Finite Fields and Their Applications最新文献_第10页

Application of the Cartier operator in coding theory 卡蒂埃算子在编码理论中的应用

IF 1 3区数学

Finite Fields and Their Applications Pub Date : 2024-03-29 DOI: 10.1016/j.ffa.2024.102419

Vahid Nourozi

引用次数: 0

Private information retrieval from locally repairable databases with colluding servers 从有串通服务器的本地可修复数据库中检索私人信息

IF 1 3区数学

Finite Fields and Their Applications Pub Date : 2024-03-28 DOI: 10.1016/j.ffa.2024.102421

Umberto Martínez-Peñas

{"title":"Private information retrieval from locally repairable databases with colluding servers","authors":"Umberto Martínez-Peñas","doi":"10.1016/j.ffa.2024.102421","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102421","url":null,"abstract":"<div>We consider information-theoretical private information retrieval (PIR) from a coded database with colluding servers. We target, for the first time, locally repairable storage codes (LRCs). We consider any number of local groups g, locality r, local distance δ and dimension k. Our main contribution is a PIR scheme for maximally recoverable (MR) LRCs based on linearized Reed–Solomon codes, which achieve the smallest field sizes among MR-LRCs for many parameter regimes. In our scheme, nodes are identified with codeword symbols and servers are identified with local groups of nodes. Only locally non-redundant information is downloaded from each server, that is, only r nodes (out of <math><mi>r</mi><mo>+</mo><mi>δ</mi><mo>−</mo><mn>1</mn></math>) are downloaded per server. The PIR scheme achieves the (download) rate <math><mi>R</mi><mo>=</mo><mo>(</mo><mi>N</mi><mo>−</mo><mi>k</mi><mo>−</mo><mi>r</mi><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mi>N</mi></math>, where <math><mi>N</mi><mo>=</mo><mi>g</mi><mi>r</mi></math> is the length of the MDS code obtained after removing the local parities, and for any t colluding servers such that <math><mi>k</mi><mo>+</mo><mi>r</mi><mi>t</mi><mo>≤</mo><mi>N</mi></math>. For an unbounded number of stored files, the obtained rate is strictly larger than those of known PIR schemes that work for any MDS code. Finally, the obtained PIR scheme can also be adapted when communication between the user and each server is performed via linear network coding, achieving the same rate as previous PIR schemes for this scenario but with polynomial finite field sizes, instead of exponential. Our rates are equal to those of PIR schemes for Reed–Solomon codes, but Reed–Solomon codes are incompatible with the MR-LRC property or linear network coding, thus our PIR scheme is less restrictive in its applications.</div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"96 ","pages":"Article 102421"},"PeriodicalIF":1.0,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1071579724000601/pdfft?md5=0fe90fcdc546f6a24d87a8e7912affb8&pid=1-s2.0-S1071579724000601-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140320783","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}

引用次数: 0

Primitive elements of finite fields Fqr avoiding affine hyperplanes for q = 4 and q = 5 q = 4 和 q = 5 时避免仿射超平面的有限域 Fqr 的基元

IF 1 3区数学

Finite Fields and Their Applications Pub Date : 2024-03-26 DOI: 10.1016/j.ffa.2024.102416

Philipp A. Grzywaczyk , Arne Winterhof

引用次数: 0

On permutation quadrinomials from Niho exponents in characteristic two 从特征二中的尼霍指数论 permutation quadrinomials from Niho exponents

IF 1 3区数学

Finite Fields and Their Applications Pub Date : 2024-03-26 DOI: 10.1016/j.ffa.2024.102418

Vincenzo Pallozzi Lavorante

引用次数: 0

New classes of permutation trinomials of F22m F22m 的新类置换三项式

IF 1 3区数学

Finite Fields and Their Applications Pub Date : 2024-03-26 DOI: 10.1016/j.ffa.2024.102414

Akshay Ankush Yadav , Indivar Gupta , Harshdeep Singh , Arvind Yadav

{"title":"New classes of permutation trinomials of F22m","authors":"Akshay Ankush Yadav , Indivar Gupta , Harshdeep Singh , Arvind Yadav","doi":"10.1016/j.ffa.2024.102414","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102414","url":null,"abstract":"<div>In recent years, there have been a lot of research towards finding conditions under which the trinomial <math><msup><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>α</mi><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>+</mo><mn>1</mn><mo>)</mo></math> permutes <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> with <math><mi>α</mi><mo>></mo><mi>β</mi></math> and r being positive integers. The authors of [6], [10], [24] have determined these conditions when <math><mi>α</mi><mo>≤</mo><mn>5</mn></math> for certain values of β and r. In this paper, we work for <math><mi>α</mi><mo>=</mo><mn>6</mn></math> and determine four new classes of such permutation trinomials. Our contribution encompasses the investigation of these unexplored classes. Additionally, we analyze their quasi-multiplicative equivalence with already known permutation trinomials for <math><mi>m</mi><mo>≥</mo><mn>1</mn></math>. Through our research, we demonstrate that two of these determined classes are new, and for others, we explicitly compute the exponent for which they become equivalent.</div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"96 ","pages":"Article 102414"},"PeriodicalIF":1.0,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140295912","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}

引用次数: 0

Irreducibility properties of Carlitz' binomial coefficients for algebraic function fields 代数函数场的卡利茨二项式系数的不可逆性质

IF 1 3区数学

Finite Fields and Their Applications Pub Date : 2024-03-26 DOI: 10.1016/j.ffa.2024.102413

Robert Tichy , Daniel Windisch

引用次数: 0

Plane curves with a large linear automorphism group in characteristic p 特性 p 中具有大线性自变群的平面曲线

IF 1 3区数学

Finite Fields and Their Applications Pub Date : 2024-03-22 DOI: 10.1016/j.ffa.2024.102402

Herivelto Borges , Gábor Korchmáros , Pietro Speziali

{"title":"Plane curves with a large linear automorphism group in characteristic p","authors":"Herivelto Borges , Gábor Korchmáros , Pietro Speziali","doi":"10.1016/j.ffa.2024.102402","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102402","url":null,"abstract":"<div>Let G be a subgroup of the three dimensional projective group <math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math> defined over a finite field <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math> of order q, viewed as a subgroup of <math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>K</mi><mo>)</mo></math> where K is an algebraic closure of <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>. For <math><mi>G</mi><mo>≅</mo><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math> and for the seven nonsporadic, maximal subgroups G of <math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math>, we investigate the (projective, irreducible) plane curves defined over K that are left invariant by G. For each, we compute the minimum degree <math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> of G-invariant curves, provide a classification of all G-invariant curves of degree <math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math>, and determine the first gap <math><mi>ε</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> in the spectrum of the degrees of all G-invariant curves. We show that the curves of degree <math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> belong to a pencil depending on G, unless they are uniquely determined by G. For most examples of plane curves left invariant by a large subgroup of <math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math>, the whole automorphism group of the curve is linear, i.e., a subgroup of <math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>K</mi><mo>)</mo></math>. Although this appears to be a general behavior, we show that the opposite case can also occur for some irreducible plane curves, that is, the curve has a large group of linear automorphisms, but its full automorphism group is nonlinear.</div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"96 ","pages":"Article 102402"},"PeriodicalIF":1.0,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140190950","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}

引用次数: 0

On a class of permutation polynomials and their inverses 关于一类置换多项式及其倒数

IF 1 3区数学

Finite Fields and Their Applications Pub Date : 2024-03-18 DOI: 10.1016/j.ffa.2024.102403

Ruikai Chen , Sihem Mesnager

引用次数: 0

On dual-containing, almost dual-containing matrix-product codes and related quantum codes 论双含、近双含矩阵积码及相关量子码

IF 1 3区数学

Finite Fields and Their Applications Pub Date : 2024-03-13 DOI: 10.1016/j.ffa.2024.102400

Meng Cao

{"title":"On dual-containing, almost dual-containing matrix-product codes and related quantum codes","authors":"Meng Cao","doi":"10.1016/j.ffa.2024.102400","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102400","url":null,"abstract":"<div>Matrix-product (MP) codes are a type of long codes formed by combining several commensurate constituent codes with a defining matrix. In this paper, we study the MP code when the defining matrix A satisfies the condition that <math><mi>A</mi><msup><mrow><mi>A</mi></mrow><mrow><mo>⊤</mo></mrow></msup></math> is <math><mo>(</mo><mi>D</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math>-monomial. We give an explicit formula for calculating the dimension of the hull of a MP code. We present the necessary and sufficient conditions for a MP code to be dual-containing (DC), almost dual-containing (ADC), self-orthogonal (SO) and almost self-orthogonal (ASO), respectively. We theoretically determine the number of all possible ways involving the relationships among the constituent codes to yield a MP code that is DC, ADC, SO and ASO, respectively. We give alternative necessary and sufficient conditions for a MP code to be ADC and ASO, respectively, and show several cases where a MP code is not ADC or ASO. We give the construction methods of DC and ADC MP codes, including those with optimal minimum distance lower bounds. We introduce the notation of τ-optimal defining (τ-OD) matrices and provide the criteria for determining whether two types of <math><mi>k</mi><mo>×</mo><mi>k</mi></math> matrices are τ-OD matrices at <math><mi>k</mi><mo>=</mo><mn>3</mn></math> and <math><mi>k</mi><mo>=</mo><mn>4</mn></math>, respectively. We give many examples of DC and ADC MP codes involving τ-OD matrices, some of which are optimal or almost optimal according to the Database [11]. By applying the generalized Steane's enlargement procedure to these DC MP codes, we obtain some good quantum codes that improve those available in the Database [7].</div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"96 ","pages":"Article 102400"},"PeriodicalIF":1.0,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140113044","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}

引用次数: 0

The extended codes of some linear codes 某些线性编码的扩展编码

IF 1 3区数学

Finite Fields and Their Applications Pub Date : 2024-03-12 DOI: 10.1016/j.ffa.2024.102401

Zhonghua Sun , Cunsheng Ding , Tingfang Chen

{"title":"The extended codes of some linear codes","authors":"Zhonghua Sun , Cunsheng Ding , Tingfang Chen","doi":"10.1016/j.ffa.2024.102401","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102401","url":null,"abstract":"<div>The classical way of extending an <math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>d</mi><mo>]</mo></math> linear code <math><mi>C</mi></math> is to add an overall parity-check coordinate to each codeword of the linear code <math><mi>C</mi></math>. This extended code, denoted by <math><mover><mrow><mi>C</mi></mrow><mo>‾</mo></mover><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></math> and called the standardly extended code of <math><mi>C</mi></math>, is a linear code with parameters <math><mo>[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>,</mo><mover><mrow><mi>d</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>]</mo></math>, where <math><mover><mrow><mi>d</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>=</mo><mi>d</mi></math> or <math><mover><mrow><mi>d</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>=</mo><mi>d</mi><mo>+</mo><mn>1</mn></math>. This is one of the two extending techniques for linear codes in the literature. The standardly extended codes of some families of binary linear codes have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for this type of extended linear codes. The second objective is to study this type of extended codes of a number of families of linear codes, including cyclic codes and nonbinary Hamming codes. Four families of distance-optimal or dimension-optimal linear codes are obtained with this extending technique. The parameters of certain extended codes of many families of linear codes are settled in this paper.</div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"96 ","pages":"Article 102401"},"PeriodicalIF":1.0,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140103696","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}

引用次数: 0