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Array-designed reversible and complementary codes over GF(4) 阵列设计的GF(4)上可逆和互补码
IF 0.6 4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-09-19 DOI: 10.1007/s00200-023-00622-6
Manabu Hagiwara, Whan-Hyuk Choi, Jon-Lark Kim
{"title":"Array-designed reversible and complementary codes over GF(4)","authors":"Manabu Hagiwara,&nbsp;Whan-Hyuk Choi,&nbsp;Jon-Lark Kim","doi":"10.1007/s00200-023-00622-6","DOIUrl":"10.1007/s00200-023-00622-6","url":null,"abstract":"<div><p>Array-designed codes are capable of correcting row and column deletions. In this paper, we introduce array-designed reversible and complementary codes over the finite field <i>GF</i>(4), which can correct row deletions and column deletions errors. Hence our codes can be useful for DNA codes. We also propose a construction method of array-designed reversible and complementary codes. As examples, we describe the tensor product of cyclic LCD codes and the tensor product of self-dual reversible codes over <i>GF</i>(4).</p></div>","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":"36 3","pages":"503 - 519"},"PeriodicalIF":0.6,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135061088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Exploring implications of Trace (Inversion) formula and Artin algebras in extremal combinatorics 迹（逆）公式和Artin代数在极值组合数学中的意义探讨
IF 0.6 4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-08-13 DOI: 10.1007/s00200-023-00619-1
Luis M. Pardo
{"title":"Exploring implications of Trace (Inversion) formula and Artin algebras in extremal combinatorics","authors":"Luis M. Pardo","doi":"10.1007/s00200-023-00619-1","DOIUrl":"10.1007/s00200-023-00619-1","url":null,"abstract":"<div><p>This note is just a modest contribution to prove several classical results in Combinatorics from notions of Duality in some Artinian <i>K</i>-algebras (mainly through the Trace Formula), where <i>K</i> is a perfect field of characteristics not equal to 2. We prove how several classic combinatorial results are particular instances of a Trace (Inversion) Formula in finite <span>(mathbb {Q})</span>-algebras. This is the case with the Exclusion-Inclusion Principle (in its general form, both with direct and reverse order associated to subsets inclusion). This approach also allows us to exhibit a basis of the space of null <i>t</i>-designs, which differs from the one described in Theorem 4 of Deza and Frankl (Combinatorica 2:341–345, 1982). Provoked by the elegant proof (which uses no induction) in Frankl and Pach (Eur J Comb 4:21–23, 1983) of the Sauer–Shelah–Perles Lemma, we produce a new one based only in duality in the <span>(mathbb {Q})</span>-algebra <span>(mathbb {Q}[V_n])</span> of polynomials functions defined on the zero-dimensional algebraic variety of subsets of the set <span>([n]:={1,2,ldots , n})</span>. All results are equally true if we replace <span>(mathbb {Q}[V_n])</span> by <span>(K[V_n])</span>, where <i>K</i> is any perfect field of characteristics <span>(not =2)</span>. The article connects results from two fields of mathematical knowledge that are not usually connected, at least not in this form. Thus, we decided to write the manuscript in a self-contained survey-like style, although it is not a survey paper at all. Readers familiar with Commutative Algebra probably know most of the proofs of the statements described in section 2. We decided to include these proofs for those potential readers not so familiar with this framework.</p></div>","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":"35 2022","pages":"71 - 118"},"PeriodicalIF":0.6,"publicationDate":"2023-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00200-023-00619-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41398532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Three new classes of optimal quinary cyclic codes with minimum distance four 三类新的最小距离为4的最优五元循环码
IF 0.6 4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-08-07 DOI: 10.1007/s00200-023-00621-7
Yan Liu, Xiwang Cao
{"title":"Three new classes of optimal quinary cyclic codes with minimum distance four","authors":"Yan Liu,&nbsp;Xiwang Cao","doi":"10.1007/s00200-023-00621-7","DOIUrl":"10.1007/s00200-023-00621-7","url":null,"abstract":"<div><p>Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an important subject of study for many years. Recently, several classes of optimal quinary cyclic codes of the forms <span>(mathcal {C}_{(0,1,e)})</span> and <span>(mathcal {C}_{(1,e,s)})</span> are presented in the literature, where <span>(s=frac{5^m-1}{2})</span> and <span>(2 le e le 5^{m}-2)</span>. In this paper, by considering the solutions of certain equations over finite fields, we give three new classes of infinite families of optimal quinary cyclic codes of the form <span>(mathcal {C}_{(1,e,s)})</span> with parameters <span>([5^{m}-1, 5^{m}-2m-2, 4])</span>. Specifically, we make progress towards an open problem proposed by Gaofei Wu et al. [17].</p></div>","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":"36 3","pages":"493 - 501"},"PeriodicalIF":0.6,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47119674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Generic Gröbner basis of a parametric ideal and its application to a comprehensive Gröbner system 参数理想的一般Gröbner基础及其在综合Gröbner系统中的应用
IF 0.6 4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-08-03 DOI: 10.1007/s00200-023-00620-8
Katsusuke Nabeshima
{"title":"Generic Gröbner basis of a parametric ideal and its application to a comprehensive Gröbner system","authors":"Katsusuke Nabeshima","doi":"10.1007/s00200-023-00620-8","DOIUrl":"10.1007/s00200-023-00620-8","url":null,"abstract":"<div><p>A relationship between a parametric ideal and its generic Gröbner basis is discussed, and a new stability condition of a Gröbner basis is given. It is shown that the ideal quotient of the parametric ideal by the ideal generated using the generic Gröbner basis provides the stability condition. Further, the stability condition is utilized to compute a comprehensive Gröbner system, and hence as an application, we obtain a new algorithm for computing comprehensive Gröbner systems. The proposed algorithm has been implemented in the computer algebra system Risa/Asir and experimented with a number of examples. Its performance (execution timings) has been compared with the Kapur-Sun-Wang’s algorithm for computing comprehensive Gröbner systems.</p></div>","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":"35 2022","pages":"55 - 70"},"PeriodicalIF":0.6,"publicationDate":"2023-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00200-023-00620-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47966809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Linear label code of a root lattice using Gröbner bases 线性标签代码的根晶格使用Gröbner的基础
IF 0.6 4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-07-28 DOI: 10.1007/s00200-023-00614-6
Malihe Aliasgari, Daniel Panario, Mohammad-Reza Sadeghi
{"title":"Linear label code of a root lattice using Gröbner bases","authors":"Malihe Aliasgari,&nbsp;Daniel Panario,&nbsp;Mohammad-Reza Sadeghi","doi":"10.1007/s00200-023-00614-6","DOIUrl":"10.1007/s00200-023-00614-6","url":null,"abstract":"<div><p>The label code of a lattice plays a key role in the characterization of the lattice. Every lattice <span>(Lambda)</span> can be described in terms of a label code <i>L</i> and an orthogonal sublattice <span>(Lambda ')</span> such that <span>(Lambda /Lambda 'cong L)</span>. We identify the binomial ideal associated to an integer lattice and then establish a relation between the ideal quotient of the lattice and its label code. Furthermore, we present the Gröbner basis of the well-known root lattice <span>(D_n)</span>. As an application of the relation <span>(I_{Lambda }=I_{Lambda '}+I_{L})</span>, where <span>(I_{Lambda },I_{Lambda '})</span> and <span>(I_L)</span> denote binomial ideals associated to <span>(Lambda ,~Lambda ')</span> and <i>L</i>, respectively, a linear label code of <span>(D_n)</span> is obtained using its Gröbner basis.</p></div>","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":"35 2022","pages":"3 - 15"},"PeriodicalIF":0.6,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48057651","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Further perspectives on elimination 关于消除的进一步观点
IF 0.7 4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-07-20 DOI: 10.1007/s00200-023-00618-2
Teo Mora
{"title":"Further perspectives on elimination","authors":"Teo Mora","doi":"10.1007/s00200-023-00618-2","DOIUrl":"10.1007/s00200-023-00618-2","url":null,"abstract":"","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":"34 5","pages":"745 - 749"},"PeriodicalIF":0.7,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44365194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Dihedral codes with 1-dimensional hulls and 1-dimensional linear complementary pairs of dihedral codes 具有一维外壳的二面体码和一维线性互补对二面体代码
IF 0.6 4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-07-19 DOI: 10.1007/s00200-023-00617-3
S. T. Dougherty, Serap Şahinkaya, Deniz Ustun
{"title":"Dihedral codes with 1-dimensional hulls and 1-dimensional linear complementary pairs of dihedral codes","authors":"S. T. Dougherty,&nbsp;Serap Şahinkaya,&nbsp;Deniz Ustun","doi":"10.1007/s00200-023-00617-3","DOIUrl":"10.1007/s00200-023-00617-3","url":null,"abstract":"<div><p>In this paper, we study dihedral codes with 1-dimensional hulls and we determine precisely when dihedral codes over finite fields with 1-dimensional hulls exist. Moreover, we show that these codes come canonically in pairs. We also introduce 1-dimensional linear complementary pairs of dihedral codes and examine the properties of this class of codes. As an application, we obtain 1-dimensional linear complementary pair of dihedral codes, which are either optimal or near optimal.</p></div>","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":"36 3","pages":"471 - 491"},"PeriodicalIF":0.6,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45259924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Constructing doubly even self-dual codes and even unimodular lattices from Hadamard matrices 从Hadamard矩阵构造双偶自对偶码和偶单模格
IF 0.6 4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-07-14 DOI: 10.1007/s00200-023-00615-5
Dean Crnković, Andrea Švob
{"title":"Constructing doubly even self-dual codes and even unimodular lattices from Hadamard matrices","authors":"Dean Crnković,&nbsp;Andrea Švob","doi":"10.1007/s00200-023-00615-5","DOIUrl":"10.1007/s00200-023-00615-5","url":null,"abstract":"<div><p>We show that from every skew-type Hadamard matrix of order 4<i>t</i> one can obtain a series of skew-type Hadamard matrices of order <span>(2^{i+2}t)</span>, <i>i</i> a positive integer, whose binary linear codes are doubly even self-dual binary codes of length <span>(2^{i+2}t)</span>. It is known that a doubly even self-dual binary code yields an even unimodular lattice. Hence, this construction of skew-type Hadamard matrices gives us a series of even unimodular lattices of rank <span>(2^{i+2}t)</span>, <i>i</i> a positive integer. Furthermore, we provide a construction of doubly even self-dual binary codes from conference graphs.</p></div>","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":"36 3","pages":"457 - 470"},"PeriodicalIF":0.6,"publicationDate":"2023-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47782694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A note on a standard model for Galois rings 伽罗瓦环标准模型的注释
IF 0.6 4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-06-22 DOI: 10.1007/s00200-023-00612-8
E. Martínez-Moro, A. Piñera-Nicolás, I. F. Rúa
{"title":"A note on a standard model for Galois rings","authors":"E. Martínez-Moro,&nbsp;A. Piñera-Nicolás,&nbsp;I. F. Rúa","doi":"10.1007/s00200-023-00612-8","DOIUrl":"10.1007/s00200-023-00612-8","url":null,"abstract":"<div><p>In this work we present a standard model for Galois rings based on the standard model of their residual fields, that is, a a sequence of Galois rings starting with <span>({mathbb Z}_{p^r})</span> that coves all the Galois rings with that characteristic ring and such that there is an algorithm producing each member of the sequence whose input is the size of the required ring.</p></div>","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":"36 3","pages":"443 - 456"},"PeriodicalIF":0.6,"publicationDate":"2023-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46075479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Correction: The paint pot problem and common multiples in monoids 更正：油漆罐问题和常见的倍数在monoids
IF 0.6 4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-06-14 DOI: 10.1007/s00200-023-00613-7
Hans Zantema, Vincent van Oostrom
{"title":"Correction: The paint pot problem and common multiples in monoids","authors":"Hans Zantema,&nbsp;Vincent van Oostrom","doi":"10.1007/s00200-023-00613-7","DOIUrl":"10.1007/s00200-023-00613-7","url":null,"abstract":"","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":"36 2","pages":"365 - 366"},"PeriodicalIF":0.6,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00200-023-00613-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135916460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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