Finite Fields and Their Applications最新文献

{"title":"Artin-Schreier towers of finite fields","authors":"Leandro Cagliero , Allen Herman , Fernando Szechtman","doi":"10.1016/j.ffa.2025.102606","DOIUrl":"10.1016/j.ffa.2025.102606","url":null,"abstract":"<div><div>Given a prime number <em>p</em>, we consider the tower of finite fields <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>=</mo><msub><mrow><mi>L</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>⊂</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>⊂</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊂</mo><mo>⋯</mo></math></span>, where each step corresponds to an Artin-Schreier extension of degree <em>p</em>, so that for <span><math><mi>i</mi><mo>≥</mo><mn>0</mn></math></span>, <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>[</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>]</mo></math></span>, where <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is a root of <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>−</mo><mi>X</mi><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><msup><mrow><mo>(</mo><msub><mrow><mi>c</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, with <span><math><msub><mrow><mi>c</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span>. We extend and strengthen to arbitrary primes prior work of Popovych for <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span> on the multiplicative order <span><math><mi>O</mi><mo>(</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span> of the given generator <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> for <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> over <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>. In particular, for <span><math><mi>i</mi><mo>≥</mo><mn>0</mn></math></span>, we show that <span><math><mi>O</mi><mo>(</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span>, except only when <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span> and <span><math><mi>i</mi><mo>=</mo><mn>1</mn></math></span>, and that <span><math><mi>O</mi><mo>(</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span> is equal to the product of the orders of <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> modulo <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow><mrow><mo>×</mo></mrow></m","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"106 ","pages":"Article 102606"},"PeriodicalIF":1.2,"publicationDate":"2025-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143611110","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":"On the existence of Galois self-dual GRS and TGRS codes","authors":"Shixin Zhu,&nbsp;Ruhao Wan","doi":"10.1016/j.ffa.2025.102608","DOIUrl":"10.1016/j.ffa.2025.102608","url":null,"abstract":"<div><div>Let <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> be a prime power and <em>e</em> be an integer with <span><math><mn>0</mn><mo>≤</mo><mi>e</mi><mo>≤</mo><mi>m</mi><mo>−</mo><mn>1</mn></math></span>. <em>e</em>-Galois self-dual codes are generalizations of Euclidean <span><math><mo>(</mo><mi>e</mi><mo>=</mo><mn>0</mn><mo>)</mo></math></span> and Hermitian (<span><math><mi>e</mi><mo>=</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> with even <em>m</em>) self-dual codes. In this paper, for a linear code <span><math><mi>C</mi></math></span> and a nonzero vector <span><math><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>, we give a sufficient and necessary condition for the dual extended code <span><math><munder><mrow><mi>C</mi></mrow><mo>_</mo></munder><mo>[</mo><mi>u</mi><mo>]</mo></math></span> of <span><math><mi>C</mi></math></span> to be <em>e</em>-Galois self-orthogonal. From this, a new systematic approach is proposed to prove the existence of <em>e</em>-Galois self-dual codes. By this method, we prove that <em>e</em>-Galois self-dual (extended) generalized Reed-Solomon (GRS) codes of length <span><math><mi>n</mi><mo>&gt;</mo><mi>min</mi><mo>⁡</mo><mo>{</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup><mo>+</mo><mn>1</mn><mo>,</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi><mo>−</mo><mi>e</mi></mrow></msup><mo>+</mo><mn>1</mn><mo>}</mo></math></span> do not exist, where <span><math><mn>1</mn><mo>≤</mo><mi>e</mi><mo>≤</mo><mi>m</mi><mo>−</mo><mn>1</mn></math></span>. Moreover, based on the non-GRS properties of twisted GRS (TGRS) codes, we show that in many cases <em>e</em>-Galois self-dual (extended) TGRS codes do not exist. Furthermore, we present a sufficient and necessary condition for <span><math><mo>(</mo><mo>⁎</mo><mo>)</mo></math></span>-TGRS codes to be Hermitian self-dual, and then construct several new classes of Hermitian self-dual <span><math><mo>(</mo><mo>+</mo><mo>)</mo></math></span>-TGRS and <span><math><mo>(</mo><mo>⁎</mo><mo>)</mo></math></span>-TGRS codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102608"},"PeriodicalIF":1.2,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143562784","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":"Polynomial p-adic low-discrepancy sequences","authors":"Christian Weiß","doi":"10.1016/j.ffa.2025.102607","DOIUrl":"10.1016/j.ffa.2025.102607","url":null,"abstract":"<div><div>The classic example of a low-discrepancy sequence in <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> is <span><math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>a</mi><mi>n</mi><mo>+</mo><mi>b</mi></math></span> with <span><math><mi>a</mi><mo>∈</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo>×</mo></mrow></msubsup></math></span> and <span><math><mi>b</mi><mo>∈</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>. Here we address the non-linear case and show that a polynomial <em>f</em> generates a low-discrepancy sequence in <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> if and only if it is a permutation polynomial <span><math><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mspace></mspace><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. By this it is possible to construct non-linear examples of low-discrepancy sequences in <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> for all primes <em>p</em>. Moreover, we prove a criterion which decides for any given polynomial in <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> with <span><math><mi>p</mi><mo>∈</mo><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo>}</mo></mrow></math></span> if it generates a low-discrepancy sequence. We also discuss connections to the theories of Poissonian pair correlations and real discrepancy.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102607"},"PeriodicalIF":1.2,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bound on the minimum distance of double circulant cubic residue codes","authors":"Minjia Shi ,&nbsp;Xinpeng Bian ,&nbsp;Ferruh Özbudak ,&nbsp;Patrick Solé","doi":"10.1016/j.ffa.2025.102605","DOIUrl":"10.1016/j.ffa.2025.102605","url":null,"abstract":"<div><div>We study a class of pure double circulant binary codes attached to the cyclotomy of order 3 with respect to a prime <span><math><mi>p</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>3</mn><mo>)</mo></math></span>. The minimum distance is bounded below by an argument involving cyclotomic numbers and Weil inequality for multiplicative character sums.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102605"},"PeriodicalIF":1.2,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143489010","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":"Beyond uniform cyclotomy","authors":"Sophie Huczynska ,&nbsp;Laura Johnson ,&nbsp;Maura B. Paterson","doi":"10.1016/j.ffa.2025.102604","DOIUrl":"10.1016/j.ffa.2025.102604","url":null,"abstract":"<div><div>Cyclotomy, the study of cyclotomic classes and cyclotomic numbers, is an area of number theory first studied by Gauss. It has natural applications in discrete mathematics and information theory. Despite this long history, there are significant limitations to what is known explicitly about cyclotomic numbers, which limits the use of cyclotomy in applications. The main explicit tool available is that of uniform cyclotomy, introduced by Baumert, Mills and Ward in 1982. In this paper, we present an extension of uniform cyclotomy which gives a direct method for evaluating all cyclotomic numbers over <span><math><mrow><mi>GF</mi></mrow><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> of order dividing <span><math><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>, for any prime power <em>q</em> and <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, which does not use character theory nor direct calculation in the field. This allows the straightforward evaluation of many cyclotomic numbers for which other methods are unknown or impractical, extending the currently limited portfolio of tools to work with cyclotomic numbers. Our methods exploit connections between cyclotomy, Singer difference sets and finite geometry.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102604"},"PeriodicalIF":1.2,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143488563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The central limit theorem for entries of random matrices with specified rank over finite fields","authors":"Chin Hei Chan,&nbsp;Maosheng Xiong","doi":"10.1016/j.ffa.2025.102603","DOIUrl":"10.1016/j.ffa.2025.102603","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> be the finite field of order <em>q</em>, and <span><math><mi>A</mi></math></span> a non-empty proper subset of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. Let <strong>M</strong> be a random <span><math><mi>m</mi><mo>×</mo><mi>n</mi></math></span> matrix of rank <em>r</em> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> taken with uniform distribution. It was proved recently by Sanna that as <span><math><mi>m</mi><mo>,</mo><mi>n</mi><mo>→</mo><mo>∞</mo></math></span> and <span><math><mi>r</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>A</mi></math></span> are fixed, the number of entries of <strong>M</strong> in <span><math><mi>A</mi></math></span> approaches a normal distribution. The question was raised as to whether or not one can still obtain a central limit theorem of some sort when <em>r</em> goes to infinity in a way controlled by <em>m</em> and <em>n</em>. In this paper we answer this question affirmatively.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102603"},"PeriodicalIF":1.2,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143478640","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":"Double circulant codes from cubic cyclotomy","authors":"Minjia Shi ,&nbsp;Xinpeng Bian ,&nbsp;Patrick Solé","doi":"10.1016/j.ffa.2025.102593","DOIUrl":"10.1016/j.ffa.2025.102593","url":null,"abstract":"<div><div>We introduce a parametrized family of binary double circulant codes, based on cubic cyclotomy. We determine the parameters for which the codes are self-dual and those for which they are LCD (Linear Complementary Dual). As a bonus, they turn out to be formally self-dual in the latter case. The main theoretical tools are the properties of cyclotomic numbers. Examples in modest length show quasi-optimal formally self-dual codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"104 ","pages":"Article 102593"},"PeriodicalIF":1.2,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143473902","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":"Foundations of additive codes over finite fields","authors":"Dipak K. Bhunia ,&nbsp;Steven T. Dougherty ,&nbsp;Cristina Fernández-Córdoba ,&nbsp;Mercè Villanueva","doi":"10.1016/j.ffa.2025.102592","DOIUrl":"10.1016/j.ffa.2025.102592","url":null,"abstract":"<div><div>Additive codes were initially introduced by Delsarte in 1973 within the context of association schemes and recently they have become of interest due to their application in constructing quantum error-correcting codes.</div><div>We give foundational results for additive codes where the elements are from a finite field, and define the orthogonality relation using group characters. We introduce a type for these additive codes and explore the notion of independence for a generating set. Additionally, we provide a definition for a generator matrix of an additive code based on its type. We also relate the type of an additive code to the type of its orthogonal. We study a family of kernels and ranks associated with these additive codes. We relate the equivalence of additive codes to their type, the family of kernels and ranks, and duality. We see how these relations contribute in the classification of additive codes. Finally, we provide a general encoding and decoding method for these codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"104 ","pages":"Article 102592"},"PeriodicalIF":1.2,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143444838","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":"Two classes of twisted generalized Reed-Solomon codes with two twists","authors":"Shudi Yang ,&nbsp;Jinlong Wang ,&nbsp;Yansheng Wu","doi":"10.1016/j.ffa.2025.102595","DOIUrl":"10.1016/j.ffa.2025.102595","url":null,"abstract":"<div><div>MDS codes, abbreviated from maximum distance separable codes, hold significant importance in coding theory owing to their good algebraic structures, alongside intriguing practical applications. In this paper, we mainly study two classes of twisted generalized Reed-Solomon codes with two twists. Specifically, some sufficient and necessary conditions for these codes to be MDS or self-orthogonal are obtained; two explicit constructions of MDS self-orthogonal codes are presented; and finally, several classes of linear complementary dual codes via twisted generalized Reed-Solomon codes are also provided.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"104 ","pages":"Article 102595"},"PeriodicalIF":1.2,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143427874","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 results on linear subspace codes","authors":"Mahak,&nbsp;Maheshanand Bhaintwal","doi":"10.1016/j.ffa.2025.102596","DOIUrl":"10.1016/j.ffa.2025.102596","url":null,"abstract":"<div><div>The notion of linearity in projective spaces was defined by Etzion and Vardy (2008) <span><span>[3]</span></span>. In this paper, we have obtained some results on linear subspace codes. We have proved that if in a linear subspace code <em>C</em> in the projective space <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, the number of one-dimensional subspaces is <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span>, then the cardinality of <em>C</em> is <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>; and if the number of the one-dimensional subspaces in <em>C</em> is <span><math><mi>n</mi><mo>−</mo><mn>2</mn></math></span> and the ambient space does not belong to <em>C</em>, then the cardinality of <em>C</em> is <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup></math></span>. We have also studied complementary linear subspace codes. An example has been given to show that a complement function can exist on a non-distributive sublattice of the projective lattice. We have also proved that a non-distributive sublattice of the projective lattice cannot be a linear subspace code.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"104 ","pages":"Article 102596"},"PeriodicalIF":1.2,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143403591","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}