Pub Date : 2025-12-12DOI: 10.1016/j.ffa.2025.102779
Zhonghao Liang, Qunying Liao
Since near maximum distance separable (NMDS) codes have good algebraic properties and excellent error-correcting capabilities, they have been widely used in various fields such as communication systems, data storage, quantum codes, and so on. In this paper, basing on the generator matrix of Roth-Lempel codes, we present two classes of NMDS codes which generalize Han's, Zheng's and Zhou's constructions in 2023 and 2025, respectively. And we also completely determine their weight distributions.
{"title":"Two classes of NMDS codes from Roth-Lempel codes","authors":"Zhonghao Liang, Qunying Liao","doi":"10.1016/j.ffa.2025.102779","DOIUrl":"10.1016/j.ffa.2025.102779","url":null,"abstract":"<div><div>Since near maximum distance separable (NMDS) codes have good algebraic properties and excellent error-correcting capabilities, they have been widely used in various fields such as communication systems, data storage, quantum codes, and so on. In this paper, basing on the generator matrix of Roth-Lempel codes, we present two classes of NMDS codes which generalize Han's, Zheng's and Zhou's constructions in 2023 and 2025, respectively. And we also completely determine their weight distributions.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102779"},"PeriodicalIF":1.2,"publicationDate":"2025-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737315","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 : 2025-12-11DOI: 10.1016/j.ffa.2025.102781
Tetsushi Ito , Ren Tatematsu , Takahiro Tsushima
The van der Geer–van der Vlugt curves are Artin–Schreier coverings of the affine line defined by linearized polynomials over finite fields. We provide several criteria for them to be maximal or minimal, i.e. attaining the upper or lower bound in the Hasse–Weil inequalities. As applications, we identify several maximal (or minimal) curves within this family. Our proofs are based on an explicit formula for the L-polynomials, recently obtained by Takeuchi and the third author.
van der Geer-van der Vlugt曲线是有限域上由线性化多项式定义的仿射线的Artin-Schreier覆盖。我们给出了它们最大或最小的几个标准，即达到Hasse-Weil不等式的上界或下界。作为应用，我们在这个族中确定了几个最大（或最小）曲线。我们的证明是基于最近由Takeuchi和第三作者获得的l多项式的显式公式。
{"title":"Criteria of maximality and minimality of van der Geer–van der Vlugt curves","authors":"Tetsushi Ito , Ren Tatematsu , Takahiro Tsushima","doi":"10.1016/j.ffa.2025.102781","DOIUrl":"10.1016/j.ffa.2025.102781","url":null,"abstract":"<div><div>The van der Geer–van der Vlugt curves are Artin–Schreier coverings of the affine line defined by linearized polynomials over finite fields. We provide several criteria for them to be maximal or minimal, i.e. attaining the upper or lower bound in the Hasse–Weil inequalities. As applications, we identify several maximal (or minimal) curves within this family. Our proofs are based on an explicit formula for the <em>L</em>-polynomials, recently obtained by Takeuchi and the third author.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102781"},"PeriodicalIF":1.2,"publicationDate":"2025-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737313","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 : 2025-12-11DOI: 10.1016/j.ffa.2025.102777
Sujata Bansal, Pramod Kumar Kewat
In this paper, we construct a family of optimal linear codes over with parameters , where e is a positive integer and . We determine the duals of these codes and establish that for , these codes are dual-containing. This property makes them suitable for the construction of CSS quantum error-correcting codes. Furthermore, we calculate the weight distribution of the duals of these codes and show that the duals are 3-weight codes. We derive the weight enumerator of these codes using the MacWilliams identities. Additionally, we establish that these codes are reversible for all . This ensures the symmetry in the code structure and facilitates them for the possible applications in DNA computing and bidirectional communication systems. The optimality, duality, and reversibility of this family of codes highlight the potential of these codes for various practical and theoretical applications in the error correction.
{"title":"A family of optimal dual-containing and reversible linear codes over F4","authors":"Sujata Bansal, Pramod Kumar Kewat","doi":"10.1016/j.ffa.2025.102777","DOIUrl":"10.1016/j.ffa.2025.102777","url":null,"abstract":"<div><div>In this paper, we construct a family of optimal linear codes over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> with parameters <span><math><mo>[</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>e</mi></mrow></msup><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>e</mi></mrow></msup><mo>−</mo><mi>e</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>]</mo></math></span>, where <em>e</em> is a positive integer and <span><math><mi>e</mi><mo>≥</mo><mn>2</mn></math></span>. We determine the duals of these codes and establish that for <span><math><mi>e</mi><mo>≥</mo><mn>3</mn></math></span>, these codes are dual-containing. This property makes them suitable for the construction of CSS quantum error-correcting codes. Furthermore, we calculate the weight distribution of the duals of these codes and show that the duals are 3-weight codes. We derive the weight enumerator of these codes using the MacWilliams identities. Additionally, we establish that these codes are reversible for all <span><math><mi>e</mi><mo>≥</mo><mn>2</mn></math></span>. This ensures the symmetry in the code structure and facilitates them for the possible applications in DNA computing and bidirectional communication systems. The optimality, duality, and reversibility of this family of codes highlight the potential of these codes for various practical and theoretical applications in the error correction.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102777"},"PeriodicalIF":1.2,"publicationDate":"2025-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737312","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 : 2025-12-11DOI: 10.1016/j.ffa.2025.102768
Sarah Arpin , Ross Bowden , James Clements , Wissam Ghantous , Jason T. LeGrow , Krystal Maughan
Supersingular elliptic curve isogeny graphs underlie isogeny-based cryptography. For isogenies of a single prime degree ℓ, their structure has been investigated graph-theoretically. We generalise the notion of ℓ-isogeny graphs to L-isogeny graphs (studied in the prime field case by Delfs and Galbraith), where L is a set of small primes dictating the allowed isogeny degrees in the graph. We analyse the graph-theoretic structure of L-isogeny graphs. Our approaches may be put into two categories: cycles and graph cuts.
On the topic of cycles, we provide: a count for the number of cycles in the L-isogeny graph with cyclic kernels using traces of Brandt matrices; an efficiently computable estimate based on this approach; and a third ideal-theoretic count for a certain subclass of L-isogeny cycles. We provide code to compute each of these three counts.
On the topic of graph cuts, we compare several algorithms to compute graph cuts which minimise a measure called the edge expansion, outlining a cryptographic motivation for doing so. Our results show that a greedy neighbour algorithm out-performs standard spectral algorithms for computing optimal graph cuts. We provide code and study explicit examples.
Furthermore, we describe several directions of active and future research.
{"title":"Cycles and cuts in supersingular L-isogeny graphs","authors":"Sarah Arpin , Ross Bowden , James Clements , Wissam Ghantous , Jason T. LeGrow , Krystal Maughan","doi":"10.1016/j.ffa.2025.102768","DOIUrl":"10.1016/j.ffa.2025.102768","url":null,"abstract":"<div><div>Supersingular elliptic curve isogeny graphs underlie isogeny-based cryptography. For isogenies of a single prime degree <em>ℓ</em>, their structure has been investigated graph-theoretically. We generalise the notion of <em>ℓ</em>-isogeny graphs to <em>L</em>-isogeny graphs (studied in the prime field case by Delfs and Galbraith), where <em>L</em> is a set of small primes dictating the allowed isogeny degrees in the graph. We analyse the graph-theoretic structure of <em>L</em>-isogeny graphs. Our approaches may be put into two categories: cycles and graph cuts.</div><div>On the topic of cycles, we provide: a count for the number of cycles in the <em>L</em>-isogeny graph with cyclic kernels using traces of Brandt matrices; an efficiently computable estimate based on this approach; and a third ideal-theoretic count for a certain subclass of <em>L</em>-isogeny cycles. We provide code to compute each of these three counts.</div><div>On the topic of graph cuts, we compare several algorithms to compute graph cuts which minimise a measure called the <em>edge expansion</em>, outlining a cryptographic motivation for doing so. Our results show that a <em>greedy neighbour</em> algorithm out-performs standard spectral algorithms for computing optimal graph cuts. We provide code and study explicit examples.</div><div>Furthermore, we describe several directions of active and future research.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102768"},"PeriodicalIF":1.2,"publicationDate":"2025-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737314","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 : 2025-12-03DOI: 10.1016/j.ffa.2025.102767
Kaimin Cheng , Arne Winterhof
Let q be a prime power and r a positive even integer. Let be the finite field with q elements and be its extension field of degree r. Let χ be a nontrivial multiplicative character of and a polynomial over with exactly one simple root in . In this paper, we improve estimates for character sums , where is either a subset of of sparse elements, with respect to some fixed basis of which contains a basis of , or a subset avoiding affine hyperplanes in general position. While such sums have been previously studied, our approach yields sharper bounds by reducing them to sums over the subfield rather than sums over general linear spaces. These estimates can be used to prove the existence of primitive elements in in the standard way.
{"title":"Multiplicative character sums over two classes of subsets of quadratic extensions of finite fields","authors":"Kaimin Cheng , Arne Winterhof","doi":"10.1016/j.ffa.2025.102767","DOIUrl":"10.1016/j.ffa.2025.102767","url":null,"abstract":"<div><div>Let <em>q</em> be a prime power and <em>r</em> a positive even integer. Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> be the finite field with <em>q</em> elements and <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span> be its extension field of degree <em>r</em>. Let <em>χ</em> be a nontrivial multiplicative character of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span> and <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> a polynomial over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span> with exactly one simple root in <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span>. In this paper, we improve estimates for character sums <span><math><munder><mo>∑</mo><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></munder><mi>χ</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>g</mi><mo>)</mo><mo>)</mo></math></span>, where <span><math><mi>G</mi></math></span> is either a subset of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span> of sparse elements, with respect to some fixed basis of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span> which contains a basis of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi><mo>/</mo><mn>2</mn></mrow></msup></mrow></msub></math></span>, or a subset avoiding affine hyperplanes in general position. While such sums have been previously studied, our approach yields sharper bounds by reducing them to sums over the subfield <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi><mo>/</mo><mn>2</mn></mrow></msup></mrow></msub></math></span> rather than sums over general linear spaces. These estimates can be used to prove the existence of primitive elements in <span><math><mi>G</mi></math></span> in the standard way.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102767"},"PeriodicalIF":1.2,"publicationDate":"2025-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145684471","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 : 2025-12-02DOI: 10.1016/j.ffa.2025.102766
I. Cardinali , L. Giuzzi
Let V be a vector space over the finite field with q elements and Λ be the image of the Segre geometry in under the Segre map. Consider the subvariety of Λ represented by the pure tensors with and such that . Regarding as a projective system of , we study the linear code arising from it. We show that is a minimal code and we determine its basic parameters, its full weight list and its linear automorphism group. We also give a geometrical characterization of its minimum and second lowest weight codewords as well as of some of the words of maximum weight.
{"title":"Linear codes arising from the point-hyperplane geometry-Part I: The Segre embedding","authors":"I. Cardinali , L. Giuzzi","doi":"10.1016/j.ffa.2025.102766","DOIUrl":"10.1016/j.ffa.2025.102766","url":null,"abstract":"<div><div>Let <em>V</em> be a vector space over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> with <em>q</em> elements and Λ be the image of the Segre geometry <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mi>V</mi><mo>)</mo><mo>⊗</mo><mrow><mi>PG</mi></mrow><mo>(</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo></math></span> in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mi>V</mi><mo>⊗</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo></math></span> under the Segre map. Consider the subvariety <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> of Λ represented by the pure tensors <span><math><mi>x</mi><mo>⊗</mo><mi>ξ</mi></math></span> with <span><math><mi>x</mi><mo>∈</mo><mi>V</mi></math></span> and <span><math><mi>ξ</mi><mo>∈</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> such that <span><math><mi>ξ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. Regarding <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> as a projective system of <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mi>V</mi><mo>⊗</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo></math></span>, we study the linear code <span><math><mi>C</mi><mo>(</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> arising from it. We show that <span><math><mi>C</mi><mo>(</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> is a minimal code and we determine its basic parameters, its full weight list and its linear automorphism group. We also give a geometrical characterization of its minimum and second lowest weight codewords as well as of some of the words of maximum weight.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102766"},"PeriodicalIF":1.2,"publicationDate":"2025-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145684470","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 : 2025-12-01DOI: 10.1016/j.ffa.2025.102760
Liqin Qian , Minjia Shi , Xiwang Cao
In this paper, we propose several classes of permutation polynomials of the form over , where , and . The permutation behavior of the proposed polynomials is investigated by the AGW criterion and determination of the number of solutions to certain equations over . Based on an effective method proposed by Mesnager (2014), we construct several classes of involutions and further obtain some self-dual bent functions by employing three permutations of satisfying an algebraic property . Finally, it is worth pointing out that there exist examples of bent functions we obtained which do not belong to .
{"title":"Permutation polynomials and involutions over the finite field F22m","authors":"Liqin Qian , Minjia Shi , Xiwang Cao","doi":"10.1016/j.ffa.2025.102760","DOIUrl":"10.1016/j.ffa.2025.102760","url":null,"abstract":"<div><div>In this paper, we propose several classes of permutation polynomials of the form <span><math><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><msup><mrow><mo>(</mo><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msubsup><msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup><mo>+</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup><mo>+</mo><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></msub></math></span>, where <span><math><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>a</mi><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>b</mi><mi>x</mi></math></span>, <span><math><mi>a</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></msub></math></span> and <span><math><mi>b</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. The permutation behavior of the proposed polynomials is investigated by the AGW criterion and determination of the number of solutions to certain equations over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></msub></math></span>. Based on an effective method proposed by Mesnager (2014), we construct several classes of involutions and further obtain some self-dual bent functions by employing three permutations of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></msub></math></span> satisfying an algebraic property <span><math><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msub><mo>)</mo></math></span>. Finally, it is worth pointing out that there exist examples of bent functions we obtained which do not belong to <span><math><mi>M</mi><msup><mrow><mi>M</mi></mrow><mrow><mi>#</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102760"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145684472","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 : 2025-12-01DOI: 10.1016/j.ffa.2025.102765
Haode Yan
In this paper, we study the duals of BCH codes and . By analyzing the number of solutions to certain equations over the multiplicative subgroup of , we determine the possible weights of codewords in and , respectively. The weight distributions of these two dual codes are derived by applying the Pless power moments. Our results provide affirmative solutions to recent conjectures.
{"title":"On two conjectures of the duals of AMDS BCH codes","authors":"Haode Yan","doi":"10.1016/j.ffa.2025.102765","DOIUrl":"10.1016/j.ffa.2025.102765","url":null,"abstract":"<div><div>In this paper, we study the duals of BCH codes <span><math><msub><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mfrac><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mo>(</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>s</mi></mrow></msup><mo>,</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>s</mi></mrow></msup><mo>+</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow></msub></math></span>. By analyzing the number of solutions to certain equations over the multiplicative subgroup <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mo>{</mo><mi>x</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>:</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>1</mn><mo>}</mo></math></span> of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>, we determine the possible weights of codewords in <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mfrac><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mrow><mo>⊥</mo></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mo>(</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>s</mi></mrow></msup><mo>,</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>s</mi></mrow></msup><mo>+</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow><mrow><mo>⊥</mo></mrow></msubsup></math></span>, respectively. The weight distributions of these two dual codes are derived by applying the Pless power moments. Our results provide affirmative solutions to recent conjectures.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102765"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145684565","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 : 2025-12-01DOI: 10.1016/j.ffa.2025.102764
Fengwei Li, Yuting Liu, Ruiyuan Jiang
Let be the finite field with q elements, where q is a power of an odd prime p. In this paper, we provide a method to construct Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes and extended GRS codes, which their support sets are roots of polynomials from affine and projective linear transformation over . Moreover, we construct three classes of quantum maximum distance separable (MDS) codes with minimum distances . Some of these quantum MDS codes have not been obtained before, and in some cases, have larger minimum distances and higher efficiency than the well-known quantum MDS codes.
{"title":"Quantum MDS codes induced by the projective linear transformation","authors":"Fengwei Li, Yuting Liu, Ruiyuan Jiang","doi":"10.1016/j.ffa.2025.102764","DOIUrl":"10.1016/j.ffa.2025.102764","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> be the finite field with <em>q</em> elements, where <em>q</em> is a power of an odd prime <em>p</em>. In this paper, we provide a method to construct Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes and extended GRS codes, which their support sets are roots of polynomials from affine and projective linear transformation over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>. Moreover, we construct three classes of quantum maximum distance separable (MDS) codes with minimum distances <span><math><mo>></mo><mfrac><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mn>1</mn></math></span>. Some of these quantum MDS codes have not been obtained before, and in some cases, have larger minimum distances and higher efficiency than the well-known quantum MDS codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102764"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145684564","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 : 2025-11-26DOI: 10.1016/j.ffa.2025.102763
Krishna Kaipa , Puspendu Pradhan
We consider the problem of classifying the lines of the projective 3-space over a finite field into orbits of the group of linear symmetries of the twisted cubic C. The problem has been solved in literature in characteristic different from 3, and in this work, we solve the problem in characteristic 3. We reduce this problem to another problem, which is the classification of binary quartic forms into -orbits. We first solve the latter problem and use to solve the former problem. We also obtain the point-line and the line-plane incidence structures of the point, line, and plane orbits.
{"title":"On the PGL2(q)-orbits of lines of PG(3,q) and binary quartic forms in characteristic three","authors":"Krishna Kaipa , Puspendu Pradhan","doi":"10.1016/j.ffa.2025.102763","DOIUrl":"10.1016/j.ffa.2025.102763","url":null,"abstract":"<div><div>We consider the problem of classifying the lines of the projective 3-space <span><math><mi>P</mi><mi>G</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> into orbits of the group <span><math><mi>P</mi><mi>G</mi><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> of linear symmetries of the twisted cubic <em>C</em>. The problem has been solved in literature in characteristic different from 3, and in this work, we solve the problem in characteristic 3. We reduce this problem to another problem, which is the classification of binary quartic forms into <span><math><mi>P</mi><mi>G</mi><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span>-orbits. We first solve the latter problem and use to solve the former problem. We also obtain the point-line and the line-plane incidence structures of the point, line, and plane orbits.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102763"},"PeriodicalIF":1.2,"publicationDate":"2025-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145624274","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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