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.
{"title":"Normal and primitive normal elements with prescribed traces in intermediate extensions of finite fields","authors":"Arpan Chandra Mazumder , Giorgos Kapetanakis , 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}
Pub Date : 2026-02-01Epub Date: 2025-10-08DOI: 10.1016/j.ffa.2025.102735
Thomas Karam
We identify a new sufficient condition on linear forms which guarantees that every subset of on which none of has full image has a density which tends to 0 with k. The condition is much weaker than the condition usually used to guarantee that takes each value of with probability close to when x is chosen uniformly at random in the Boolean cube . The density is at most quasipolynomially small in k, a bound that is necessarily close to sharp.
{"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}
Pub Date : 2026-02-01Epub Date: 2025-10-07DOI: 10.1016/j.ffa.2025.102737
Bo-Hae Im , Hansol Kim
For a field K of characteristic , let be an elliptic curve defined over the function field in two variables s and t. For a non-negative positive integer e and a positive integer N which is not divisible by p, we prove that if , then the automorphism group of the normal extension over is isomorphic to . Applying this result, we also determine the automorphism group of the normal extension for a general field K of characteristic .
{"title":"Automorphism groups of the fields of definition of torsion points of elliptic curves in characteristic ≥5","authors":"Bo-Hae Im , 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}
Pub Date : 2026-02-01Epub Date: 2025-10-16DOI: 10.1016/j.ffa.2025.102740
Xue Jia , Fengwei Li , Huan Sun , Qin Yue
In this paper, we investigate polynomials of the form , where , p is a prime, and k divides n. By introducing a new approach based on the projective general linear group, we show that the number of zeros of in belongs to , and provide explicit criteria on b for each case. We also count the number of such polynomials corresponding to each possible number of zeros. Moreover, for the cases where has at least one zero, we determine its complete irreducible factorization over .
{"title":"Irreducible factorizations of polynomials xpk+1−bx+b over a finite field","authors":"Xue Jia , Fengwei Li , Huan Sun , 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}
Pub Date : 2026-02-01Epub Date: 2025-09-22DOI: 10.1016/j.ffa.2025.102729
Peter Beelen , Maria Montanucci , Lara Vicino
In this article, we explicitly determine the Weierstrass semigroup at any place and the full automorphism group of a known -maximal function field , which is realised as a Galois subfield of the Hermitian function field and has the third largest genus, for . This completes the work contained in [3] and [4], where the cases and , respectively, were studied. Like for these other two cases, the problem of determining the uniqueness of the function field , 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 [11] and second largest genera [1], [7]. Similarly to what observed in [3] and [4], also in the case of we find that many different types of Weierstrass semigroups appear, and that the set of Weierstrass places contains also non--rational places. We also determine , which turns out to be exactly the automorphism group inherited from the Hermitian function field, apart from the case .
{"title":"Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, q≡0(mod3)","authors":"Peter Beelen , Maria Montanucci , 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}
Pub Date : 2026-02-01Epub Date: 2025-11-03DOI: 10.1016/j.ffa.2025.102751
Jong Yoon Hyun , Zhao Hu
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.
{"title":"Recursive construction of projective two-weight linear codes","authors":"Jong Yoon Hyun , 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}
Pub Date : 2026-02-01Epub Date: 2025-10-30DOI: 10.1016/j.ffa.2025.102746
Ferdinand Ihringer , Andrey Kupavskii
The famous Erdős-Rado sunflower conjecture suggests that an s-sunflower-free family of k-element sets has size at most for some absolute constant C. In this note, we investigate the analog problem for k-spaces over the field with q elements. For , we show that the largest s-sunflower-free family satisfies For , we show that Our lower bounds rely on an iterative construction that uses lifted maximum rank-distance (MRD) codes.
{"title":"The Erdős-Rado sunflower problem for vector spaces","authors":"Ferdinand Ihringer , 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}
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.
{"title":"Trace duality and additive complementary pairs of additive cyclic codes over finite chain rings","authors":"Sanjit Bhowmick , Kuntal Deka , Alexandre Fotue Tabue , 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}
Pub Date : 2026-02-01Epub Date: 2025-10-30DOI: 10.1016/j.ffa.2025.102749
Ferruh Özbudak, İlknur Öztürk
We prove that the third generalized covering radius of binary primitive double-error-correcting BCH codes of length is 7 if is an even integer. We also prove that the third generalized covering radius of binary primitive double-error-correcting BCH codes of length is either 6 or 7 if 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.
{"title":"The third generalized covering radius for binary primitive double-error-correcting BCH codes","authors":"Ferruh Özbudak, 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}
Pub Date : 2026-02-01Epub Date: 2025-11-03DOI: 10.1016/j.ffa.2025.102755
Bogdan Nica
We obtain transformation formulas for quadratic character sums with quartic and cubic polynomial arguments.
得到了具有四次和三次多项式参数的二次特征和的变换公式。
{"title":"On quadratic character sums over quartics","authors":"Bogdan Nica","doi":"10.1016/j.ffa.2025.102755","DOIUrl":"10.1016/j.ffa.2025.102755","url":null,"abstract":"<div><div>We obtain transformation formulas for quadratic character sums with quartic and cubic polynomial arguments.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102755"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466208","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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