Pub Date : 2020-10-02DOI: 10.13069/jacodesmath.1056492
Martin Jur'avs, M. Ursul
We show that the set of all commuting probabilities in finite rings is a subset of the set of all commuting probabilities in finite nilpotent groups of class $le2$. We believe that these two sets are equal; we prove they are equal, when restricted to groups and rings with odd number of elements.
{"title":"On commuting probabilities in finite groups and rings","authors":"Martin Jur'avs, M. Ursul","doi":"10.13069/jacodesmath.1056492","DOIUrl":"https://doi.org/10.13069/jacodesmath.1056492","url":null,"abstract":"We show that the set of all commuting probabilities in finite rings is a subset of the set of all commuting probabilities in finite nilpotent groups of class $le2$. We believe that these two sets are equal; we prove they are equal, when restricted to groups and rings with odd number of elements.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46943101","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 : 2020-09-04DOI: 10.13069/jacodesmath.790751
T. Amouzegar, R. Tribak
Let $R$ be a ring. In this article, we introduce and study relative dual Baer property. We characterize $R$-modules $M$ which are $R_R$-dual Baer, where $R$ is a commutative principal ideal domain. It is shown that over a right noetherian right hereditary ring $R$, an $R$-module $M$ is $N$-dual Baer for all $R$-modules $N$ if and only if $M$ is an injective $R$-module. It is also shown that for $R$-modules $M_1$, $M_2$, $ldots$, $M_n$ such that $M_i$ is $M_j$-projective for all $i > j in {1,2,ldots, n}$, an $R$-module $N$ is $bigoplus_{i=1}^nM_i$-dual Baer if and only if $N$ is $M_i$-dual Baer for all $iin {1,2,ldots,n}$. We prove that an $R$-module $M$ is dual Baer if and only if $S=End_R(M)$ is a Baer ring and $IM=r_M(l_S(IM))$ for every right ideal $I$ of $S$.
{"title":"Some results on relative dual Baer property","authors":"T. Amouzegar, R. Tribak","doi":"10.13069/jacodesmath.790751","DOIUrl":"https://doi.org/10.13069/jacodesmath.790751","url":null,"abstract":"Let $R$ be a ring. In this article, we introduce and study relative dual Baer property. We characterize $R$-modules $M$ which are $R_R$-dual Baer, where $R$ is a commutative principal ideal domain. It is shown that over a right noetherian right hereditary ring $R$, an $R$-module $M$ is $N$-dual Baer for all $R$-modules $N$ if and only if $M$ is an injective $R$-module. It is also shown that for $R$-modules $M_1$, $M_2$, $ldots$, $M_n$ such that $M_i$ is $M_j$-projective for all $i > j in {1,2,ldots, n}$, an $R$-module $N$ is $bigoplus_{i=1}^nM_i$-dual Baer if and only if $N$ is $M_i$-dual Baer for all $iin {1,2,ldots,n}$. We prove that an $R$-module $M$ is dual Baer if and only if $S=End_R(M)$ is a Baer ring and $IM=r_M(l_S(IM))$ for every right ideal $I$ of $S$.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43669327","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 : 2020-05-01DOI: 10.13069/jacodesmath.729422
S. Sedghi, N. Shobe
The degree distance was introduced by Dobrynin, Kochetova and Gutman as a weighted version of the Wiener index. In this paper, we investigate the degree distance and Gutman index of complete, and strong product graphs by using the adjacency and distance matrices of a graph.
{"title":"Degree distance and Gutman index of two graph products","authors":"S. Sedghi, N. Shobe","doi":"10.13069/jacodesmath.729422","DOIUrl":"https://doi.org/10.13069/jacodesmath.729422","url":null,"abstract":"The degree distance was introduced by Dobrynin, Kochetova and Gutman as a weighted version of the Wiener index. In this paper, we investigate the degree distance and Gutman index of complete, and strong product graphs by using the adjacency and distance matrices of a graph.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47149618","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 : 2020-05-01DOI: 10.13069/jacodesmath.729440
A. J. Ferrari, A. A. Andrade, R. R. Araujo, J. Interlando
In this work, we present a explicit trace forms for maximal real subfields of cyclotomic fields as tools for constructing algebraic lattices in Euclidean space with optimal center density. We also obtain a closed formula for the Gram matrix of algebraic lattices obtained from these subfields. The obtained lattices are rotated versions of the lattices Λ9,Λ10Lambda_9, Lambda_{10}Λ9,Λ10 and Λ11Lambda_{11}Λ11 and they are images of Zmathbb{Z}Z-submodules of rings of integers under the twisted homomorphism, and these constructions, as algebraic lattices, are new in the literature. We also obtain algebraic lattices in odd dimensions up to 777 over real subfields, calculate their minimum product distance and compare with those known in literatura, since lattices constructed over real subfields have full diversity.
{"title":"Trace forms of certain subfields of cyclotomic fields and applications","authors":"A. J. Ferrari, A. A. Andrade, R. R. Araujo, J. Interlando","doi":"10.13069/jacodesmath.729440","DOIUrl":"https://doi.org/10.13069/jacodesmath.729440","url":null,"abstract":"In this work, we present a explicit trace forms for maximal real subfields of cyclotomic fields as tools for constructing algebraic lattices in Euclidean space with optimal center density. We also obtain a closed formula for the Gram matrix of algebraic lattices obtained from these subfields. The obtained lattices are rotated versions of the lattices Λ9,Λ10Lambda_9, Lambda_{10}Λ9,Λ10 and Λ11Lambda_{11}Λ11 and they are images of Zmathbb{Z}Z-submodules of rings of integers under the twisted homomorphism, and these constructions, as algebraic lattices, are new in the literature. We also obtain algebraic lattices in odd dimensions up to 777 over real subfields, calculate their minimum product distance and compare with those known in literatura, since lattices constructed over real subfields have full diversity.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42166522","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 : 2019-09-13DOI: 10.13069/jacodesmath.617239
Ibrahim Ozbek, F. Temiz, I. Siap
Secret sharing scheme is an efficient method to hide secret key or secret image by partitioning it into parts such that some predetermined subsets of partitions can recover the secret but remaining subsets cannot. In 1979, the pioneer construction on this area was given by Shamir and Blakley independently. After these initial studies, Asmuth-Bloom and Mignotte have proposed a different $(k,n)$ threshold modular secret sharing scheme by using the Chinese remainder theorem. In this study, we explore the generalization of Mignotte's scheme to Euclidean domains for which we obtain some promising results. Next, we propose new algorithms to construct threshold secret image sharing schemes by using Mignotte's scheme over polynomial rings. Finally, we compare our proposed scheme to the existing ones and we show that this new method is more efficient and it has higher security.
{"title":"A generalization of the Mignotte’s scheme over Euclidean domains and applications to secret image sharing","authors":"Ibrahim Ozbek, F. Temiz, I. Siap","doi":"10.13069/jacodesmath.617239","DOIUrl":"https://doi.org/10.13069/jacodesmath.617239","url":null,"abstract":"Secret sharing scheme is an efficient method to hide secret key or secret image by partitioning it into parts such that some predetermined subsets of partitions can recover the secret but remaining subsets cannot. In 1979, the pioneer construction on this area was given by Shamir and Blakley independently. After these initial studies, Asmuth-Bloom and Mignotte have proposed a different $(k,n)$ threshold modular secret sharing scheme by using the Chinese remainder theorem. In this study, we explore the generalization of Mignotte's scheme to Euclidean domains for which we obtain some promising results. Next, we propose new algorithms to construct threshold secret image sharing schemes by using Mignotte's scheme over polynomial rings. Finally, we compare our proposed scheme to the existing ones and we show that this new method is more efficient and it has higher security.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44925293","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 : 2019-09-13DOI: 10.13069/jacodesmath.617232
D. Bikov, I. Bouyukliev, S. Bouyuklieva
The aim of this paper is to construct S-boxes of different sizes with good cryptographic properties. An algebraic construction for bijective S-boxes is described. It uses quasi-cyclic representations of the binary simplex code. Good S-boxes of sizes 4, 6, 8, 9, 10, 11, 12, 14, 15, 16 and 18 are obtained.
{"title":"Bijective S-boxes of different sizes obtained from quasi-cyclic codes","authors":"D. Bikov, I. Bouyukliev, S. Bouyuklieva","doi":"10.13069/jacodesmath.617232","DOIUrl":"https://doi.org/10.13069/jacodesmath.617232","url":null,"abstract":"The aim of this paper is to construct S-boxes of different sizes with good cryptographic properties. An algebraic construction for bijective S-boxes is described. It uses quasi-cyclic representations of the binary simplex code. Good S-boxes of sizes 4, 6, 8, 9, 10, 11, 12, 14, 15, 16 and 18 are obtained.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41735656","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 : 2019-09-13DOI: 10.13069/jacodesmath.617244
Tushar Bag, H. Islam, O. Prakash, A. Upadhyay
For odd prime $p$, this paper studies $(1+(p-2)u)$-constacyclic codes over the ring $R= mathbb{Z}_{p} [u,v]/langle u^2-u,v^2-v,uv-vurangle$. We show that the Gray images of $(1+(p-2)u)$-constacyclic codes over $R$ are cyclic and permutation equivalent to a quasi cyclic code over $mathbb{Z}_{p}$. We derive the generators for $(1+(p-2)u)$-constacyclic and principally generated $(1+(p-2)u)$-constacyclic codes over $R$. Among others, we extend our results for skew $(1+(p-2)u)$-constacyclic codes over $R$ and exhibit the relation between skew $(1+(p-2)u)$-constacyclic codes with the other linear codes. Finally, as an application of our study, we compute several non trivial linear codes by using the Gray images of $(1+(p-2)u)$-constacyclic codes over this ring $R$.
{"title":"A note on constacyclic and skew constacyclic codes over the ring $mathbb{Z}_{p} [u,v]/langle u^2-u,v^2-v,uv-vurangle$","authors":"Tushar Bag, H. Islam, O. Prakash, A. Upadhyay","doi":"10.13069/jacodesmath.617244","DOIUrl":"https://doi.org/10.13069/jacodesmath.617244","url":null,"abstract":"For odd prime $p$, this paper studies $(1+(p-2)u)$-constacyclic codes over the ring $R= mathbb{Z}_{p} [u,v]/langle u^2-u,v^2-v,uv-vurangle$. We show that the Gray images of $(1+(p-2)u)$-constacyclic codes over $R$ are cyclic and permutation equivalent to a quasi cyclic code over $mathbb{Z}_{p}$. We derive the generators for $(1+(p-2)u)$-constacyclic and principally generated $(1+(p-2)u)$-constacyclic codes over $R$. Among others, we extend our results for skew $(1+(p-2)u)$-constacyclic codes over $R$ and exhibit the relation between skew $(1+(p-2)u)$-constacyclic codes with the other linear codes. Finally, as an application of our study, we compute several non trivial linear codes by using the Gray images of $(1+(p-2)u)$-constacyclic codes over this ring $R$.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44264437","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 : 2019-09-13DOI: 10.13069/jacodesmath.617235
J. McCullough, Heather Newman
Let $Delta$ be an abstract simplicial complex. We study classical homological error correcting codes associated to $Delta$, which generalize the cycle codes of simple graphs. It is well-known that cycle codes of graphs do not yield asymptotically good families of codes. We show that asymptotically good families of codes do exist for homological codes associated to simplicial complexes of dimension at least $2$. We also prove general bounds and formulas for (co-)cycle and (co-)boundary codes for arbitrary simplicial complexes over arbitrary fields.
{"title":"Asymptotically good homological error correcting codes","authors":"J. McCullough, Heather Newman","doi":"10.13069/jacodesmath.617235","DOIUrl":"https://doi.org/10.13069/jacodesmath.617235","url":null,"abstract":"Let $Delta$ be an abstract simplicial complex. We study classical homological error correcting codes associated to $Delta$, which generalize the cycle codes of simple graphs. It is well-known that cycle codes of graphs do not yield asymptotically good families of codes. We show that asymptotically good families of codes do exist for homological codes associated to simplicial complexes of dimension at least $2$. We also prove general bounds and formulas for (co-)cycle and (co-)boundary codes for arbitrary simplicial complexes over arbitrary fields.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42373426","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 : 2019-05-08DOI: 10.13069/JACODESMATH.560404
Omar Tout
We use the combinatorial way to give an explicit expression for the product of the class of cycles of length three with an arbitrary class of cycles. In addition, an explicit formula for the coefficient of an arbitrary class in the expansion of the product of an arbitrary class by the class of cycles of length three is given.
{"title":"Some explicit expressions for the structure coefficients of the center of the symmetric group algebra involving cycles of length three","authors":"Omar Tout","doi":"10.13069/JACODESMATH.560404","DOIUrl":"https://doi.org/10.13069/JACODESMATH.560404","url":null,"abstract":"We use the combinatorial way to give an explicit expression for the product of the class of cycles of length three with an arbitrary class of cycles. In addition, an explicit formula for the coefficient of an arbitrary class in the expansion of the product of an arbitrary class by the class of cycles of length three is given.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47319163","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 : 2019-05-08DOI: 10.13069/JACODESMATH.561316
Rachelle R. Bouchat, Tricia Muldoon Brown
This paper considers a class of monomial ideals, called domino ideals, whose generating sets correspond to the sets of domino tilings of a $2times n$ tableau. The multi-graded Betti numbers are shown to be in one-to-one correspondence with equivalence classes of sets of tilings. It is well-known that the number of domino tilings of a $2times n$ tableau is given by a Fibonacci number. Using the bijection, this relationship is further expanded to show the relationship between the Fibonacci numbers and the graded Betti numbers of the corresponding domino ideal.
{"title":"Fibonacci numbers and resolutions of domino ideals","authors":"Rachelle R. Bouchat, Tricia Muldoon Brown","doi":"10.13069/JACODESMATH.561316","DOIUrl":"https://doi.org/10.13069/JACODESMATH.561316","url":null,"abstract":"This paper considers a class of monomial ideals, called domino ideals, whose generating sets correspond to the sets of domino tilings of a $2times n$ tableau. The multi-graded Betti numbers are shown to be in one-to-one correspondence with equivalence classes of sets of tilings. It is well-known that the number of domino tilings of a $2times n$ tableau is given by a Fibonacci number. Using the bijection, this relationship is further expanded to show the relationship between the Fibonacci numbers and the graded Betti numbers of the corresponding domino ideal.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44540466","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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