Finite Fields and Their Applications最新文献

{"title":"On dual-containing, almost dual-containing matrix-product codes and related quantum codes","authors":"Meng Cao","doi":"10.1016/j.ffa.2024.102400","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102400","url":null,"abstract":"<div><p>Matrix-product (MP) codes are a type of long codes formed by combining several commensurate constituent codes with a defining matrix. In this paper, we study the MP code when the defining matrix <em>A</em> satisfies the condition that <span><math><mi>A</mi><msup><mrow><mi>A</mi></mrow><mrow><mo>⊤</mo></mrow></msup></math></span> is <span><math><mo>(</mo><mi>D</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math></span>-monomial. We give an explicit formula for calculating the dimension of the hull of a MP code. We present the necessary and sufficient conditions for a MP code to be dual-containing (DC), almost dual-containing (ADC), self-orthogonal (SO) and almost self-orthogonal (ASO), respectively. We theoretically determine the number of all possible ways involving the relationships among the constituent codes to yield a MP code that is DC, ADC, SO and ASO, respectively. We give alternative necessary and sufficient conditions for a MP code to be ADC and ASO, respectively, and show several cases where a MP code is not ADC or ASO. We give the construction methods of DC and ADC MP codes, including those with optimal minimum distance lower bounds. We introduce the notation of <em>τ</em>-optimal defining (<em>τ</em>-OD) matrices and provide the criteria for determining whether two types of <span><math><mi>k</mi><mo>×</mo><mi>k</mi></math></span> matrices are <em>τ</em>-OD matrices at <span><math><mi>k</mi><mo>=</mo><mn>3</mn></math></span> and <span><math><mi>k</mi><mo>=</mo><mn>4</mn></math></span>, respectively. We give many examples of DC and ADC MP codes involving <em>τ</em>-OD matrices, some of which are optimal or almost optimal according to the Database <span>[11]</span>. By applying the generalized Steane's enlargement procedure to these DC MP codes, we obtain some good quantum codes that improve those available in the Database <span>[7]</span>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140113044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The extended codes of some linear codes","authors":"Zhonghua Sun ,&nbsp;Cunsheng Ding ,&nbsp;Tingfang Chen","doi":"10.1016/j.ffa.2024.102401","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102401","url":null,"abstract":"<div><p>The classical way of extending an <span><math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>d</mi><mo>]</mo></math></span> linear code <span><math><mi>C</mi></math></span> is to add an overall parity-check coordinate to each codeword of the linear code <span><math><mi>C</mi></math></span>. This extended code, denoted by <span><math><mover><mrow><mi>C</mi></mrow><mo>‾</mo></mover><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></math></span> and called the standardly extended code of <span><math><mi>C</mi></math></span>, is a linear code with parameters <span><math><mo>[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>,</mo><mover><mrow><mi>d</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>]</mo></math></span>, where <span><math><mover><mrow><mi>d</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>=</mo><mi>d</mi></math></span> or <span><math><mover><mrow><mi>d</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>=</mo><mi>d</mi><mo>+</mo><mn>1</mn></math></span>. This is one of the two extending techniques for linear codes in the literature. The standardly extended codes of some families of binary linear codes have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for this type of extended linear codes. The second objective is to study this type of extended codes of a number of families of linear codes, including cyclic codes and nonbinary Hamming codes. Four families of distance-optimal or dimension-optimal linear codes are obtained with this extending technique. The parameters of certain extended codes of many families of linear codes are settled in this paper.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140103696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Non-Abelian extensions of degree p3 and p4 in characteristic p > 2","authors":"Grant Moles","doi":"10.1016/j.ffa.2024.102399","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102399","url":null,"abstract":"<div><p>This paper describes in terms of Artin-Schreier equations field extensions whose Galois group is isomorphic to any of the four non-cyclic groups of order <span><math><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> or the ten non-Abelian groups of order <span><math><msup><mrow><mi>p</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span>, <em>p</em> an odd prime, over a field of characteristic <em>p</em>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140113043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Diagonal hypersurfaces and elliptic curves over finite fields and hypergeometric functions","authors":"Sulakashna ,&nbsp;Rupam Barman","doi":"10.1016/j.ffa.2024.102397","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102397","url":null,"abstract":"<div><p>Let <span><math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>λ</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>k</mi></mrow></msubsup></math></span> denote the family of diagonal hypersurface over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> given by<span><span><span><math><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>λ</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>k</mi></mrow></msubsup><mo>:</mo><msubsup><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo>=</mo><mi>λ</mi><mi>d</mi><msubsup><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></msubsup><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>d</mi><mo>−</mo><mi>k</mi></mrow></msubsup><mo>,</mo></mrow></math></span></span></span> where <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>, <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>d</mi><mo>−</mo><mn>1</mn></math></span>, and <span><math><mi>gcd</mi><mo>⁡</mo><mo>(</mo><mi>d</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. Let <span><math><mi>#</mi><msubsup><mrow><mi>D</mi></mrow><mrow><mi>λ</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>k</mi></mrow></msubsup></math></span> denote the number of points on <span><math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>λ</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>k</mi></mrow></msubsup></math></span> in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span>. It is easy to see that <span><math><mi>#</mi><msubsup><mrow><mi>D</mi></mrow><mrow><mi>λ</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>k</mi></mrow></msubsup></math></span> is equal to the number of distinct zeros of the polynomial <span><math><msup><mrow><mi>y</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>−</mo><mi>d</mi><mi>λ</mi><msup><mrow><mi>y</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>+</mo><mn>1</mn><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>y</mi><mo>]</mo></math></span> in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. In this article, we prove that <span><math><mi>#</mi><msubsup><mrow><mi>D</mi></mrow><mrow><mi>λ</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>k</mi></mrow></msubsup></math></span> is also equal to the number of distinct zeros of the polynomial <span><math><msup><mrow><mi>y</mi></mrow><mrow><mi>d</mi><mo>−</mo><mi>k</mi></mrow></msup><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>y</mi><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msup><mo>−</mo><msup><mrow><mo>(</mo><mi>d</mi><mi>λ</mi><mo>)</mo></mrow><mrow><mo>−</mo><mi>d</mi></mrow></msup></math></span> in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. We express the number of distinct zeros of the polynomial <span><","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140041673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A structure theorem for the restricted sum of four squares","authors":"Wei Wang ,&nbsp;Weijia Wang ,&nbsp;Hao Zhang","doi":"10.1016/j.ffa.2024.102398","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102398","url":null,"abstract":"<div><p>Let <em>p</em> be an odd prime. We show that each solution of the system of congruence equations <span><math><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>≡</mo><mn>0</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>p</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>≡</mo><mn>0</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>p</mi><mo>)</mo></math></span> corresponds to precisely four solutions of the system of Diophantine equations <span><math><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>=</mo><mi>p</mi></math></span> that are pairwise orthogonal over <span><math><mi>Z</mi></math></span>, partially answering a conjecture proposed in Wang et al. <span>[10]</span>. The result was obtained by counting the number of solutions of both equations using Gaussian sum and modular forms, and the classical Cayley transformation.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140014355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A class of double-twisted generalized Reed-Solomon codes","authors":"Canze Zhu ,&nbsp;Qunying Liao","doi":"10.1016/j.ffa.2024.102395","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102395","url":null,"abstract":"<div><p>In this paper, let <em>q</em> be a prime power, we focus on a class of double-twisted generalized Reed-Solomon code <span><math><mi>C</mi></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. We give a sufficient and necessary condition for <span><math><mi>C</mi></math></span> to be MDS or AMDS, and prove that <span><math><mi>C</mi></math></span> is non-GRS by calculating the Schur square of its dual code. Furthermore, we present a sufficient and necessary condition for <span><math><mi>C</mi></math></span> to be self-dual, and then construct several classes of self-dual NMDS or non-GRS MDS codes.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140014354","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Transverse linear subspaces to hypersurfaces over finite fields","authors":"Shamil Asgarli ,&nbsp;Lian Duan ,&nbsp;Kuan-Wen Lai","doi":"10.1016/j.ffa.2024.102396","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102396","url":null,"abstract":"<div><p>Ballico proved that a smooth projective variety <em>X</em> of degree <em>d</em> and dimension <em>m</em> over a finite field of <em>q</em> elements admits a smooth hyperplane section if <span><math><mi>q</mi><mo>≥</mo><mi>d</mi><msup><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>m</mi></mrow></msup></math></span>. In this paper, we refine this criterion for higher codimensional linear sections on smooth hypersurfaces and for hyperplane sections on Frobenius classical hypersurfaces. We also prove a similar result for the existence of reduced hyperplane sections on reduced hypersurfaces.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139985896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Constructions and decoding of GC-balanced codes for edit errors","authors":"Kenan Wu,&nbsp;Shu Liu","doi":"10.1016/j.ffa.2024.102391","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102391","url":null,"abstract":"<div><p>DNA-based storage has been a promising technique of data storage, due to its high density and long duration. During synthesizing and sequencing of DNA storage, edit errors including insertions, deletions and substitutions are introduced inevitably. An effective way to reduce the error probability is to limit the content of G and C in DNA sequences to around 50%, which is called GC-balanced. To deal with edit errors, DNA sequences are also expected to have error-correcting capabilities. In this paper, GC globally balanced and GC locally balanced error-correcting codes are explicitly constructed, respectively. Inspired by repetition codes, the proposed codes are able to correct multiple edit errors. Furthermore, an efficient decoding algorithm applied for both codes is derived when only one kind of edit error occur.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139985766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Some new classes of additive MDS and almost MDS codes over finite fields","authors":"Monika Yadav ,&nbsp;Anuradha Sharma","doi":"10.1016/j.ffa.2024.102394","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102394","url":null,"abstract":"<div><p>In this paper, we introduce and study two new classes of additive codes over finite fields, <em>viz.</em> additive generalized Reed-Solomon (additive GRS) codes and additive generalized twisted Reed-Solomon (additive GTRS) codes, which are extensions of linear generalized Reed-Solomon (GRS) codes and twisted Reed-Solomon (GTRS) codes, respectively. Unlike linear GRS codes, additive GRS codes are not maximum distance separable (MDS) codes and the dual of an additive GRS code need not be an additive GRS code in general. We derive necessary and sufficient conditions under which an additive GRS code is MDS. We further apply this result to identify several new classes of additive MDS codes and a class of additive MDS codes whose dual codes are also MDS within the family of additive GRS codes. We also identify several new classes of additive codes that are either MDS or almost MDS within the family of additive GTRS codes. We also obtain several classes of additive TRS codes that are not monomially equivalent to additive RS codes. Besides this, we identify classes of monomially inequivalent additive MDS TRS codes and additive MDS RS codes, whose dual codes are also MDS. We also provide methods to construct additive MDS self-orthogonal, self-dual, and ACD codes through additive GRS and GTRS codes. Based on additive MDS codes whose dual codes are also MDS, we present a perfect threshold secret-sharing scheme that can detect cheating, identify a certain number of cheaters among the participants, and correctly recover the secret.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139986194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"AG codes on flag bundles over a curve","authors":"Tohru Nakashima","doi":"10.1016/j.ffa.2024.102392","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102392","url":null,"abstract":"<div><p>In the present paper, we construct codes from the flag bundle associated to a vector bundle <em>E</em> over a curve. Our code may be considered as a relative version of the codes on the flag variety studied by F. Rodier. We investigate the dimension and the minimum distance of such relative Rodier codes using intersection theory. For this purpose, we exploit the invariants of vector bundles which control the asymptotic behavior of semistability of <em>E</em> under pull-back by Frobenius morphisms</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139935910","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}