{"title":"On the functions which are CCZ-equivalent but not EA-equivalent to quadratic functions over Fpn","authors":"Jaeseong Jeong , Namhun Koo , Soonhak Kwon","doi":"10.1016/j.ffa.2025.102574","DOIUrl":"10.1016/j.ffa.2025.102574","url":null,"abstract":"<div><div>For a given function <em>F</em> from <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> to itself, determining whether there exists a function which is CCZ-equivalent but EA-inequivalent to <em>F</em> is a very important and interesting problem. For example, Kölsch <span><span>[33]</span></span> showed that there is no function which is CCZ-equivalent but EA-inequivalent to the inverse function. On the other hand, for the cases of Gold function <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>i</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></msup></math></span> and <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mrow><mi>Tr</mi></mrow><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>9</mn></mrow></msup><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>, Budaghyan, Carlet and Pott (respectively, Budaghyan, Carlet and Leander) <span><span>[12]</span></span>, <span><span>[14]</span></span> found functions which are CCZ-equivalent but EA-inequivalent to <em>F</em>. In this paper, when a given function <em>F</em> has a component function which has a linear structure, we present functions which are CCZ-equivalent to <em>F</em>, and if suitable conditions are satisfied, the constructed functions are shown to be EA-inequivalent to <em>F</em>. As a consequence, for every quadratic function <em>F</em> on <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> (<span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>) with nonlinearity greater than 0 and differential uniformity not exceeding <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msup></math></span>, we explicitly construct functions which are CCZ-equivalent but EA-inequivalent to <em>F</em>. Also for every non-planar quadratic function on <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> <span><math><mo>(</mo><mi>p</mi><mo>></mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>≥</mo><mn>4</mn><mo>)</mo></math></span> with <span><math><mo>|</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>|</mo><mo>≤</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> and differential uniformity not exceeding <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msup></math></span>, we explicitly construct functions which are CCZ-equivalent but EA-inequivalent to <em>F</em>. As an application, for a proper divisor <em>m</em> of <em>n</em>, we present many examples","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102574"},"PeriodicalIF":1.2,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139997","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 p-rank of singular curves and their smooth models","authors":"Sadık Terzi","doi":"10.1016/j.ffa.2025.102578","DOIUrl":"10.1016/j.ffa.2025.102578","url":null,"abstract":"<div><div>In this paper, we are concerned with the computation of the <em>p</em>-rank and <em>a</em>-number of singular curves and their smooth models. We consider a pair <span><math><mi>X</mi><mo>,</mo><msup><mrow><mi>X</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of proper curves over an algebraically closed field <em>k</em> of characteristic <em>p</em>, where <span><math><msup><mrow><mi>X</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is a singular curve which lies on a smooth projective variety, particularly on smooth projective surfaces <em>S</em> (with <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>=</mo><mi>q</mi><mo>(</mo><mi>S</mi><mo>)</mo></math></span>) and <em>X</em> is the smooth model of <span><math><msup><mrow><mi>X</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>. We determine the <em>p</em>-rank of <em>X</em> by using the exact sequence of group schemes relating the Jacobians <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>J</mi></mrow><mrow><msup><mrow><mi>X</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub></math></span>. As an application, we determine a relation about the fundamental invariants <em>p</em>-rank and <em>a</em>-number of a family of singular curves and their smooth models. Moreover, we calculate <em>a</em>-number and find lower bound for <em>p</em>-rank of a family of smooth curves.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102578"},"PeriodicalIF":1.2,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143140004","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}
Daniele Bartoli , Francesco Ghiandoni , Alessandro Giannoni , Giuseppe Marino
{"title":"A new family of 2-scattered subspaces and related MRD codes","authors":"Daniele Bartoli , Francesco Ghiandoni , Alessandro Giannoni , Giuseppe Marino","doi":"10.1016/j.ffa.2025.102572","DOIUrl":"10.1016/j.ffa.2025.102572","url":null,"abstract":"<div><div>Scattered subspaces and <em>h</em>-scattered subspaces have been extensively studied in recent decades for both theoretical purposes and their connections to various applications. While numerous constructions of scattered subspaces exist, relatively few are known about <em>h</em>-scattered subspaces with <span><math><mi>h</mi><mo>≥</mo><mn>2</mn></math></span>. In this paper, we establish the existence of maximum 2-scattered <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-subspaces in <span><math><mi>V</mi><mo>(</mo><mi>r</mi><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>)</mo></math></span> whenever <span><math><mi>r</mi><mo>≥</mo><mn>3</mn></math></span>, <span><math><mi>r</mi><mo>≠</mo><mn>5</mn></math></span>, and <em>q</em> is an odd power of 2. Additionally, we explore the corresponding MRD codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102572"},"PeriodicalIF":1.2,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143140002","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":"On the distribution of quadratic residues","authors":"I.D. Shkredov","doi":"10.1016/j.ffa.2025.102577","DOIUrl":"10.1016/j.ffa.2025.102577","url":null,"abstract":"<div><div>In our paper, we apply additive–combinatorial methods to study the distribution of the set of squares <span><math><mi>R</mi></math></span> in the prime field. We obtain the best upper bound on the number of gaps in <span><math><mi>R</mi></math></span> at the moment and generalize this result for sets with small doubling.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102577"},"PeriodicalIF":1.2,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139255","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":"Regular sets of lines in rank 3 polar spaces","authors":"Ferdinand Ihringer , Morgan Rodgers","doi":"10.1016/j.ffa.2024.102569","DOIUrl":"10.1016/j.ffa.2024.102569","url":null,"abstract":"<div><div>There are 6 families of finite polar spaces of rank 3. The set of lines in a rank 3 polar space forms a rank 5 association scheme. We determine the regular sets of minimal size in several of these polar spaces, and describe some interesting examples. We also give a new family of Cameron–Liebler sets of generators in the polar spaces <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><mn>10</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> when <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>h</mi></mrow></msup></math></span> using a regular set of lines in <span><math><mi>O</mi><mo>(</mo><mn>7</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102569"},"PeriodicalIF":1.2,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143140003","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":"Small solutions of generic ternary quadratic congruences to general moduli","authors":"Stephan Baier, Aishik Chattopadhyay","doi":"10.1016/j.ffa.2025.102571","DOIUrl":"10.1016/j.ffa.2025.102571","url":null,"abstract":"<div><div>We study small non-trivial solutions of quadratic congruences of the form <span><math><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>3</mn></mrow></msub><msubsup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>≡</mo><mn>0</mn><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>q</mi></math></span>, with <em>q</em> being an odd natural number, in an average sense. This extends previous work of the authors in which they considered the case of prime power moduli <em>q</em>. Above, <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> is arbitrary but fixed and <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> is variable, and we assume that <span><math><mo>(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>α</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><mi>q</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. We show that for all <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> modulo <em>q</em> which are coprime to <em>q</em> except for a small number of <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>'s, an asymptotic formula for the number of solutions <span><math><mo>(</mo><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></math></span> to the congruence <span><math><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>3</mn></mrow></msub><msubsup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>≡</mo><mn>0</mn><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>q</mi></math></span> with <span><math><mi>max</mi><mo></mo><mo>{</mo><mo>|</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><mo>,</mo><mo>|</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>|</mo><mo>,</mo><mo>|</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>|</mo><mo>}</mo><mo>≤</mo><mi>N</mi></math></span> and <span><math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><mi>q</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span> holds if <span><math><mi>N</mi><mo>≥</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>11</mn><mo>/</mo><mn>24</mn><mo>+</mo><mi>ε</mi></mrow></msup></math></span> and <em>q</em>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102571"},"PeriodicalIF":1.2,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143140006","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":"(2p + 1)-class association schemes from the generalized Maiorana-McFarland class","authors":"Nurdagül Anbar , Tekgül Kalaycı , Wilfried Meidl , Ferruh Özbudak","doi":"10.1016/j.ffa.2024.102568","DOIUrl":"10.1016/j.ffa.2024.102568","url":null,"abstract":"<div><div>In several articles, it has been shown that the preimage set partition of weakly regular (vectorial) bent functions, which are vectorial dual-bent, give rise to association schemes. The first construction of association schemes from certain partitions obtained from non-weakly regular bent functions, namely from ternary generalized Maiorana-McFarland functions, is presented in Özbudak and Pelen (2022) <span><span>[32]</span></span>.</div><div>In this article, association schemes are obtained from generalized Maiorana-McFarland bent functions from <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>×</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></msub><mo>×</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></msub></math></span> to <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, which are constructed from <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span> bent functions from <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> to <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> with certain properties. The obtained schemes are in general <span><math><mo>(</mo><mn>2</mn><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-class association schemes. In the case that <span><math><mi>n</mi><mo>=</mo><mn>1</mn></math></span> respectively in one case for <span><math><mi>n</mi><mo>=</mo><mn>2</mn></math></span>, the association schemes reduce to <span><math><mo>(</mo><mn>3</mn><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></math></span>-class association schemes respectively to 2<em>p</em>-class association schemes. For <span><math><mi>p</mi><mo>=</mo><mn>3</mn></math></span>, these schemes are the 5-class and 6-class association schemes obtained by Özbudak and Pelen. Therefore, the construction in this article substantially generalizes these earlier constructions. Also note that for <span><math><mi>n</mi><mo>=</mo><mn>1</mn></math></span> respectively <span><math><mi>n</mi><mo>=</mo><mn>2</mn></math></span>, the construction is based in bent functions from <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> to <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> respectively from <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> to <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, for which the choices are very limited.</div><div>Depending on the choice of the bent functions used for the construction, the resulting generalized Maiorana-McFarland function may be weakly reg","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102568"},"PeriodicalIF":1.2,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143140007","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 sumsets of units and exceptional units in residue class rings","authors":"Siao Hong","doi":"10.1016/j.ffa.2024.102566","DOIUrl":"10.1016/j.ffa.2024.102566","url":null,"abstract":"<div><div>Let <span><math><mi>n</mi><mo>≥</mo><mn>1</mn><mo>,</mo><mi>e</mi><mo>≥</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and <em>c</em> be integers such that <span><math><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>k</mi></math></span>. An integer <em>u</em> is called a unit in the ring <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of residue classes modulo <em>n</em> if <span><math><mi>gcd</mi><mo></mo><mo>(</mo><mi>u</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. Let <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span> be the multiplicative group of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. A unit <em>u</em> is called an exceptional unit in the ring <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> if <span><math><mn>1</mn><mo>−</mo><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. We write <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>⁎</mo><mo>⁎</mo></mrow></msubsup></math></span> for the set of all exceptional units of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We denote by <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>e</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> the number of representations of the element <span><math><mi>c</mi><mo>∈</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> as the sum of <em>e</em>-th powers of <em>t</em> units and <em>e</em>-th powers of <span><math><mi>k</mi><mo>−</mo><mi>t</mi></math></span> exceptional units in the ring <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. When <span><math><mi>t</mi><mo>=</mo><mi>k</mi></math></span>, Brauer determined the number <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>c</mi><mo>,</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> which answers a question of Rademacher. Mollahajiaghaei gave a formula for <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>c</mi><mo>,</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. When <span><math><mi>t</mi><mo>=</mo><mn>0</mn></math></span>, Sander presented a formula for <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>0</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>c</mi><mo>,</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, and later on Yang and Zhao got an exact formula for <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>k</mi><mo>,</mo><mi>c</mi><mo>,</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102566"},"PeriodicalIF":1.2,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143135116","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 complexity of elliptic normal bases","authors":"Daniel Panario , Mohamadou Sall , Qiang Wang","doi":"10.1016/j.ffa.2024.102570","DOIUrl":"10.1016/j.ffa.2024.102570","url":null,"abstract":"<div><div>We study the complexity (that is, the weight of the multiplication table) of the elliptic normal bases introduced by Couveignes and Lercier. We give an upper bound on the complexity of these elliptic normal bases, and we analyze the weight of some specific vectors related to the multiplication table of those bases. This analysis leads us to some perspectives on the search for low complexity normal bases from elliptic periods.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102570"},"PeriodicalIF":1.2,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143135117","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":"More classes of permutation pentanomials over finite fields with even characteristic","authors":"Tongliang Zhang , Lijing Zheng","doi":"10.1016/j.ffa.2024.102567","DOIUrl":"10.1016/j.ffa.2024.102567","url":null,"abstract":"<div><div>Let <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span>. In this paper, we propose several classes of permutation pentanomials of the form <span><math><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup><mo>+</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>+</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mspace></mspace><mo>(</mo><mn>2</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>4</mn><mo>)</mo></math></span> from some certain linearized polynomial <span><math><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> by using multivariate method and some techniques to determine the solutions of some equations. Furthermore, two classes of permutation pentanomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> for <em>n</em> satisfying <span><math><mn>3</mn><mo>|</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn></math></span> are also constructed based on some bijections over the unit circle <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>τ</mi></mrow></msub></math></span> of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> with order <span><math><mi>τ</mi><mo>=</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>q</mi><mo>+</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102567"},"PeriodicalIF":1.2,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139958","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}
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