{"title":"Some classes of permutation pentanomials","authors":"Zhiguo Ding , Michael E. Zieve","doi":"10.1016/j.ffa.2025.102619","DOIUrl":"10.1016/j.ffa.2025.102619","url":null,"abstract":"<div><div>For each prime <span><math><mi>p</mi><mo>≠</mo><mn>3</mn></math></span> and each power <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>, we present two large classes of permutation polynomials 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> of the form <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>r</mi></mrow></msup><mi>B</mi><mo>(</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> which have at most five terms, where <span><math><mi>B</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is a polynomial with coefficients in <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>}</mo></math></span>. The special case <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span> of our results comprises a vast generalization of 76 recent results and conjectures in the literature. In case <span><math><mi>p</mi><mo>></mo><mn>2</mn></math></span>, no instances of our permutation polynomials have appeared in the literature, and the construction of such polynomials had been posed as an open problem. Our proofs are short and involve no computations, in contrast to the proofs of many of the special cases of our results which were published previously.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"106 ","pages":"Article 102619"},"PeriodicalIF":1.2,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143705967","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 self-dual completely regular codes with covering radius ρ ≤ 3","authors":"J. Borges , V. Zinoviev","doi":"10.1016/j.ffa.2025.102617","DOIUrl":"10.1016/j.ffa.2025.102617","url":null,"abstract":"<div><div>We give a complete classification of self-dual completely regular codes with covering radius <span><math><mi>ρ</mi><mo>≤</mo><mn>3</mn></math></span>. For <span><math><mi>ρ</mi><mo>=</mo><mn>1</mn></math></span> the results are almost trivial. For <span><math><mi>ρ</mi><mo>=</mo><mn>2</mn></math></span>, by using properties of the more general class of uniformly packed codes in the wide sense, we show that there are two sporadic such codes, of length 8, and an infinite family, of length 4, apart from the direct sum of two self-dual completely regular codes with <span><math><mi>ρ</mi><mo>=</mo><mn>1</mn></math></span>, each one. For <span><math><mi>ρ</mi><mo>=</mo><mn>3</mn></math></span>, in some cases, we use similar techniques to the ones used for <span><math><mi>ρ</mi><mo>=</mo><mn>2</mn></math></span>. However, for some other cases we use different methods, namely, the Pless power moments which allow to us to discard several possibilities. We show that there are only two self-dual completely regular codes with <span><math><mi>ρ</mi><mo>=</mo><mn>3</mn></math></span> and <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>, which are both ternary: the extended ternary Golay code and the direct sum of three ternary Hamming codes of length 4. Therefore, any self-dual completely regular code with <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span> and <span><math><mi>ρ</mi><mo>=</mo><mn>3</mn></math></span> is ternary and has length 12.</div><div>We provide the intersection arrays for all such codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102617"},"PeriodicalIF":1.2,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704076","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 differential uniformity of the power functions xpn+52 over Fpn","authors":"Wenping Yuan , Xiaoni Du , Huan Zhou , Xingbin Qiao","doi":"10.1016/j.ffa.2025.102622","DOIUrl":"10.1016/j.ffa.2025.102622","url":null,"abstract":"<div><div>Cryptographic functions with low differential uniformity have important applications in designing S-box in the block ciphers. In this paper, we mainly investigate the differential uniformity <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>F</mi></mrow></msub></math></span> on a new class of power mappings <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mfrac><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span> 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> with <em>p</em> being an odd prime and <em>n</em> being a positive integer. More precisely, for <span><math><mi>p</mi><mo>=</mo><mn>3</mn></math></span>, the differential uniformity and the differential spectrum of <em>F</em> have been determined explicitly. The results indicate that <em>F</em> is a locally-PN function with differentially <span><math><mfrac><mrow><msup><mrow><mn>3</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math></span>-uniform when <em>n</em> is odd and a locally-APN function with differentially <span><math><mfrac><mrow><msup><mrow><mn>3</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math></span>-uniform when <em>n</em> is even. Then, for <span><math><mi>p</mi><mo>=</mo><mn>5</mn></math></span>, we prove that <em>F</em> is APN for even <em>n</em> and <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>=</mo><mn>6</mn></math></span> for odd <em>n</em> through specific differential equations and quadratic character over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>5</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>. The method is different from the existing one. Moreover, for primes <span><math><mi>p</mi><mo>></mo><mn>5</mn></math></span>, we show that <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>≤</mo><mn>5</mn></math></span> when <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>≡</mo><mn>3</mn><mspace></mspace><mo>(</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mspace></mspace><mn>4</mn><mo>)</mo></math></span> and <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>≤</mo><mn>8</mn></math></span> when <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mspace></mspace><mn>4</mn><mo>)</mo></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102622"},"PeriodicalIF":1.2,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143716154","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":"Zeros of a system of diagonal polynomials over finite fields","authors":"Yulu Feng","doi":"10.1016/j.ffa.2025.102623","DOIUrl":"10.1016/j.ffa.2025.102623","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 of characteristic <em>p</em>, having <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>a</mi></mrow></msup></math></span> elements and let <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span> be the unit group of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. Let <span><math><mi>N</mi><mo>(</mo><mi>V</mi><mo>)</mo></math></span> be the number of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-rational points of the affine algebraic variety defined by the simultaneous vanishing of the diagonal polynomials <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub></mrow></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>n</mi></mrow></msub><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi><mi>n</mi></mrow></msub></mrow></msubsup><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>, <span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>r</mi></math></span>, where <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>, <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math></span> is a nonnegative integer for <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>r</mi><mo>,</mo><mn>1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>n</mi></math></span>. By using properties of Teichmüller representations and the Stickelberger relation applied by Ax and Wan, we show that<span><span><span><math><msub><mrow><mi>ord</mi></mrow><mrow><mi>q</mi></mrow></msub><mi>N</mi><mo>(</mo><mi>V</mi><mo>)</mo><mo>≥</mo><mo>⌈</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mfrac><mrow><mn>1</mn></mrow><mrow><munder><mi>max</mi><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>r</mi></mrow></munder><mo></mo><mo>{</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>}</mo></mrow></mfrac><mo>⌉</mo><mo>−</mo><mi>r</mi></math></span></span></span> if <span><math><msub><mrow><mi>max</mi></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>r</mi></mrow></msub><mo></mo><mo>{</mo><msub><mrow><mi>d</mi></mrow><","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"106 ","pages":"Article 102623"},"PeriodicalIF":1.2,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704040","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":"Fagundes-Mello conjecture over a finite field","authors":"Qian Chen , Yingyu Luo , Yu Wang","doi":"10.1016/j.ffa.2025.102620","DOIUrl":"10.1016/j.ffa.2025.102620","url":null,"abstract":"<div><div>The Fagundes-Mello conjecture asserts that the image of every multilinear polynomial on upper triangular matrix algebras over a field is a vector space, which is an important variation of the old and famous Lvov-Kaplansky conjecture. The Fagundes-Mello conjecture has been solved over an infinite field by Gargate and Mello in 2022. In the present paper, we shall give a result on evaluations of homogenous polynomials with commutative variables over a finite field. As an application, we give a positive answer of the Fagundes-Mello conjecture under a mild condition on the ground field, which covers all results on the Fagundes-Mello conjecture.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102620"},"PeriodicalIF":1.2,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143696631","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":"An elementary proof of Bridy's theorem","authors":"Eric Rowland , Manon Stipulanti , Reem Yassawi","doi":"10.1016/j.ffa.2025.102621","DOIUrl":"10.1016/j.ffa.2025.102621","url":null,"abstract":"<div><div>Christol's theorem states that a power series with coefficients in a finite field is algebraic if and only if its coefficient sequence is automatic. A natural question is how the size of a polynomial describing such a sequence relates to the size of an automaton describing the same sequence. Bridy used tools from algebraic geometry to bound the size of the minimal automaton for a sequence, given its minimal polynomial. We produce a new proof of Bridy's bound by embedding algebraic sequences as diagonals of rational functions.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102621"},"PeriodicalIF":1.2,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143696632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"BRK-type sets over finite fields","authors":"Charlotte Trainor","doi":"10.1016/j.ffa.2025.102624","DOIUrl":"10.1016/j.ffa.2025.102624","url":null,"abstract":"<div><div>A Besicovitch-Rado-Kinney (BRK) set in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is a Borel set that contains a <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-dimensional sphere of radius <em>r</em>, for each <span><math><mi>r</mi><mo>></mo><mn>0</mn></math></span>. It is known that such sets have Hausdorff dimension <em>n</em> from the work of Kolasa and Wolff. In this paper, we consider an analogous problem over a finite field, <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. We define BRK-type sets in <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>, and establish lower bounds on the size of such sets using techniques introduced by Dvir's proof of the finite field Kakeya conjecture.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102624"},"PeriodicalIF":1.2,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143696630","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":"Non-isomorphic maximal function fields of genus q − 1","authors":"Jonathan Niemann","doi":"10.1016/j.ffa.2025.102618","DOIUrl":"10.1016/j.ffa.2025.102618","url":null,"abstract":"<div><div>The classification of maximal function fields over a finite field is a difficult open problem, and even determining isomorphism classes among known function fields is challenging in general. We study a particular family of maximal function fields defined over a finite field with <span><math><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> elements, where <em>q</em> is the power of an odd prime. When <span><math><mi>d</mi><mo>:</mo><mo>=</mo><mo>(</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></math></span> is a prime, this family is known to contain a large number of non-isomorphic function fields of the same genus and with the same automorphism group. We compute the automorphism group and isomorphism classes also in the case where <em>d</em> is not a prime.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"106 ","pages":"Article 102618"},"PeriodicalIF":1.2,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143684818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Characterizations of NMDS codes and a proof of the Geng-Yang-Zhang-Zhou conjecture","authors":"Shiyuan Qiang, Huakai Wei, Shaofang Hong","doi":"10.1016/j.ffa.2025.102616","DOIUrl":"10.1016/j.ffa.2025.102616","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 of <em>q</em> elements, where <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> with <em>p</em> being a prime number and <em>m</em> being a positive integer. Let <span><math><msub><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi>h</mi><mo>)</mo></mrow></msub></math></span> be a class of BCH codes of length <em>n</em> and designed distance <em>δ</em>. A linear code <span><math><mi>C</mi></math></span> is said to be maximum distance separable (MDS) if the minimum distance <span><math><mi>d</mi><mo>=</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></span>. If <span><math><mi>d</mi><mo>=</mo><mi>n</mi><mo>−</mo><mi>k</mi></math></span>, then <span><math><mi>C</mi></math></span> is called an almost MDS (AMDS) code. Moreover, if both of <span><math><mi>C</mi></math></span> and its dual code <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⊥</mo></mrow></msup></math></span> are AMDS, then <span><math><mi>C</mi></math></span> is called a near MDS (NMDS) code. In <span><span>[9]</span></span>, Geng, Yang, Zhang and Zhou proved that the BCH code <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><mn>4</mn><mo>)</mo></mrow></msub></math></span> is an almost MDS code, where <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span> and <em>m</em> is an odd integer, and they also showed that its parameters is <span><math><mo>[</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>−</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>]</mo></math></span>. Furthermore, they proposed a conjecture stating that the dual code <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><mn>4</mn><mo>)</mo></mrow><mrow><mo>⊥</mo></mrow></msubsup></math></span> is also an AMDS code with parameters <span><math><mo>[</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>,</mo><mi>q</mi><mo>−</mo><mn>3</mn><mo>]</mo></math></span>. In this paper, we introduce the concept of subset code and use it together with the MacWilliams identity to establish characterizations for the dual code of an AMDS code to be an AMDS code. Then by this criteria, we show that the Geng-Yang-Zhang-Zhou conjecture is true. Our result together with the Geng-Yang-Zhang-Zhou theorem implies that the BCH code <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><mn>4</mn><mo>)</mo></mrow></msub></math></span> is an NMDS code.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102616"},"PeriodicalIF":1.2,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143619240","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":"Central and non-central metacyclic codes","authors":"Seema Chahal, Sugandha Maheshwary","doi":"10.1016/j.ffa.2025.102615","DOIUrl":"10.1016/j.ffa.2025.102615","url":null,"abstract":"<div><div>In this article, the primitive central idempotents of finite semisimple group algebra of split metacyclic groups are explicitly written and their supports are evaluated. This is used to derive substantial information on the distance of the corresponding central codes. The dimensions and bases for these codes are also obtained. Some central and non-central good codes are produced.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"106 ","pages":"Article 102615"},"PeriodicalIF":1.2,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143620350","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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