Pub Date : 2025-11-10DOI: 10.1016/j.ffa.2025.102758
Xiujing Zheng, Sujuan Huang, Shixin Zhu
The linear l-intersection pairs of codes are a generalization of linear complementary dual (LCD) codes, hulls of codes and linear complementary pairs (LCPs) of codes. Matrix-product codes are extended versions derived from shorter codes through matrix-product techniques. In this paper, we investigate linear l-intersection pairs of matrix-product codes. The characterization of these pairs can be achieved by examining the dimension of the intersection between their respective constituent codes. For the dimension part of the conjecture for linear l-intersection pairs of codes proposed by Guenda et al. (Des Codes Cryptogr. 88: 133-152, 2020), we prove that if the conjecture holds for prime lengths, then its dimension part holds. As a practical application, linear l-intersection pairs of matrix-product codes are utilized to the constructions of asymmetric quantum error-correcting (AQEC) codes and asymmetric entanglement-assisted quantum error-correcting (AEAQEC) codes. Some instances exhibit favorable parameters.
线性l交码对是线性互补对偶码、码壳和码的线性互补对的推广。矩阵积代码是通过矩阵积技术从较短的代码衍生而来的扩展版本。本文研究了矩阵积码的线性l交对。这些对的特征可以通过检查它们各自组成代码之间的交集的维度来实现。对于Guenda et al. （Des codes Cryptogr. 88: 133- 152,2020）提出的线性l交码对猜想的维数部分，证明了如果该猜想对于素数长度成立，则其维数部分成立。在实际应用中，将线性l交对矩阵积码用于构造非对称量子纠错码和非对称纠缠辅助量子纠错码。有些实例显示出有利的参数。
{"title":"Linear l-intersection pairs of matrix-product codes and their applications","authors":"Xiujing Zheng, Sujuan Huang, Shixin Zhu","doi":"10.1016/j.ffa.2025.102758","DOIUrl":"10.1016/j.ffa.2025.102758","url":null,"abstract":"<div><div>The linear <em>l</em>-intersection pairs of codes are a generalization of linear complementary dual (LCD) codes, hulls of codes and linear complementary pairs (LCPs) of codes. Matrix-product codes are extended versions derived from shorter codes through matrix-product techniques. In this paper, we investigate linear <em>l</em>-intersection pairs of matrix-product codes. The characterization of these pairs can be achieved by examining the dimension of the intersection between their respective constituent codes. For the dimension part of the conjecture for linear <em>l</em>-intersection pairs of codes proposed by Guenda et al. (Des Codes Cryptogr. <strong>88</strong>: 133-152, 2020), we prove that if the conjecture holds for prime lengths, then its dimension part holds. As a practical application, linear <em>l</em>-intersection pairs of matrix-product codes are utilized to the constructions of asymmetric quantum error-correcting (AQEC) codes and asymmetric entanglement-assisted quantum error-correcting (AEAQEC) codes. Some instances exhibit favorable parameters.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102758"},"PeriodicalIF":1.2,"publicationDate":"2025-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145519892","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 : 2025-11-07DOI: 10.1016/j.ffa.2025.102756
Robert S. Coulter , Paul Hearding
Given a subset S of a finite field, an S-complete mapping is a polynomial for which is a permutation polynomial over the finite field for each . In this paper, we introduce a new method for constructing permutation polynomials and use it to establish a class of S-complete mappings with “large” S.
{"title":"On S-complete mappings for large S","authors":"Robert S. Coulter , Paul Hearding","doi":"10.1016/j.ffa.2025.102756","DOIUrl":"10.1016/j.ffa.2025.102756","url":null,"abstract":"<div><div>Given a subset <em>S</em> of a finite field, an <em>S</em>-complete mapping is a polynomial <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> for which <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>+</mo><mi>c</mi><mi>X</mi></math></span> is a permutation polynomial over the finite field for each <span><math><mi>c</mi><mo>∈</mo><mi>S</mi></math></span>. In this paper, we introduce a new method for constructing permutation polynomials and use it to establish a class of <em>S</em>-complete mappings with “large” <em>S</em>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102756"},"PeriodicalIF":1.2,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466211","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 : 2025-11-05DOI: 10.1016/j.ffa.2025.102752
Tong Lin , Qiang Wang
Using arbitrary bases for the finite field over , we obtain the generalized Möbius transformations (GMTs), which are a class of bijections between the projective geometry and the set of roots of unity , where is any integer. We also introduce a class of projective polynomials, using the properties of which we determine the inverses of the GMTs. Moreover, we study the roots of those projective polynomials, which lead to a three-way correspondence between partitions of and . Through this correspondence and the GMTs, we construct permutation polynomials of index over .
{"title":"Permutation polynomials and finite projective spaces","authors":"Tong Lin , Qiang Wang","doi":"10.1016/j.ffa.2025.102752","DOIUrl":"10.1016/j.ffa.2025.102752","url":null,"abstract":"<div><div>Using arbitrary bases for the finite field <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> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, we obtain the generalized Möbius transformations (GMTs), which are a class of bijections between the projective geometry <span><math><mtext>PG</mtext><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> and the set of roots of unity <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msub><mo>⊆</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>, where <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> is any integer. We also introduce a class of projective polynomials, using the properties of which we determine the inverses of the GMTs. Moreover, we study the roots of those projective polynomials, which lead to a three-way correspondence between partitions of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>,</mo><msub><mrow><mi>μ</mi></mrow><mrow><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msub></math></span> and <span><math><mtext>PG</mtext><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>. Through this correspondence and the GMTs, we construct permutation polynomials of index <span><math><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></mfrac></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></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102752"},"PeriodicalIF":1.2,"publicationDate":"2025-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466210","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 : 2025-11-04DOI: 10.1016/j.ffa.2025.102757
Mohammed Rahmani , Abderrahmane Nitaj , Mhammed Ziane
Let be an RSA modulus, and be an integer. Two recently algebraic variants of the RSA cryptosystem use a public exponent e for encryption, and a private exponent d for decryption with , where . In this paper, we propose an attack on the two variants using Coppersmith's method and lattice basis reduction. Our attack breaks the systems when d is less than an explicit bound that depends only on n and N. We analyze the security of the RSA variants characterized by the equation . Specifically, we propose a novel attack utilizing lattice-based methods and Coppersmith's technique, when the prime numbers p and q share an amount of their least significant bits. This enables the efficient recovery of the primes p and q in polynomial time.
{"title":"Cryptanalysis of some algebraic variants of the RSA cryptosystem","authors":"Mohammed Rahmani , Abderrahmane Nitaj , Mhammed Ziane","doi":"10.1016/j.ffa.2025.102757","DOIUrl":"10.1016/j.ffa.2025.102757","url":null,"abstract":"<div><div>Let <span><math><mi>N</mi><mo>=</mo><mi>p</mi><mi>q</mi></math></span> be an RSA modulus, and <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span> be an integer. Two recently algebraic variants of the RSA cryptosystem use a public exponent <em>e</em> for encryption, and a private exponent <em>d</em> for decryption with <span><math><mi>e</mi><mi>d</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><msub><mrow><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>=</mo><mrow><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>. In this paper, we propose an attack on the two variants using Coppersmith's method and lattice basis reduction. Our attack breaks the systems when <em>d</em> is less than an explicit bound that depends only on <em>n</em> and <em>N</em>. We analyze the security of the RSA variants characterized by the equation <span><math><mi>e</mi><mi>d</mi><mo>−</mo><mi>k</mi><msub><mrow><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. Specifically, we propose a novel attack utilizing lattice-based methods and Coppersmith's technique, when the prime numbers <em>p</em> and <em>q</em> share an amount of their least significant bits. This enables the efficient recovery of the primes <em>p</em> and <em>q</em> in polynomial time.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102757"},"PeriodicalIF":1.2,"publicationDate":"2025-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466204","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 : 2025-11-04DOI: 10.1016/j.ffa.2025.102754
Tonghui Zhang , Pinhui Ke , Zuling Chang
Three classes of binary linear codes with at most four nonzero weights were constructed in this paper, in which two of them are projective three-weight codes. As applications, s-sum sets for any odd were constructed.
{"title":"Construction of three class of at most four-weight binary linear codes and their applications","authors":"Tonghui Zhang , Pinhui Ke , Zuling Chang","doi":"10.1016/j.ffa.2025.102754","DOIUrl":"10.1016/j.ffa.2025.102754","url":null,"abstract":"<div><div>Three classes of binary linear codes with at most four nonzero weights were constructed in this paper, in which two of them are projective three-weight codes. As applications, <em>s</em>-sum sets for any odd <span><math><mi>s</mi><mo>></mo><mn>1</mn></math></span> were constructed.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102754"},"PeriodicalIF":1.2,"publicationDate":"2025-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466207","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 : 2025-11-03DOI: 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":"2025-11-03","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 : 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":"2025-11-03","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}
Pub Date : 2025-10-31DOI: 10.1016/j.ffa.2025.102747
Jorge Morales
The relation between cycles of indefinite binary quadratic forms over and continued fractions is classical and well-known. We describe a similar relation for binary quadratic forms over the polynomial ring , where q is a power of an odd prime. In this context, the cycles of the classical theory are replaced by orbits of the metacyclic group acting on the set of reduced forms of a given discriminant, where each orbit corresponds to a proper equivalence class.
{"title":"Continued fractions and indefinite binary quadratic forms over Fq[t]","authors":"Jorge Morales","doi":"10.1016/j.ffa.2025.102747","DOIUrl":"10.1016/j.ffa.2025.102747","url":null,"abstract":"<div><div>The relation between cycles of indefinite binary quadratic forms over <span><math><mi>Z</mi></math></span> and continued fractions is classical and well-known. We describe a similar relation for binary quadratic forms over the polynomial ring <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>t</mi><mo>]</mo></math></span>, where <em>q</em> is a power of an odd prime. In this context, the cycles of the classical theory are replaced by orbits of the metacyclic group <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>⋊</mo><mi>Z</mi></math></span> acting on the set of reduced forms of a given discriminant, where each orbit corresponds to a proper equivalence class.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102747"},"PeriodicalIF":1.2,"publicationDate":"2025-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416452","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 : 2025-10-30DOI: 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":"2025-10-30","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}
Pub 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":"2025-10-30","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}
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