Finite Fields and Their Applications最新文献

{"title":"Evaluations of Eisenstein sums","authors":"Ron Evans ,&nbsp;Mark Van Veen","doi":"10.1016/j.ffa.2025.102698","DOIUrl":"10.1016/j.ffa.2025.102698","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> be a field of <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> elements. For<span><span><span><math><mrow><mi>k</mi><mo>=</mo><mn>9</mn><mo>,</mo><mn>11</mn><mo>,</mo><mn>13</mn><mo>,</mo><mn>14</mn><mo>,</mo><mn>15</mn><mo>,</mo><mn>16</mn><mo>,</mo><mn>18</mn><mo>,</mo><mn>20</mn><mo>,</mo><mn>22</mn><mo>,</mo><mn>23</mn><mo>,</mo><mn>26</mn><mo>,</mo><mn>30</mn><mo>,</mo><mn>46</mn></mrow></math></span></span></span> we evaluate Eisenstein sums of order <em>k</em> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> when <em>r</em> is the smallest positive integer for which <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>k</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102698"},"PeriodicalIF":1.2,"publicationDate":"2025-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596554","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":"Closed formulas for the generators of all constacyclic codes and for the factorization of Xn − 1, the n-th cyclotomic polynomial and every composition of the form f(Xn) over a finite field for arbitrary positive integers n","authors":"Anna-Maurin Graner","doi":"10.1016/j.ffa.2025.102695","DOIUrl":"10.1016/j.ffa.2025.102695","url":null,"abstract":"<div><div>In this paper we present the solution to four 150 year-old closely related problems in the study of polynomials and algebraic codes over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. We give <em>one closed formula each</em> for the factorization of the polynomials <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mi>a</mi></math></span> for arbitrary <span><math><mi>a</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>, <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn></math></span>, the <em>n</em>-th cyclotomic polynomial and the composition <span><math><mi>f</mi><mo>(</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> for arbitrary monic irreducible polynomials <span><math><mi>f</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>X</mi><mo>]</mo></math></span>, <span><math><mi>f</mi><mo>≠</mo><mi>X</mi></math></span>, for <em>arbitrary positive integers n.</em> With a new perspective on these problems we show that the factorization of <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mi>a</mi></math></span> has a beautiful underlying structure which is completely determined by the order of <em>a</em> in the multiplicative group <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>.</div><div>The binomial <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mi>a</mi></math></span> and the composition <span><math><mi>f</mi><mo>(</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> were first considered over prime fields by Joseph Alfred Serret in 1866. In recent years, the factorization of <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mi>a</mi></math></span> has been studied extensively due to the fact that its factors are the generators of the popular constacyclic codes over finite fields. The factorization of <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn></math></span> and of its famous factor, the <em>n</em>-th cyclotomic polynomial, was first studied over prime fields by Carl Friedrich Gauss (among others) in the middle of the 19th century. Since then, many mathematicians were fascinated by this factorization and nowadays it is needed for the construction of the cyclic codes over finite fields.</div><div>Many formulas for the factorization of the four polynomials for special choices of <em>n</em> and <em>a</em> were obtained by a large number of mathematicians. Our formulas naturally extend, cover, combine and complete all of these partial solutions.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102695"},"PeriodicalIF":1.2,"publicationDate":"2025-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596934","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":"Square values of several polynomials over a finite field","authors":"Kaloyan Slavov","doi":"10.1016/j.ffa.2025.102696","DOIUrl":"10.1016/j.ffa.2025.102696","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> be polynomials in <em>n</em> variables with coefficients in a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. We estimate the number of points <span><math><munder><mrow><mi>x</mi></mrow><mo>_</mo></munder></math></span> in <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> such that each value <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><munder><mrow><mi>x</mi></mrow><mo>_</mo></munder><mo>)</mo></math></span> is a nonzero square in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. The error term is especially small when the <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> define smooth projective quadrics with nonsingular intersections. We improve the error term in a recent work by Asgarli–Yip on mutual position of smooth quadrics.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102696"},"PeriodicalIF":1.2,"publicationDate":"2025-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596553","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":"Bounds on the size of (r,δ)-locally repairable codes for fixed values q and d","authors":"Wenqin Zhang ,&nbsp;Chen Yuan ,&nbsp;Yuan Luo ,&nbsp;Nian Li","doi":"10.1016/j.ffa.2025.102697","DOIUrl":"10.1016/j.ffa.2025.102697","url":null,"abstract":"<div><div>Erasure codes can strengthen fault-tolerance and reliability in distributed storage systems. One of these, locally repairable codes (LRCs), plays a crucial role. A locally repairable code with locality <em>r</em> (called <em>r</em>-LRC) can recover any coded symbol by accessing at most <em>r</em> other coded symbols. The original definition of LRCs is used to repair a single failed node. To address the practical concern of multiple node failures, the concept of LRCs with locality <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span> was introduced by Prakash et al. (2012) which can be seen as a generalization of <em>r</em>-LRCs. Since then, the bounds and constructions of <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-LRCs have been extensively studied.</div><div>This paper is dedicated to both the bounds and constructions of <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-LRCs with disjoint local repair groups over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, where the parameters <em>r</em>, <em>δ</em>, the alphabet size <em>q</em>, and the minimum distance <em>d</em> are fixed constants, while the code length <em>n</em> tends to infinity. Inspired by the method of the classical Gilbert-Varshamov (GV) bound, we first derive an asymptotic Gilbert-Varshamov-type bound for <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-LRCs in this regime. We manage to show that this GV-type bound works as a threshold for random linear <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-LRCs with disjoint local repair groups using the first and second moment methods. As a corollary, such a random linear <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-LRC has a high probability of attaining the GV-type bound. As an analogue to the classic GV-type bound, we present two constructions of <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-LRCs that each beats this GV-type bound. One construction is obtained from a straightforward concatenation of an outer BCH code and an inner <span><math><mo>[</mo><mi>r</mi><mo>+</mo><mi>δ</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>r</mi><mo>]</mo></math></span> MDS code. Another construction is based on the Kronecker product of two matrices. To complement our results, the case that <em>n</em> is large but finite is also considered. In this regime, we provide an explicit upper bound for the binary <em>r</em>-LRCs, which is an improvement over the one in Ma and Ge (2019). Furthermore, this bound is shown to be tight for some specific parameters.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102697"},"PeriodicalIF":1.2,"publicationDate":"2025-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596550","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":"New constructions of permutation polynomials of the form x+γTrqq3(h(x)) over finite fields with even characteristic","authors":"Sha Jiang,&nbsp;Kangquan Li,&nbsp;Longjiang Qu","doi":"10.1016/j.ffa.2025.102694","DOIUrl":"10.1016/j.ffa.2025.102694","url":null,"abstract":"<div><div>Permutation polynomials over finite fields have applications in many areas of mathematics and engineering. Particularly, permutation polynomials of the form <span><math><mi>x</mi><mo>+</mo><mi>γ</mi><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>q</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msubsup><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> have been studied for a long time. In this paper, we further investigate permutation polynomials of the form <span><math><mi>x</mi><mo>+</mo><mi>γ</mi><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>q</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msubsup><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> over finite fields with even characteristic. For one thing, by choosing functions <span><math><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> with a low <em>q</em>-degree, we propose four classes of permutation polynomials of the form <span><math><mi>x</mi><mo>+</mo><mi>γ</mi><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>q</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msubsup><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msub></math></span>. For the other thing, we give seven classes of permutations of the form <span><math><mi>x</mi><mo>+</mo><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>q</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msubsup><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> with binomials <span><math><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msub></math></span>. Finally, we also show that the permutation polynomials constructed in this paper are not quasi-multiplicative equivalent to the known permutation polynomials.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102694"},"PeriodicalIF":1.2,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579385","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":"Counting irreducible polynomials with restricted coefficients","authors":"Kaimin Cheng","doi":"10.1016/j.ffa.2025.102691","DOIUrl":"10.1016/j.ffa.2025.102691","url":null,"abstract":"<div><div>Let <em>q</em> be a prime power, and let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> denote the finite field with <em>q</em> elements. Consider a positive integer <em>n</em>, and let <span><math><mi>R</mi><mo>=</mo><msubsup><mrow><mo>{</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msubsup></math></span> be a family of subsets of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. Define <span><math><mi>N</mi><mo>(</mo><mi>R</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> as the number of monic irreducible polynomials of degree <em>n</em> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> where the coefficient of each non-leading term <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msup></math></span> lies in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. In this paper, we provide an asymptotic formula for <span><math><mi>N</mi><mo>(</mo><mi>R</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span>, extending a result of Porritt to a more general case.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102691"},"PeriodicalIF":1.2,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570385","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":"Expansion properties of polynomials over finite fields","authors":"Nuno Arala ,&nbsp;Sam Chow","doi":"10.1016/j.ffa.2025.102687","DOIUrl":"10.1016/j.ffa.2025.102687","url":null,"abstract":"<div><div>We establish expansion properties for suitably generic polynomials of degree <em>d</em> in <span><math><mi>d</mi><mo>+</mo><mn>1</mn></math></span> variables over finite fields. In particular, we show that if <span><math><mi>P</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>]</mo></math></span> is a polynomial of degree <em>d</em>, whose coefficients avoid the zero locus of some explicit polynomial of degree <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>)</mo></math></span>, and <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>⊆</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> are suitably large, then <span><math><mo>|</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>|</mo><mo>=</mo><mi>q</mi><mo>−</mo><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>. Our methods rely on a higher-degree extension of a result of Vinh on point–line incidences over a finite field.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102687"},"PeriodicalIF":1.2,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570241","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":"Projective symplectic groups of genus one and two","authors":"Rezhna M. Hussein,&nbsp;Haval M. Mohammed Salih","doi":"10.1016/j.ffa.2025.102689","DOIUrl":"10.1016/j.ffa.2025.102689","url":null,"abstract":"<div><div>In this article, we provide a comprehensive classification of all primitive genus one and genus two systems of the finite group <em>G</em> with <span><math><mi>P</mi><mi>S</mi><mi>p</mi><mo>(</mo><mn>4</mn><mo>,</mo><mi>q</mi><mo>)</mo><mo>≤</mo><mi>G</mi><mo>≤</mo><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>P</mi><mi>S</mi><mi>p</mi><mo>(</mo><mn>4</mn><mo>,</mo><mi>q</mi><mo>)</mo><mo>)</mo></math></span>, where <em>q</em> is a prime power. Also, we use computational tools to show that <em>G</em> possesses no genus <em>g</em> group if <span><math><mi>q</mi><mo>&gt;</mo><mn>5</mn></math></span> where <span><math><mi>g</mi><mo>=</mo><mn>1</mn></math></span>, and 2.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102689"},"PeriodicalIF":1.2,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144569057","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":"A new construction of maximally recoverable codes with hierarchical locality","authors":"Rajendra Prasad Rajpurohit,&nbsp;Maheshanand Bhaintwal","doi":"10.1016/j.ffa.2025.102686","DOIUrl":"10.1016/j.ffa.2025.102686","url":null,"abstract":"<div><div>In this paper, we present a novel construction of maximally recoverable codes with two-level hierarchical locality using a parity-check matrix approach. The construction given in this paper utilizes Gabidulin codes for mid-level heavy parities and linearized Reed-Solomon codes for global heavy parities. When the number of local sets is small, this construction performs better than the previously known constructions as the field size required in our construction is smaller for such cases, making it useful for practical scenarios in distributed data storage systems. We also consider a special case of our construction when the number of global parities is fixed and is equal to 1. In this case, our construction performs better when the number of local sets is small and the number of mid-level parities is even.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102686"},"PeriodicalIF":1.2,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570348","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 the set of stable primes for postcritically infinite maps over number fields","authors":"Joachim König","doi":"10.1016/j.ffa.2025.102693","DOIUrl":"10.1016/j.ffa.2025.102693","url":null,"abstract":"<div><div>Many interesting questions in arithmetic dynamics revolve, in one way or another, around the (local and/or global) reducibility behavior of iterates of a polynomial. We show that for very general families of integer polynomials <em>f</em> (and, more generally, rational functions over number fields), the set of stable primes, i.e., primes modulo which all iterates of <em>f</em> are irreducible, is a density zero set. Compared to previous results, our families cover a much wider ground, and in particular apply to 100% of polynomials of any given odd degree, thus adding evidence to the conjecture that polynomials with a “large” set of stable primes are necessarily of a very specific shape, and in particular are necessarily postcritically finite.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102693"},"PeriodicalIF":1.2,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570240","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}