Finite Fields and Their Applications最新文献

{"title":"Normal and primitive normal elements with prescribed traces in intermediate extensions of finite fields","authors":"Arpan Chandra Mazumder ,&nbsp;Giorgos Kapetanakis ,&nbsp;Dhiren Kumar Basnet","doi":"10.1016/j.ffa.2025.102745","DOIUrl":"10.1016/j.ffa.2025.102745","url":null,"abstract":"<div><div>In this article, we study the existence and distribution of elements in finite field extensions with prescribed traces in several intermediate extensions that are also either normal or primitive normal. In the former case, we fully characterize the conditions under which such elements exist and provide an explicit enumeration of these elements. In the latter case we provide asymptotic results.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102745"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416455","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 small densities defined without pseudorandomness","authors":"Thomas Karam","doi":"10.1016/j.ffa.2025.102735","DOIUrl":"10.1016/j.ffa.2025.102735","url":null,"abstract":"<div><div>We identify a new sufficient condition on linear forms <span><math><msub><mrow><mi>ϕ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>:</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>→</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> which guarantees that every subset of <span><math><msup><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span> on which none of <span><math><msub><mrow><mi>ϕ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> has full image has a density which tends to 0 with <em>k</em>. The condition is much weaker than the condition usually used to guarantee that <span><math><mo>(</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> takes each value of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math></span> with probability close to <span><math><msup><mrow><mi>p</mi></mrow><mrow><mo>−</mo><mi>k</mi></mrow></msup></math></span> when <em>x</em> is chosen uniformly at random in the Boolean cube <span><math><msup><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span>. The density is at most quasipolynomially small in <em>k</em>, a bound that is necessarily close to sharp.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102735"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267946","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":"Automorphism groups of the fields of definition of torsion points of elliptic curves in characteristic ≥5","authors":"Bo-Hae Im ,&nbsp;Hansol Kim","doi":"10.1016/j.ffa.2025.102737","DOIUrl":"10.1016/j.ffa.2025.102737","url":null,"abstract":"<div><div>For a field <em>K</em> of characteristic <span><math><mi>p</mi><mo>≥</mo><mn>5</mn></math></span>, let <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>:</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mi>s</mi><mi>x</mi><mo>+</mo><mi>t</mi></math></span> be an elliptic curve defined over the function field <span><math><mi>K</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></math></span> in two variables <em>s</em> and <em>t</em>. For a non-negative positive integer <em>e</em> and a positive integer <em>N</em> which is not divisible by <em>p</em>, we prove that if <span><math><mi>K</mi><mo>⊇</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>alg</mi></mrow></msubsup></math></span>, then the automorphism group of the normal extension <span><math><mi>K</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mrow><mo>(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mrow><mo>[</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup><mi>N</mi><mo>]</mo></mrow><mo>)</mo></mrow></math></span> over <span><math><mi>K</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></math></span> is isomorphic to <span><math><msup><mrow><mo>(</mo><mi>Z</mi><mo>/</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup><mi>Z</mi><mo>)</mo></mrow><mrow><mo>×</mo></mrow></msup><mo>×</mo><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mspace></mspace><mrow><mo>(</mo><mi>Z</mi><mo>/</mo><mi>N</mi><mi>Z</mi><mo>)</mo></mrow></math></span>. Applying this result, we also determine the automorphism group of the normal extension <span><math><mi>K</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mrow><mo>(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mrow><mo>[</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup><mi>N</mi><mo>]</mo></mrow><mo>)</mo></mrow></math></span> for a general field <em>K</em> of characteristic <span><math><mi>p</mi><mo>≥</mo><mn>5</mn></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102737"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267941","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":"Irreducible factorizations of polynomials xpk+1−bx+b over a finite field","authors":"Xue Jia ,&nbsp;Fengwei Li ,&nbsp;Huan Sun ,&nbsp;Qin Yue","doi":"10.1016/j.ffa.2025.102740","DOIUrl":"10.1016/j.ffa.2025.102740","url":null,"abstract":"<div><div>In this paper, we investigate polynomials of the form <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>b</mi></math></span>, where <span><math><mn>0</mn><mo>≠</mo><mi>b</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>, <em>p</em> is a prime, and <em>k</em> divides <em>n</em>. By introducing a new approach based on the projective general linear group, we show that the number of zeros of <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> in <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> belongs to <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>+</mo><mn>1</mn><mo>}</mo></math></span>, and provide explicit criteria on <em>b</em> for each case. We also count the number of such polynomials corresponding to each possible number of zeros. Moreover, for the cases where <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> has at least one zero, we determine its complete irreducible factorization over <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>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102740"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321165","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":"Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, q≡0(mod3)","authors":"Peter Beelen ,&nbsp;Maria Montanucci ,&nbsp;Lara Vicino","doi":"10.1016/j.ffa.2025.102729","DOIUrl":"10.1016/j.ffa.2025.102729","url":null,"abstract":"<div><div>In this article, we explicitly determine the Weierstrass semigroup at any place and the full automorphism group of a known <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>-maximal function field <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, which is realised as a Galois subfield of the Hermitian function field and has the third largest genus, for <span><math><mi>q</mi><mo>≡</mo><mn>0</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>3</mn><mo>)</mo></math></span>. This completes the work contained in <span><span>[3]</span></span> and <span><span>[4]</span></span>, where the cases <span><math><mi>q</mi><mo>≡</mo><mn>2</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>3</mn><mo>)</mo></math></span> and <span><math><mi>q</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>3</mn><mo>)</mo></math></span>, respectively, were studied. Like for these other two cases, the problem of determining the uniqueness of the function field <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, with respect to the value of its genus, is still open. The knowledge of the Weierstrass semigroups may be instrumental in finding a solution to this problem, as it happened to be the case for the function fields with the largest <span><span>[11]</span></span> and second largest genera <span><span>[1]</span></span>, <span><span>[7]</span></span>. Similarly to what observed in <span><span>[3]</span></span> and <span><span>[4]</span></span>, also in the case of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> we find that many different types of Weierstrass semigroups appear, and that the set of Weierstrass places contains also non-<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>-rational places. We also determine <span><math><mrow><mi>Aut</mi></mrow><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span>, which turns out to be exactly the automorphism group inherited from the Hermitian function field, apart from the case <span><math><mi>q</mi><mo>=</mo><mn>3</mn></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102729"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145110037","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":"Recursive construction of projective two-weight linear codes","authors":"Jong Yoon Hyun ,&nbsp;Zhao Hu","doi":"10.1016/j.ffa.2025.102751","DOIUrl":"10.1016/j.ffa.2025.102751","url":null,"abstract":"<div><div>In this paper, we develop a construction method that uses given projective two-weight linear codes to recursively produce new ones. Numerous constructions of projective two-weight linear codes are provided building upon well-known projective two-weight linear codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102751"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466209","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 Erdős-Rado sunflower problem for vector spaces","authors":"Ferdinand Ihringer ,&nbsp;Andrey Kupavskii","doi":"10.1016/j.ffa.2025.102746","DOIUrl":"10.1016/j.ffa.2025.102746","url":null,"abstract":"<div><div>The famous Erdős-Rado sunflower conjecture suggests that an <em>s</em>-sunflower-free family of <em>k</em>-element sets has size at most <span><math><msup><mrow><mo>(</mo><mi>C</mi><mi>s</mi><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msup></math></span> for some absolute constant <em>C</em>. In this note, we investigate the analog problem for <em>k</em>-spaces over the field with <em>q</em> elements. For <span><math><mi>s</mi><mo>≥</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></span>, we show that the largest <em>s</em>-sunflower-free family <span><math><mi>F</mi></math></span> satisfies<span><span><span><math><mn>1</mn><mo>≤</mo><mo>|</mo><mi>F</mi><mo>|</mo><mo>/</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>(</mo><mi>s</mi><mo>−</mo><mn>1</mn><mo>)</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mi>k</mi></mrow></msup><mo>≤</mo><msup><mrow><mo>(</mo><mi>q</mi><mo>/</mo><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msup><mo>.</mo></math></span></span></span> For <span><math><mi>s</mi><mo>≤</mo><mi>k</mi></math></span>, we show that<span><span><span><math><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></mrow></msup><mo>≤</mo><mo>|</mo><mi>F</mi><mo>|</mo><mo>/</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>(</mo><mi>s</mi><mo>−</mo><mn>1</mn><mo>)</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mi>k</mi></mrow></msup><mo>≤</mo><msup><mrow><mo>(</mo><mi>q</mi><mo>/</mo><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msup><mo>.</mo></math></span></span></span> Our lower bounds rely on an iterative construction that uses lifted maximum rank-distance (MRD) codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102746"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416453","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":"Trace duality and additive complementary pairs of additive cyclic codes over finite chain rings","authors":"Sanjit Bhowmick ,&nbsp;Kuntal Deka ,&nbsp;Alexandre Fotue Tabue ,&nbsp;Edgar Martínez-Moro","doi":"10.1016/j.ffa.2025.102732","DOIUrl":"10.1016/j.ffa.2025.102732","url":null,"abstract":"<div><div>This paper investigates the algebraic structure of complementary pairs of additive cyclic codes over a finite commutative chain ring of odd characteristic. We demonstrate that for every additive complementary pair of additive codes, both constituent codes are free modules. Moreover, we present a necessary and sufficient condition for a pair of additive codes over a finite commutative chain ring of odd characteristic to form an additive complementary pair. Finally, we show that, in the case of a complementary pair of additive cyclic codes over a finite chain ring of odd characteristic, one of the codes is permutation equivalent to the trace dual of the other.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102732"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145221462","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 third generalized covering radius for binary primitive double-error-correcting BCH codes","authors":"Ferruh Özbudak,&nbsp;İlknur Öztürk","doi":"10.1016/j.ffa.2025.102749","DOIUrl":"10.1016/j.ffa.2025.102749","url":null,"abstract":"<div><div>We prove that the third generalized covering radius of binary primitive double-error-correcting BCH codes of length <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn></math></span> is 7 if <span><math><mi>m</mi><mo>≥</mo><mn>8</mn></math></span> is an even integer. We also prove that the third generalized covering radius of binary primitive double-error-correcting BCH codes of length <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn></math></span> is either 6 or 7 if <span><math><mi>m</mi><mo>≥</mo><mn>9</mn></math></span> is an odd integer. We use some methods derived from the theory of algebraic curves over finite fields in our proofs and we obtain some further related results.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102749"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416451","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":"Finite fields whose members are the sum of a potent and a 4-potent","authors":"Stephen D. Cohen ,&nbsp;Peter V. Danchev ,&nbsp;Tomás Oliveira e Silva","doi":"10.1016/j.ffa.2025.102739","DOIUrl":"10.1016/j.ffa.2025.102739","url":null,"abstract":"<div><div>We classify those finite fields <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> whose members are the sum of an <em>n</em>-potent element with <span><math><mi>n</mi><mo>&gt;</mo><mn>1</mn></math></span> and a 4-potent element. It is shown that there are precisely ten non-trivial pairs <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> for which this is the case. This continues a recent publication by Abyzov et al. (2024) <span><span>[1]</span></span> in which the tripotent version was examined in-depth, inasmuch as it extends recent results in this seam of research established by Abyzov and Tapkin (2024) <span><span>[4]</span></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102739"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145362887","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}