Pub Date : 2025-01-15DOI: 10.1016/j.ffa.2024.102569
Ferdinand Ihringer , Morgan Rodgers
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 when using a regular set of lines in .
{"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}
Pub Date : 2025-01-13DOI: 10.1016/j.ffa.2025.102571
Stephan Baier, Aishik Chattopadhyay
<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>
{"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}
<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
{"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}
Pub Date : 2025-01-07DOI: 10.1016/j.ffa.2024.102566
Siao Hong
<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
{"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}
Pub Date : 2025-01-06DOI: 10.1016/j.ffa.2024.102570
Daniel Panario , Mohamadou Sall , Qiang Wang
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.
{"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}
Pub Date : 2025-01-03DOI: 10.1016/j.ffa.2024.102567
Tongliang Zhang , Lijing Zheng
Let . In this paper, we propose several classes of permutation pentanomials of the form over from some certain linearized polynomial by using multivariate method and some techniques to determine the solutions of some equations. Furthermore, two classes of permutation pentanomials over for n satisfying are also constructed based on some bijections over the unit circle of with order .
{"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}
Pub Date : 2024-12-11DOI: 10.1016/j.ffa.2024.102547
Daniel Smith-Tone , Cristina Tone
Cheng et al. (2014) [6] introduced a substantial improvement to the Miura-Hashimoto-Takagi algorithm for solving sufficiently underdetermined systems of multivariate polynomial equations. This improvement claimed to make the algorithm polynomial time for instances satisfying , where m is the number of equations and n is the number of variables. While experimentally, the algorithm seems to work, we have uncovered a subtle error in the proof of time complexity for the algorithm. Due to the fact that there have been multiple proposals for algorithms based on this and related algorithms, as well as the recent submission to NIST's call for additional post-quantum digital signatures of a more modern “provably secure” version of the famous UOV digital signature algorithm based on the foundational structure of this algorithm, our observation may highlight a concerning theoretical deficiency in this area of research.
In this work, we provide a tight justification for the polynomial time complexity of the algorithm (with a very minor tweak), thereby justifying the complexity of enhancements based upon it as well. At the heart of this justification is a precise calculation of the probability of recovering a maximal depth path in polynomially many steps within a possibly exponentially large search tree. While this algorithmic problem is generic, we find that the parameters relevant for the application to the above algorithm are extremal and poorly studied. Thus, our analysis serves to clarify the boundary behavior of such search algorithms with respect to complexity classes.
{"title":"A correct justification for the CHMT algorithm for solving underdetermined multivariate systems","authors":"Daniel Smith-Tone , Cristina Tone","doi":"10.1016/j.ffa.2024.102547","DOIUrl":"10.1016/j.ffa.2024.102547","url":null,"abstract":"<div><div>Cheng et al. (2014) <span><span>[6]</span></span> introduced a substantial improvement to the Miura-Hashimoto-Takagi algorithm for solving sufficiently underdetermined systems of multivariate polynomial equations. This improvement claimed to make the algorithm polynomial time for instances satisfying <span><math><mi>n</mi><mo>≥</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span>, where <em>m</em> is the number of equations and <em>n</em> is the number of variables. While experimentally, the algorithm seems to work, we have uncovered a subtle error in the proof of time complexity for the algorithm. Due to the fact that there have been multiple proposals for algorithms based on this and related algorithms, as well as the recent submission to NIST's call for additional post-quantum digital signatures of a more modern “provably secure” version of the famous UOV digital signature algorithm based on the foundational structure of this algorithm, our observation may highlight a concerning theoretical deficiency in this area of research.</div><div>In this work, we provide a tight justification for the polynomial time complexity of the algorithm (with a very minor tweak), thereby justifying the complexity of enhancements based upon it as well. At the heart of this justification is a precise calculation of the probability of recovering a maximal depth path in polynomially many steps within a possibly exponentially large search tree. While this algorithmic problem is generic, we find that the parameters relevant for the application to the above algorithm are extremal and poorly studied. Thus, our analysis serves to clarify the boundary behavior of such search algorithms with respect to complexity classes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102547"},"PeriodicalIF":1.2,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139956","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}
Error correcting codes have recently gained more attention due to their applications in quantum resistant cryptography. Their suitability depends on their indistinguishability from random codes. In that sense, the study of the square code of a particular code provides a tool for distinguishing random codes from not random ones.
With this motivation, the square codes of some semisimple bilateral group codes, as abelian and dihedral ones, are studied in this paper. For this purpose, bilateral group codes are described as evaluation codes by means of the absolutely irreducible characters of the group. Finally, some results on self-duality and self-orthogonality are recovered under this alternative point of view.
{"title":"On the square code of group codes","authors":"Alejandro Piñera Nicolás , Ignacio Fernández Rúa , Adriana Suárez Corona","doi":"10.1016/j.ffa.2024.102548","DOIUrl":"10.1016/j.ffa.2024.102548","url":null,"abstract":"<div><div>Error correcting codes have recently gained more attention due to their applications in quantum resistant cryptography. Their suitability depends on their indistinguishability from random codes. In that sense, the study of the square code of a particular code provides a tool for distinguishing random codes from not random ones.</div><div>With this motivation, the square codes of some semisimple bilateral group codes, as abelian and dihedral ones, are studied in this paper. For this purpose, bilateral group codes are described as evaluation codes by means of the absolutely irreducible characters of the group. Finally, some results on self-duality and self-orthogonality are recovered under this alternative point of view.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102548"},"PeriodicalIF":1.2,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139957","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}
Pub Date : 2024-11-27DOI: 10.1016/j.ffa.2024.102546
Alp Bassa , Ricardo Menares
We give iterative constructions for irreducible polynomials over of degree for all , starting from irreducible polynomials of degree n. The iterative constructions correspond modulo fractional linear transformations to compositions with power functions . The R-transform introduced by Cohen is recovered as a particular case corresponding to , hence we obtain a generalization of Cohen's R-transform () to arbitrary degrees . Important properties like self-reciprocity and invariance of roots under certain automorphisms are deduced from invariance under multiplication by appropriate roots of unity. Extending to quadratic extensions of we recover and generalize a recursive construction of Panario, Reis and Wang.
我们从 n 度的不可还原多项式出发,给出了所有 r≥0 的 n⋅tr Fq 上不可还原多项式的迭代构造。科恩引入的 R 变换作为与 x2 相对应的特殊情况被复原,因此我们得到了科恩 R 变换 (t=2) 对任意度 t≥2 的推广。从与适当的合一根相乘的不变性推导出了自还原性和根在某些自动形态下的不变性等重要性质。扩展到 Fq 的二次扩展,我们恢复并推广了帕纳里奥、雷斯和王的递归构造。
{"title":"The R-transform as power map and its generalizations to higher degree","authors":"Alp Bassa , Ricardo Menares","doi":"10.1016/j.ffa.2024.102546","DOIUrl":"10.1016/j.ffa.2024.102546","url":null,"abstract":"<div><div>We give iterative constructions for irreducible polynomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> of degree <span><math><mi>n</mi><mo>⋅</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> for all <span><math><mi>r</mi><mo>≥</mo><mn>0</mn></math></span>, starting from irreducible polynomials of degree <em>n</em>. The iterative constructions correspond modulo fractional linear transformations to compositions with power functions <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msup></math></span>. The <em>R</em>-transform introduced by Cohen is recovered as a particular case corresponding to <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, hence we obtain a generalization of Cohen's <em>R</em>-transform (<span><math><mi>t</mi><mo>=</mo><mn>2</mn></math></span>) to arbitrary degrees <span><math><mi>t</mi><mo>≥</mo><mn>2</mn></math></span>. Important properties like self-reciprocity and invariance of roots under certain automorphisms are deduced from invariance under multiplication by appropriate roots of unity. Extending to quadratic extensions of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> we recover and generalize a recursive construction of Panario, Reis and Wang.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"102 ","pages":"Article 102546"},"PeriodicalIF":1.2,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722321","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}
Pub Date : 2024-11-26DOI: 10.1016/j.ffa.2024.102544
Nour Alnajjarine , Michel Lavrauw
This paper is a contribution towards a solution for the longstanding open problem of classifying linear systems of conics over finite fields initiated by L. E. Dickson in 1908, through his study of the projective equivalence classes of pencils of conics in , for q odd. In this paper a set of complete invariants is determined for the projective equivalence classes of webs and of squabs of conics in , both for q odd and even. Our approach is mainly geometric, and involves a comprehensive study of the geometric and combinatorial properties of the Veronese surface in . The main contribution is the determination of the distribution of the different types of hyperplanes incident with the K-orbit representatives of points and lines of , where , is the subgroup of stabilizing the Veronese surface.
本文是对 L. E. Dickson 于 1908 年通过研究 PG(2,q) 中圆锥的铅笔的投影等价类(q 为奇数)而提出的有限域上圆锥的线性系统分类这一长期未决问题的一个解决方案的贡献。本文为 PG(2,q) 中圆锥的网状和方形的投影等价类确定了一组完整的不变式,无论是 q 为奇数还是偶数。我们的方法主要是几何方法,涉及对 PG(5,q) 中维罗尼斯曲面的几何和组合性质的全面研究。我们的主要贡献是确定了与 PG(5,q) 的 K 轨道代表点和线相关的不同类型超平面的分布,其中 K≅PGL(3,q) 是 PGL(6,q) 的子群,稳定了 Veronese 曲面。
{"title":"Webs and squabs of conics over finite fields","authors":"Nour Alnajjarine , Michel Lavrauw","doi":"10.1016/j.ffa.2024.102544","DOIUrl":"10.1016/j.ffa.2024.102544","url":null,"abstract":"<div><div>This paper is a contribution towards a solution for the longstanding open problem of classifying linear systems of conics over finite fields initiated by L. E. Dickson in 1908, through his study of the projective equivalence classes of pencils of conics in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, for <em>q</em> odd. In this paper a set of complete invariants is determined for the projective equivalence classes of webs and of squabs of conics in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, both for <em>q</em> odd and even. Our approach is mainly geometric, and involves a comprehensive study of the geometric and combinatorial properties of the Veronese surface in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>5</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>. The main contribution is the determination of the distribution of the different types of hyperplanes incident with the <em>K</em>-orbit representatives of points and lines of <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>5</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, where <span><math><mi>K</mi><mo>≅</mo><mrow><mi>PGL</mi></mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, is the subgroup of <span><math><mrow><mi>PGL</mi></mrow><mo>(</mo><mn>6</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> stabilizing the Veronese surface.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"102 ","pages":"Article 102544"},"PeriodicalIF":1.2,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722322","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}