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Beyond uniform cyclotomy 除均匀睫状体切开术外
IF 1.2 3区 数学
Finite Fields and Their Applications Pub Date : 2025-02-26 DOI: 10.1016/j.ffa.2025.102604
Sophie Huczynska , Laura Johnson , Maura B. Paterson
{"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}
引用次数: 0
The central limit theorem for entries of random matrices with specified rank over finite fields 有限域上定秩随机矩阵的中心极限定理
IF 1.2 3区 数学
Finite Fields and Their Applications Pub Date : 2025-02-25 DOI: 10.1016/j.ffa.2025.102603
Chin Hei Chan, Maosheng Xiong
{"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}
引用次数: 0
Double circulant codes from cubic cyclotomy 立方环切开术的双循环代码
IF 1.2 3区 数学
Finite Fields and Their Applications Pub Date : 2025-02-24 DOI: 10.1016/j.ffa.2025.102593
Minjia Shi , Xinpeng Bian , Patrick Solé
{"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}
引用次数: 0
Foundations of additive codes over finite fields 有限域上加性码的基础
IF 1.2 3区 数学
Finite Fields and Their Applications Pub Date : 2025-02-19 DOI: 10.1016/j.ffa.2025.102592
Dipak K. Bhunia , Steven T. Dougherty , Cristina Fernández-Córdoba , Mercè Villanueva
{"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}
引用次数: 0
Two classes of twisted generalized Reed-Solomon codes with two twists 两类双扭转广义Reed-Solomon码
IF 1.2 3区 数学
Finite Fields and Their Applications Pub Date : 2025-02-18 DOI: 10.1016/j.ffa.2025.102595
Shudi Yang , Jinlong Wang , Yansheng Wu
{"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}
引用次数: 0
Some results on linear subspace codes 关于线性子空间码的一些结果
IF 1.2 3区 数学
Finite Fields and Their Applications Pub Date : 2025-02-14 DOI: 10.1016/j.ffa.2025.102596
Mahak, Maheshanand Bhaintwal
{"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}
引用次数: 0
Permutation polynomials of the form xrh(xq−1) over Fq2 with even characteristics 具有偶特征的形式为xrh(xq−1)/ Fq2的置换多项式
IF 1.2 3区 数学
Finite Fields and Their Applications Pub Date : 2025-02-14 DOI: 10.1016/j.ffa.2025.102594
Amritanshu Rai, Rohit Gupta
{"title":"Permutation polynomials of the form xrh(xq−1) over Fq2 with even characteristics","authors":"Amritanshu Rai,&nbsp;Rohit Gupta","doi":"10.1016/j.ffa.2025.102594","DOIUrl":"10.1016/j.ffa.2025.102594","url":null,"abstract":"<div><div>Let <em>r</em> be a positive integer, <em>q</em> be a prime power and <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>. In this paper, we investigate six classes of permutation polynomials of the form <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msup><mi>h</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> with <em>q</em> even, for specific values of <em>r</em> and for some choices of <span><math><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span>. One of them is a generalization of a known class of permutation polynomials, while another two deal with some open problems.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"104 ","pages":"Article 102594"},"PeriodicalIF":1.2,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143403590","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}
引用次数: 0
Four families of q-ary self-orthogonal codes via the defining-set construction 通过定义集构造得到四族q元自正交码
IF 1.2 3区 数学
Finite Fields and Their Applications Pub Date : 2025-02-13 DOI: 10.1016/j.ffa.2025.102586
Jiayuan Zhang, Xiaoshan Kai, Ping Li, Shixin Zhu
{"title":"Four families of q-ary self-orthogonal codes via the defining-set construction","authors":"Jiayuan Zhang,&nbsp;Xiaoshan Kai,&nbsp;Ping Li,&nbsp;Shixin Zhu","doi":"10.1016/j.ffa.2025.102586","DOIUrl":"10.1016/j.ffa.2025.102586","url":null,"abstract":"<div><div>Self-orthogonal codes have a wide range of applications in various fields, especially communication and cryptography. In this paper, we construct four families of linear codes over finite fields via a defining-set construction. The weight distributions of these codes are determined. We show that most of these codes are self-orthogonal and reach the Grismer bound.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"104 ","pages":"Article 102586"},"PeriodicalIF":1.2,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143403641","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}
引用次数: 0
On the construction of nonbinary LCD quadratic residue and double quadratic residue codes 非二进制LCD二次残码和双二次残码的构造
IF 1.2 3区 数学
Finite Fields and Their Applications Pub Date : 2025-02-11 DOI: 10.1016/j.ffa.2025.102591
Arezoo Soufi Karbaski , Taher Abualrub , Irfan Siap , Hashem Bordbar
{"title":"On the construction of nonbinary LCD quadratic residue and double quadratic residue codes","authors":"Arezoo Soufi Karbaski ,&nbsp;Taher Abualrub ,&nbsp;Irfan Siap ,&nbsp;Hashem Bordbar","doi":"10.1016/j.ffa.2025.102591","DOIUrl":"10.1016/j.ffa.2025.102591","url":null,"abstract":"<div><div>Linear complementary dual (LCD) codes are an important class of error-correcting codes because of their applications in many areas such as their applications in cryptography and secret sharing <span><span>[1]</span></span>, <span><span>[4]</span></span>, <span><span>[7]</span></span>. In this paper, we construct a large class of nonbinary LCD codes from the class of nonbinary quadratic residue (QR) codes and nonbinary double QR codes. We have also introduced the class of extended quasi quadratic residue (QQR) codes and construct self-orthogonal codes from these codes. As an application of our study, we have presented an optimal ternary self-orthogonal code of parameters <span><math><mo>[</mo><mn>24</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>12</mn><mo>]</mo></math></span>. We have also constructed examples of self-orthogonal codes with parameters <span><math><msub><mrow><mo>[</mo><mn>2</mn><mrow><mo>(</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mfrac><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>2</mn><mi>d</mi><mo>]</mo></mrow><mrow><mi>p</mi></mrow></msub></math></span> and also examples of LCD codes with parameters <span><math><msub><mrow><mo>[</mo><mn>2</mn><mi>q</mi><mo>,</mo><mfrac><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>2</mn><mi>d</mi><mo>]</mo></mrow><mrow><mi>p</mi></mrow></msub></math></span> and <span><math><msub><mrow><mo>[</mo><mn>2</mn><mi>q</mi><mo>,</mo><mfrac><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>2</mn><mi>d</mi><mo>]</mo></mrow><mrow><mi>p</mi></mrow></msub></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> for different values of primes <em>p</em> and <em>q</em>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102591"},"PeriodicalIF":1.2,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143386537","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}
引用次数: 0
Homogenization of binary linear codes and their applications 二进制线性码的均匀化及其应用
IF 1.2 3区 数学
Finite Fields and Their Applications Pub Date : 2025-02-06 DOI: 10.1016/j.ffa.2025.102589
Jong Yoon Hyun , Nilay Kumar Mondal , Yoonjin Lee
{"title":"Homogenization of binary linear codes and their applications","authors":"Jong Yoon Hyun ,&nbsp;Nilay Kumar Mondal ,&nbsp;Yoonjin Lee","doi":"10.1016/j.ffa.2025.102589","DOIUrl":"10.1016/j.ffa.2025.102589","url":null,"abstract":"<div><div>We introduce a new technique, called <em>homogenization</em>, for a systematic construction of augmented codes of binary linear codes, using the defining set approach in connection to multi-variable functions. We explicitly determine the parameters and the weight distribution of the homogenized codes when the defining set is either a simplicial complex generated by any finite number of elements, or the difference of two simplicial complexes, each of which is generated by a single maximal element. Using this homogenization technique, we produce several infinite families of optimal codes, self-orthogonal codes, minimal codes, and self-complementary codes. As applications, we obtain some best known <em>quantum error-correcting codes</em>, infinite families of <em>intersecting codes</em> (used in the construction of covering arrays), and we compute the <em>Trellis complexity</em> (required for decoding) for several families of codes as well.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102589"},"PeriodicalIF":1.2,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143272335","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}
引用次数: 0
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