomposition of their semisimple group algebras. And as a by-product obtain the unit groups.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46689007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-20DOI: 10.13069/JACODESMATH.935980
T. Vetrík
We generalize several topological indices and introduce the general degree distance of a connected graph $G$. For $a, b in mathbb{R}$, the general degree distance $DD_{a,b} (G) = sum_{ v in V(G)} [deg_{G}(v)]^a S^b_{G} (v)$, where $V(G)$ is the vertex set of $G$, $deg_G (v)$ is the degree of a vertex $v$, $S^b_{G} (v) = sum_{ w in V(G) setminus { v } } [d_{G} (v,w) ]^{b}$ and $d_{G} (v,w)$ is the distance between $v$ and $w$ in $G$. We present some sharp bounds on the general degree distance for multipartite graphs and trees of given order, graphs of given order and chromatic number, graphs of given order and vertex connectivity, and graphs of given order and number of pendant vertices.
{"title":"General degree distance of graphs","authors":"T. Vetrík","doi":"10.13069/JACODESMATH.935980","DOIUrl":"https://doi.org/10.13069/JACODESMATH.935980","url":null,"abstract":"We generalize several topological indices and introduce the general degree distance of a connected graph $G$. For $a, b in mathbb{R}$, the general degree distance $DD_{a,b} (G) = sum_{ v in V(G)} [deg_{G}(v)]^a S^b_{G} (v)$, where $V(G)$ is the vertex set of $G$, $deg_G (v)$ is the degree of a vertex $v$, $S^b_{G} (v) = sum_{ w in V(G) setminus { v } } [d_{G} (v,w) ]^{b}$ and $d_{G} (v,w)$ is the distance between $v$ and $w$ in $G$. We present some sharp bounds on the general degree distance for multipartite graphs and trees of given order, graphs of given order and chromatic number, graphs of given order and vertex connectivity, and graphs of given order and number of pendant vertices.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49160174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-16DOI: 10.13069/JACODESMATH.935947
Nanami Bono, Maya Fuji̇i̇, T. Maruta
In coding theory, the problem of finding the shortest linear codes for a fixed set of parameters is central. Given the dimension $k$, the minimum weight $d$, and the order $q$ of the finite field $bF_q$ over which the code is defined, the function $n_q(k, d)$ specifies the smallest length $n$ for which an $[n, k, d]_q$ code exists. The problem of determining the values of this function is known as the problem of optimal linear codes. Using the geometric methods through projective geometry, we determine $n_q(4,d)$ for some values of $d$ by constructing new codes and by proving the nonexistence of linear codes with certain parameters.
{"title":"On optimal linear codes of dimension 4","authors":"Nanami Bono, Maya Fuji̇i̇, T. Maruta","doi":"10.13069/JACODESMATH.935947","DOIUrl":"https://doi.org/10.13069/JACODESMATH.935947","url":null,"abstract":"In coding theory, the problem of finding the shortest linear codes for a fixed set of parameters is central. Given the dimension $k$, the minimum weight $d$, and the order $q$ of the finite field $bF_q$ over which the code is defined, the function $n_q(k, d)$ specifies the smallest length $n$ for which an $[n, k, d]_q$ code exists. The problem of determining the values of this function is known as the problem of optimal linear codes. Using the geometric methods through projective geometry, we determine $n_q(4,d)$ for some values of $d$ by constructing new codes and by proving the nonexistence of linear codes with certain parameters.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44473526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-16DOI: 10.13069/JACODESMATH.935951
Adrian Korban
In this paper, we extend the work done on $G$-codes over formal power series rings and finite chain rings $mathbb{F}_q[t]/(t^i)$, to composite $G$-codes over the same alphabets. We define composite $G$-codes over the infinite ring $R_infty$ as ideals in the group ring $R_infty G.$ We show that the dual of a composite $G$-code is again a composite $G$-code in this setting. We extend the known results on projections and lifts of $G$-codes over the finite chain rings and over the formal power series rings to composite $G$-codes. Additionally, we extend some known results on $gamma$-adic $G$-codes over $R_infty$ to composite $G$-codes and study these codes over principal ideal rings.
{"title":"Composite G-codes over formal power series rings and finite chain rings","authors":"Adrian Korban","doi":"10.13069/JACODESMATH.935951","DOIUrl":"https://doi.org/10.13069/JACODESMATH.935951","url":null,"abstract":"In this paper, we extend the work done on $G$-codes over formal power series rings and finite chain rings $mathbb{F}_q[t]/(t^i)$, to composite $G$-codes over the same alphabets. We define composite $G$-codes over the infinite ring $R_infty$ as ideals in the group ring $R_infty G.$ We show that the dual of a composite $G$-code is again a composite $G$-code in this setting. We extend the known results on projections and lifts of $G$-codes over the finite chain rings and over the formal power series rings to composite $G$-codes. Additionally, we extend some known results on $gamma$-adic $G$-codes over $R_infty$ to composite $G$-codes and study these codes over principal ideal rings.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48314577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-16DOI: 10.13069/JACODESMATH.938105
S. Visweswaran, P. T. Lalchandani
The rings considered in this article are commutative with identity. For an ideal $I$ of a ring $R$, we denote the annihilator of $I$ in $R$ by $Ann(I)$. An ideal $I$ of a ring $R$ is said to be an exact annihilating ideal if there exists a non-zero ideal $J$ of $R$ such that $Ann(I) = J$ and $Ann(J) = I$. For a ring $R$, we denote the set of all exact annihilating ideals of $R$ by $mathbb{EA}(R)$ and $mathbb{EA}(R)backslash {(0)}$ by $mathbb{EA}(R)^{*}$. Let $R$ be a ring such that $mathbb{EA}(R)^{*}neq emptyset$. With $R$, in [Exact Annihilating-ideal graph of commutative rings, {it J. Algebra and Related Topics} {bf 5}(1) (2017) 27-33] P.T. Lalchandani introduced and investigated an undirected graph called the exact annihilating-ideal graph of $R$, denoted by $mathbb{EAG}(R)$ whose vertex set is $mathbb{EA}(R)^{*}$ and distinct vertices $I$ and $J$ are adjacent if and only if $Ann(I) = J$ and $Ann(J) = I$. In this article, we continue the study of the exact annihilating-ideal graph of a ring. In Section 2 , we prove some basic properties of exact annihilating ideals of a commutative ring and we provide several examples. In Section 3, we determine the structure of $mathbb{EAG}(R)$, where either $R$ is a special principal ideal ring or $R$ is a reduced ring which admits only a finite number of minimal prime ideals.
{"title":"The exact annihilating-ideal graph of a commutative ring","authors":"S. Visweswaran, P. T. Lalchandani","doi":"10.13069/JACODESMATH.938105","DOIUrl":"https://doi.org/10.13069/JACODESMATH.938105","url":null,"abstract":"The rings considered in this article are commutative with identity. For an ideal $I$ of a ring $R$, we denote the annihilator of $I$ in $R$ by $Ann(I)$. An ideal $I$ of a ring $R$ is said to be an exact annihilating ideal if there exists a non-zero ideal $J$ of $R$ such that $Ann(I) = J$ and $Ann(J) = I$. For a ring $R$, we denote the set of all exact annihilating ideals of $R$ by $mathbb{EA}(R)$ and $mathbb{EA}(R)backslash {(0)}$ by $mathbb{EA}(R)^{*}$. Let $R$ be a ring such that $mathbb{EA}(R)^{*}neq emptyset$. With $R$, in [Exact Annihilating-ideal graph of commutative rings, {it J. Algebra and Related Topics} {bf 5}(1) (2017) 27-33] P.T. Lalchandani introduced and investigated an undirected graph called the exact annihilating-ideal graph of $R$, denoted by $mathbb{EAG}(R)$ whose vertex set is $mathbb{EA}(R)^{*}$ and distinct vertices $I$ and $J$ are adjacent if and only if $Ann(I) = J$ and $Ann(J) = I$. In this article, we continue the study of the exact annihilating-ideal graph of a ring. In Section 2 , we prove some basic properties of exact annihilating ideals of a commutative ring and we provide several examples. In Section 3, we determine the structure of $mathbb{EAG}(R)$, where either $R$ is a special principal ideal ring or $R$ is a reduced ring which admits only a finite number of minimal prime ideals.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46448632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-15DOI: 10.13069/JACODESMATH.867617
K. Sowndhariya, A. Muthusamy
For any integer $kgeq 3$ , we define sunlet graph of order $2k$, denoted by $L_{2k}$, as the graph consisting of a cycle of length $k$ together with $k$ pendant vertices, each adjacent to exactly one vertex of the cycle. In this paper, we give necessary and sufficient conditions for the existence of $L_{8}$-decomposition of tensor product and wreath product of complete graphs.
{"title":"Decomposition of product graphs into sunlet graphs of order eight","authors":"K. Sowndhariya, A. Muthusamy","doi":"10.13069/JACODESMATH.867617","DOIUrl":"https://doi.org/10.13069/JACODESMATH.867617","url":null,"abstract":"For any integer $kgeq 3$ , we define sunlet graph of order $2k$, denoted by $L_{2k}$, as the graph consisting of a cycle of length $k$ together with $k$ pendant vertices, each adjacent to exactly one vertex of the cycle. In this paper, we give necessary and sufficient conditions for the existence of $L_{8}$-decomposition of tensor product and wreath product of complete graphs.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43091885","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-15DOI: 10.13069/JACODESMATH.858732
Ben Paul Bautista Dela Cruz, J. M. Lampos, H. S. Palines, V. Sison
In this paper, we consider the well-known unital embedding from $FF_{q^k}$ into $M_k(FF_q)$ seen as a map of vector spaces over $FF_q$ and apply this map in a linear block code of rate $rho/ell$ over $FF_{q^k}$. This natural extension gives rise to a rank-metric code with $k$ rows, $kell$ columns, dimension $rho$ and minimum distance $k$ that satisfies the Singleton bound. Given a specific skeleton code, this rank-metric code can be seen as a Ferrers diagram rank-metric code by appending zeros on the left side so that it has length $n-k$. The generalized lift of this Ferrers diagram rank-metric code is a Grassmannian code. By taking the union of a family of the generalized lift of Ferrers diagram rank-metric codes, a Grassmannian code with length $n$, cardinality $frac{q^n-1}{q^k-1}$, minimum injection distance $k$ and dimension $k$ that satisfies the anticode upper bound can be constructed.
{"title":"A new construction of anticode-optimal Grassmannian codes","authors":"Ben Paul Bautista Dela Cruz, J. M. Lampos, H. S. Palines, V. Sison","doi":"10.13069/JACODESMATH.858732","DOIUrl":"https://doi.org/10.13069/JACODESMATH.858732","url":null,"abstract":"In this paper, we consider the well-known unital embedding from $FF_{q^k}$ into $M_k(FF_q)$ seen as a map of vector spaces over $FF_q$ and apply this map in a linear block code of rate $rho/ell$ over $FF_{q^k}$. This natural extension gives rise to a rank-metric code with $k$ rows, $kell$ columns, dimension $rho$ and minimum distance $k$ that satisfies the Singleton bound. Given a specific skeleton code, this rank-metric code can be seen as a Ferrers diagram rank-metric code by appending zeros on the left side so that it has length $n-k$. The generalized lift of this Ferrers diagram rank-metric code is a Grassmannian code. By taking the union of a family of the generalized lift of Ferrers diagram rank-metric codes, a Grassmannian code with length $n$, cardinality $frac{q^n-1}{q^k-1}$, minimum injection distance $k$ and dimension $k$ that satisfies the anticode upper bound can be constructed.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48196611","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-15DOI: 10.13069/JACODESMATH.864902
N. Aydin, Y. Cengellenmis, A. Dertli
We introduce skew cyclic codes over the finite ring $R$, where $u^{2}=0,v^{2}=v,w^{2}=w,uv=vu,uw=wu,vw=wv$ and use them to construct reversible DNA codes. The 4-mers are matched with the elements of this ring. The reversibility problem for DNA 4-bases is solved and some examples are provided.
{"title":"Reversible DNA codes from skew cyclic codes over a ring of order 256","authors":"N. Aydin, Y. Cengellenmis, A. Dertli","doi":"10.13069/JACODESMATH.864902","DOIUrl":"https://doi.org/10.13069/JACODESMATH.864902","url":null,"abstract":"We introduce skew cyclic codes over the finite ring $R$, where $u^{2}=0,v^{2}=v,w^{2}=w,uv=vu,uw=wu,vw=wv$ and use them to construct reversible DNA codes. The 4-mers are matched with the elements of this ring. The reversibility problem for DNA 4-bases is solved and some examples are provided.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43282632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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