Pub Date : 2023-07-28DOI: 10.1007/s00200-023-00614-6
Malihe Aliasgari, Daniel Panario, Mohammad-Reza Sadeghi
The label code of a lattice plays a key role in the characterization of the lattice. Every lattice (Lambda) can be described in terms of a label code L and an orthogonal sublattice (Lambda ') such that (Lambda /Lambda 'cong L). 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 (D_n). As an application of the relation (I_{Lambda }=I_{Lambda '}+I_{L}), where (I_{Lambda },I_{Lambda '}) and (I_L) denote binomial ideals associated to (Lambda ,~Lambda ') and L, respectively, a linear label code of (D_n) is obtained using its Gröbner basis.
{"title":"Linear label code of a root lattice using Gröbner bases","authors":"Malihe Aliasgari, Daniel Panario, 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}
Pub Date : 2023-07-20DOI: 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}
Pub Date : 2023-07-19DOI: 10.1007/s00200-023-00617-3
S. Dougherty, S. Șahinkaya, Deniz Ustun
{"title":"Dihedral codes with 1-dimensional hulls and 1-dimensional linear complementary pairs of dihedral codes","authors":"S. Dougherty, S. Șahinkaya, Deniz Ustun","doi":"10.1007/s00200-023-00617-3","DOIUrl":"https://doi.org/10.1007/s00200-023-00617-3","url":null,"abstract":"","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":" ","pages":""},"PeriodicalIF":0.7,"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}
Pub Date : 2023-07-14DOI: 10.1007/s00200-023-00615-5
D. Crnković, Andrea Švob
{"title":"Constructing doubly even self-dual codes and even unimodular lattices from Hadamard matrices","authors":"D. Crnković, Andrea Švob","doi":"10.1007/s00200-023-00615-5","DOIUrl":"https://doi.org/10.1007/s00200-023-00615-5","url":null,"abstract":"","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":" ","pages":""},"PeriodicalIF":0.7,"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}
Pub Date : 2023-06-14DOI: 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, Vincent van Oostrom","doi":"10.1007/s00200-023-00613-7","DOIUrl":"https://doi.org/10.1007/s00200-023-00613-7","url":null,"abstract":"","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-08DOI: 10.1007/s00200-023-00610-w
M. A. Moreno-Frías, J. Rosales
{"title":"Frobenius pseudo-variety of numerical semigroups with a given multiplicity and ratio","authors":"M. A. Moreno-Frías, J. Rosales","doi":"10.1007/s00200-023-00610-w","DOIUrl":"https://doi.org/10.1007/s00200-023-00610-w","url":null,"abstract":"","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48868476","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}
Pub Date : 2023-05-31DOI: 10.1007/s00200-023-00611-9
Kezia Saini, M. Garg
{"title":"On the higher-order nonlinearity of a new class of biquadratic Maiorana–McFarland type bent functions","authors":"Kezia Saini, M. Garg","doi":"10.1007/s00200-023-00611-9","DOIUrl":"https://doi.org/10.1007/s00200-023-00611-9","url":null,"abstract":"","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47431444","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}
Pub Date : 2023-05-19DOI: 10.1007/s00200-023-00608-4
Tim Alderson, Benjamin Morine
A combinatorial problem concerning the maximum size of the (Hamming) weight set of an $$[n,k]_q$$ linear code was recently introduced. Codes attaining the established upper bound are the Maximum Weight Spectrum (MWS) codes. Those $$[n,k]_q$$ codes with the same weight set as $$mathbb {F}_q^n$$ are called Full Weight Spectrum (FWS) codes. FWS codes are necessarily “short”, whereas MWS codes are necessarily “long”. For fixed k, q the values of n for which an $$[n,k]_q$$ -FWS code exists are completely determined, but the determination of the minimum length M(H, k, q) of an $$[n,k]_q$$ -MWS code remains an open problem. The current work broadens discussion first to general coordinate-wise weight functions, and then specifically to the Lee weight and a Manhattan like weight. In the general case we provide bounds on n for which an FWS code exists, and bounds on n for which an MWS code exists. When specializing to the Lee or to the Manhattan setting we are able to completely determine the parameters of FWS codes. As with the Hamming case, we are able to provide an upper bound on $$M({mathscr {L}},k,q)$$ (the minimum length of Lee MWS codes), and pose the determination of $$M({mathscr {L}},k,q)$$ as an open problem. On the other hand, with respect to the Manhattan weight we completely determine the parameters of MWS codes.
{"title":"MWS and FWS codes for coordinate-wise weight functions","authors":"Tim Alderson, Benjamin Morine","doi":"10.1007/s00200-023-00608-4","DOIUrl":"https://doi.org/10.1007/s00200-023-00608-4","url":null,"abstract":"A combinatorial problem concerning the maximum size of the (Hamming) weight set of an $$[n,k]_q$$ linear code was recently introduced. Codes attaining the established upper bound are the Maximum Weight Spectrum (MWS) codes. Those $$[n,k]_q$$ codes with the same weight set as $$mathbb {F}_q^n$$ are called Full Weight Spectrum (FWS) codes. FWS codes are necessarily “short”, whereas MWS codes are necessarily “long”. For fixed k, q the values of n for which an $$[n,k]_q$$ -FWS code exists are completely determined, but the determination of the minimum length M(H, k, q) of an $$[n,k]_q$$ -MWS code remains an open problem. The current work broadens discussion first to general coordinate-wise weight functions, and then specifically to the Lee weight and a Manhattan like weight. In the general case we provide bounds on n for which an FWS code exists, and bounds on n for which an MWS code exists. When specializing to the Lee or to the Manhattan setting we are able to completely determine the parameters of FWS codes. As with the Hamming case, we are able to provide an upper bound on $$M({mathscr {L}},k,q)$$ (the minimum length of Lee MWS codes), and pose the determination of $$M({mathscr {L}},k,q)$$ as an open problem. On the other hand, with respect to the Manhattan weight we completely determine the parameters of MWS codes.","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135576353","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}
Pub Date : 2023-05-17DOI: 10.1007/s00200-023-00609-3
Boran Kim
{"title":"Correction: Determination for minimum symbol-pair and RT weights via torsional degrees of repeated-root cyclic codes","authors":"Boran Kim","doi":"10.1007/s00200-023-00609-3","DOIUrl":"10.1007/s00200-023-00609-3","url":null,"abstract":"","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":"35 4","pages":"589 - 589"},"PeriodicalIF":0.6,"publicationDate":"2023-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135861225","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}
Pub Date : 2023-05-13DOI: 10.1007/s00200-023-00606-6
H. Zantema, Vincent van Oostrom
{"title":"The paint pot problem and common multiples in monoids","authors":"H. Zantema, Vincent van Oostrom","doi":"10.1007/s00200-023-00606-6","DOIUrl":"https://doi.org/10.1007/s00200-023-00606-6","url":null,"abstract":"","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48208169","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}
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