Let r be a positive divisor of and a rational function of degree sum d over with some restrictions, where the degree sum of a rational function is the sum of the degrees of and . In this article, we discuss the existence of triples over , where are primitive and is an r-primitive element of . In particular, this implies the existence of -primitive points on the surfaces of the form . As an example, we apply our results on the unit sphere over .
{"title":"Fq-primitive points on varieties over finite fields","authors":"Soniya Takshak , Giorgos Kapetanakis , Rajendra Kumar Sharma","doi":"10.1016/j.ffa.2025.102582","DOIUrl":"10.1016/j.ffa.2025.102582","url":null,"abstract":"<div><div>Let <em>r</em> be a positive divisor of <span><math><mi>q</mi><mo>−</mo><mn>1</mn></math></span> and <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> a rational function of degree sum <em>d</em> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> with some restrictions, where the degree sum of a rational function <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>/</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> is the sum of the degrees of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span>. In this article, we discuss the existence of triples <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>f</mi><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, where <span><math><mi>α</mi><mo>,</mo><mi>β</mi></math></span> are primitive and <span><math><mi>f</mi><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span> is an <em>r</em>-primitive element of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. In particular, this implies the existence of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-primitive points on the surfaces of the form <span><math><msup><mrow><mi>z</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span>. As an example, we apply our results on the unit sphere over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102582"},"PeriodicalIF":1.2,"publicationDate":"2025-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143140000","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 : 2025-01-24DOI: 10.1016/j.ffa.2025.102580
Marko Orel , Draženka Višnjić
<div><div>Let <span><math><msub><mrow><mi>SGL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> be the set of all invertible <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> symmetric matrices over the binary field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Let <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the graph with the vertex set <span><math><msub><mrow><mi>SGL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> where a pair of matrices <span><math><mo>{</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>}</mo></math></span> form an edge if and only if <span><math><mrow><mi>rank</mi></mrow><mo>(</mo><mi>A</mi><mo>−</mo><mi>B</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. In particular, <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> is the well-known Coxeter graph. The distance function <span><math><mi>d</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span> in <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is described for all matrices <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msub><mrow><mi>SGL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. The diameter of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is computed. For odd <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>, it is shown that each matrix <span><math><mi>A</mi><mo>∈</mo><msub><mrow><mi>SGL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> such that <span><math><mi>d</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>I</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> and <span><math><mrow><mi>rank</mi></mrow><mo>(</mo><mi>A</mi><mo>−</mo><mi>I</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> where <em>I</em> is the identity matrix induces a self-dual code in <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></math></span>. Conversely, each self-dual code <em>C</em> induces a family <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>C</mi></mrow></msub></math></span> of such matrices <em>A</em>. The families given by distinct self-dual codes are disjoint. The identification <span><math><mi>C</mi><mo>↔</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>C</mi></mrow></msub></math></span> provides a graph theoretical description of self-dual codes. A result of Janusz (2007) is reproved and strengthened by showing that the orthogo
{"title":"The distance function on Coxeter-like graphs and self-dual codes","authors":"Marko Orel , Draženka Višnjić","doi":"10.1016/j.ffa.2025.102580","DOIUrl":"10.1016/j.ffa.2025.102580","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>SGL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> be the set of all invertible <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> symmetric matrices over the binary field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Let <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the graph with the vertex set <span><math><msub><mrow><mi>SGL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> where a pair of matrices <span><math><mo>{</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>}</mo></math></span> form an edge if and only if <span><math><mrow><mi>rank</mi></mrow><mo>(</mo><mi>A</mi><mo>−</mo><mi>B</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. In particular, <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> is the well-known Coxeter graph. The distance function <span><math><mi>d</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span> in <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is described for all matrices <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msub><mrow><mi>SGL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. The diameter of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is computed. For odd <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>, it is shown that each matrix <span><math><mi>A</mi><mo>∈</mo><msub><mrow><mi>SGL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> such that <span><math><mi>d</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>I</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> and <span><math><mrow><mi>rank</mi></mrow><mo>(</mo><mi>A</mi><mo>−</mo><mi>I</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> where <em>I</em> is the identity matrix induces a self-dual code in <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></math></span>. Conversely, each self-dual code <em>C</em> induces a family <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>C</mi></mrow></msub></math></span> of such matrices <em>A</em>. The families given by distinct self-dual codes are disjoint. The identification <span><math><mi>C</mi><mo>↔</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>C</mi></mrow></msub></math></span> provides a graph theoretical description of self-dual codes. A result of Janusz (2007) is reproved and strengthened by showing that the orthogo","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102580"},"PeriodicalIF":1.2,"publicationDate":"2025-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139998","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}
Pub Date : 2025-01-24DOI: 10.1016/j.ffa.2025.102584
Iván Blanco-Chacón , Alberto F. Boix , Marcus Greferath , Erik Hieta–Aho
We extend Ravagnani's MacWilliams duality theory to the setting of rank metric codes over finite chain rings, relating the sequences of q-binomial moments of a rank metric code over this class of rings with those of its dual.
{"title":"MacWilliams duality for rank metric codes over finite chain rings","authors":"Iván Blanco-Chacón , Alberto F. Boix , Marcus Greferath , Erik Hieta–Aho","doi":"10.1016/j.ffa.2025.102584","DOIUrl":"10.1016/j.ffa.2025.102584","url":null,"abstract":"<div><div>We extend Ravagnani's MacWilliams duality theory to the setting of rank metric codes over finite chain rings, relating the sequences of <em>q</em>-binomial moments of a rank metric code over this class of rings with those of its dual.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102584"},"PeriodicalIF":1.2,"publicationDate":"2025-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139995","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 : 2025-01-22DOI: 10.1016/j.ffa.2025.102575
Can Xiang , Chunming Tang , Wenwei Qiu
Spreading sequences have recently received a lot of attention, as some of these sequences are used to design spreading sequence sets with low correlation and low peak-to-average power ratio (PAPR for short) and which have very important applications in communication systems. It was recently reported that a small amount of work on constructing binary spreading sequence sets with low correlation and low PAPR has been done. However, till now only one work on constructing p-ary spreading sequence sets with low correlation and low PAPR for odd prime p has been done by using special functions in Liu et al. (2023) [11], and it is, in general, hard to design spreading sequence sets with low correlation and low PAPR. In this paper, we investigate this idea further by using some quadratic functions over finite fields, thereby obtain three classes of p-ary spreading sequence sets, and explicitly determine their parameters. The parameters of these p-ary spreading sequence sets are new and flexible. Furthermore, the results of this paper show that these obtained p-ary spreading sequence sets have low correlation and PAPR.
{"title":"Three new classes of spreading sequence sets with low correlation and PAPR","authors":"Can Xiang , Chunming Tang , Wenwei Qiu","doi":"10.1016/j.ffa.2025.102575","DOIUrl":"10.1016/j.ffa.2025.102575","url":null,"abstract":"<div><div>Spreading sequences have recently received a lot of attention, as some of these sequences are used to design spreading sequence sets with low correlation and low peak-to-average power ratio (PAPR for short) and which have very important applications in communication systems. It was recently reported that a small amount of work on constructing binary spreading sequence sets with low correlation and low PAPR has been done. However, till now only one work on constructing <em>p</em>-ary spreading sequence sets with low correlation and low PAPR for odd prime <em>p</em> has been done by using special functions in Liu et al. (2023) <span><span>[11]</span></span>, and it is, in general, hard to design spreading sequence sets with low correlation and low PAPR. In this paper, we investigate this idea further by using some quadratic functions over finite fields, thereby obtain three classes of <em>p</em>-ary spreading sequence sets, and explicitly determine their parameters. The parameters of these <em>p</em>-ary spreading sequence sets are new and flexible. Furthermore, the results of this paper show that these obtained <em>p</em>-ary spreading sequence sets have low correlation and PAPR.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102575"},"PeriodicalIF":1.2,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139994","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 : 2025-01-21DOI: 10.1016/j.ffa.2025.102583
Dingxin Zhang
We show that if V is a subvariety of the affine N-space defined by polynomials of degree at most d, then the sum of its ℓ-adic Betti numbers does not exceed . This answers a question of N. Katz.
{"title":"On sums of Betti numbers of affine varieties","authors":"Dingxin Zhang","doi":"10.1016/j.ffa.2025.102583","DOIUrl":"10.1016/j.ffa.2025.102583","url":null,"abstract":"<div><div>We show that if <em>V</em> is a subvariety of the affine <em>N</em>-space defined by polynomials of degree at most <em>d</em>, then the sum of its <em>ℓ</em>-adic Betti numbers does not exceed <span><math><mn>2</mn><msup><mrow><mo>(</mo><mi>N</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn><mi>N</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mrow><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>N</mi></mrow></msup></math></span>. This answers a question of N. Katz.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102583"},"PeriodicalIF":1.2,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139254","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 : 2025-01-21DOI: 10.1016/j.ffa.2025.102576
Sihem Mesnager , Huawei Wu
Let q be an odd prime power and let be the finite field with elements. In this paper, we first present a method to determine the differential spectrum of the power function on , which provides an alternative proof of the results established by Man et al. (2022) [23]. When the characteristic of is 3, we also determine the value distribution of the Walsh spectrum of F, showing that it is 4-valued, and use the obtained result to determine the weight distribution of a 4-weight cyclic code.
{"title":"On the differential and Walsh spectra of x2q+1 over Fq2","authors":"Sihem Mesnager , Huawei Wu","doi":"10.1016/j.ffa.2025.102576","DOIUrl":"10.1016/j.ffa.2025.102576","url":null,"abstract":"<div><div>Let <em>q</em> be an odd prime power and let <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> be the finite field with <span><math><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> elements. In this paper, we first present a method to determine the differential spectrum of the power function <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> on <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>, which provides an alternative proof of the results established by Man et al. (2022) <span><span>[23]</span></span>. When the characteristic of <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> is 3, we also determine the value distribution of the Walsh spectrum of <em>F</em>, showing that it is 4-valued, and use the obtained result to determine the weight distribution of a 4-weight cyclic code.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102576"},"PeriodicalIF":1.2,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143140001","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 : 2025-01-21DOI: 10.1016/j.ffa.2025.102581
Hai-Liang Wu , Li-Yuan Wang
<div><div>In this paper, we mainly consider arithmetic properties of the cyclotomic matrix <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>[</mo><msub><mrow><mi>J</mi></mrow><mrow><mi>p</mi></mrow></msub><msup><mrow><mo>(</mo><msup><mrow><mi>χ</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msup><mo>,</mo><msup><mrow><mi>χ</mi></mrow><mrow><mi>k</mi><mi>j</mi></mrow></msup><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>]</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>−</mo><mi>k</mi><mo>)</mo><mo>/</mo><mi>k</mi></mrow></msub></math></span>, where <em>p</em> is an odd prime, <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo><</mo><mi>p</mi><mo>−</mo><mn>1</mn></math></span> is a divisor of <span><math><mi>p</mi><mo>−</mo><mn>1</mn></math></span>, <em>χ</em> is a generator of the group of all multiplicative characters of the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msup><mrow><mi>χ</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msup><mo>,</mo><msup><mrow><mi>χ</mi></mrow><mrow><mi>k</mi><mi>j</mi></mrow></msup><mo>)</mo></math></span> is Jacobi sum over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>. By using the Gross-Koblitz formula and some <em>p</em>-adic tools, we first prove that<span><span><span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>det</mi><mo></mo><msub><mrow><mi>B</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>≡</mo><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mfrac><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>p</mi><mo>+</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi><mo>!</mo></mrow></mfrac><mo>)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mfrac><mrow><mn>1</mn></mrow><mrow><mo>(</mo><mn>2</mn><mi>k</mi><mo>)</mo><mo>!</mo></mrow></mfrac><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>p</mi><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><mi>p</mi><mo>−</mo><mn>1</mn><mo>=</mo><mi>k</mi><mi>n</mi></math></span>. By establishing some theories on almost circulant matrices, we show that<span><span><span><math><mi>det</mi><mo></mo><msub><mrow><mi>B</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mfrac><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>p</mi><mo>+</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>p</mi></mro
{"title":"The Gross-Koblitz formula and almost circulant matrices related to Jacobi sums","authors":"Hai-Liang Wu , Li-Yuan Wang","doi":"10.1016/j.ffa.2025.102581","DOIUrl":"10.1016/j.ffa.2025.102581","url":null,"abstract":"<div><div>In this paper, we mainly consider arithmetic properties of the cyclotomic matrix <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>[</mo><msub><mrow><mi>J</mi></mrow><mrow><mi>p</mi></mrow></msub><msup><mrow><mo>(</mo><msup><mrow><mi>χ</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msup><mo>,</mo><msup><mrow><mi>χ</mi></mrow><mrow><mi>k</mi><mi>j</mi></mrow></msup><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>]</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>−</mo><mi>k</mi><mo>)</mo><mo>/</mo><mi>k</mi></mrow></msub></math></span>, where <em>p</em> is an odd prime, <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo><</mo><mi>p</mi><mo>−</mo><mn>1</mn></math></span> is a divisor of <span><math><mi>p</mi><mo>−</mo><mn>1</mn></math></span>, <em>χ</em> is a generator of the group of all multiplicative characters of the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msup><mrow><mi>χ</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msup><mo>,</mo><msup><mrow><mi>χ</mi></mrow><mrow><mi>k</mi><mi>j</mi></mrow></msup><mo>)</mo></math></span> is Jacobi sum over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>. By using the Gross-Koblitz formula and some <em>p</em>-adic tools, we first prove that<span><span><span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>det</mi><mo></mo><msub><mrow><mi>B</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>≡</mo><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mfrac><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>p</mi><mo>+</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi><mo>!</mo></mrow></mfrac><mo>)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mfrac><mrow><mn>1</mn></mrow><mrow><mo>(</mo><mn>2</mn><mi>k</mi><mo>)</mo><mo>!</mo></mrow></mfrac><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>p</mi><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><mi>p</mi><mo>−</mo><mn>1</mn><mo>=</mo><mi>k</mi><mi>n</mi></math></span>. By establishing some theories on almost circulant matrices, we show that<span><span><span><math><mi>det</mi><mo></mo><msub><mrow><mi>B</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mfrac><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>p</mi><mo>+</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>p</mi></mro","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102581"},"PeriodicalIF":1.2,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139256","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 : 2025-01-21DOI: 10.1016/j.ffa.2025.102579
Zheng Xu , Yi Ouyang , Zijian Zhou
For two supersingular elliptic curves E and defined over , let be the superspecial abelian surface with the principal polarization . We determine local structure of the vertices in the -isogeny graph of principally polarized superspecial abelian surfaces where either E or is defined over . We also present a simple new proof of the main theorem in [26].
{"title":"Neighborhood of vertices in the isogeny graph of principally polarized superspecial abelian surfaces","authors":"Zheng Xu , Yi Ouyang , Zijian Zhou","doi":"10.1016/j.ffa.2025.102579","DOIUrl":"10.1016/j.ffa.2025.102579","url":null,"abstract":"<div><div>For two supersingular elliptic curves <em>E</em> and <span><math><msup><mrow><mi>E</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> defined over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>, let <span><math><mo>[</mo><mi>E</mi><mo>×</mo><msup><mrow><mi>E</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>]</mo></math></span> be the superspecial abelian surface with the principal polarization <span><math><mo>{</mo><mn>0</mn><mo>}</mo><mo>×</mo><msup><mrow><mi>E</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>+</mo><mi>E</mi><mo>×</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span>. We determine local structure of the vertices <span><math><mo>[</mo><mi>E</mi><mo>×</mo><msup><mrow><mi>E</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>]</mo></math></span> in the <span><math><mo>(</mo><mi>ℓ</mi><mo>,</mo><mi>ℓ</mi><mo>)</mo></math></span>-isogeny graph of principally polarized superspecial abelian surfaces where either <em>E</em> or <span><math><msup><mrow><mi>E</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is defined over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>. We also present a simple new proof of the main theorem in <span><span>[26]</span></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102579"},"PeriodicalIF":1.2,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139996","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 : 2025-01-20DOI: 10.1016/j.ffa.2025.102573
Mu Yuan , Longjiang Qu , Kangquan Li , Xiaoqiang Wang
The implicit function theory has many applications in continuous functions as a powerful tool. This paper initiates the research on handling functions over finite fields with characteristic even from an implicit viewpoint, and exploring the applications of implicit functions in cryptographic functions and linear error-correcting codes. The implicit function over finite fields is defined by the zeros of a bivariate polynomial . First, we provide basic concepts and constructions of implicit functions. Second, some strong cryptographic functions are constructed by implicit expressions, including semi-bent (or near-bent) balanced Boolean functions and 4-differentially uniform involution without fixed points. Moreover, we construct some optimal linear codes and minimal codes by using constructed implicitly defined functions. In our proof, some algebra and algebraic curve techniques over finite fields are used. Finally, some problems for future work are provided.
{"title":"Implicit functions over finite fields and their applications to good cryptographic functions and linear codes","authors":"Mu Yuan , Longjiang Qu , Kangquan Li , Xiaoqiang Wang","doi":"10.1016/j.ffa.2025.102573","DOIUrl":"10.1016/j.ffa.2025.102573","url":null,"abstract":"<div><div>The implicit function theory has many applications in continuous functions as a powerful tool. This paper initiates the research on handling functions over finite fields with characteristic even from an implicit viewpoint, and exploring the applications of implicit functions in cryptographic functions and linear error-correcting codes. The implicit function <span><math><mmultiscripts><mrow><mi>G</mi></mrow><mprescripts></mprescripts><none></none><mrow><mi>S</mi></mrow></mmultiscripts></math></span> over finite fields is defined by the zeros of a bivariate polynomial <span><math><mi>G</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span>. First, we provide basic concepts and constructions of implicit functions. Second, some strong cryptographic functions are constructed by implicit expressions, including semi-bent (or near-bent) balanced Boolean functions and 4-differentially uniform involution without fixed points. Moreover, we construct some optimal linear codes and minimal codes by using constructed implicitly defined functions. In our proof, some algebra and algebraic curve techniques over finite fields are used. Finally, some problems for future work are provided.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102573"},"PeriodicalIF":1.2,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139999","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 : 2025-01-20DOI: 10.1016/j.ffa.2025.102574
Jaeseong Jeong , Namhun Koo , Soonhak Kwon
<div><div>For a given function <em>F</em> from <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> to itself, determining whether there exists a function which is CCZ-equivalent but EA-inequivalent to <em>F</em> is a very important and interesting problem. For example, Kölsch <span><span>[33]</span></span> showed that there is no function which is CCZ-equivalent but EA-inequivalent to the inverse function. On the other hand, for the cases of Gold function <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>i</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></msup></math></span> and <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mrow><mi>Tr</mi></mrow><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>9</mn></mrow></msup><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>, Budaghyan, Carlet and Pott (respectively, Budaghyan, Carlet and Leander) <span><span>[12]</span></span>, <span><span>[14]</span></span> found functions which are CCZ-equivalent but EA-inequivalent to <em>F</em>. In this paper, when a given function <em>F</em> has a component function which has a linear structure, we present functions which are CCZ-equivalent to <em>F</em>, and if suitable conditions are satisfied, the constructed functions are shown to be EA-inequivalent to <em>F</em>. As a consequence, for every quadratic function <em>F</em> on <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> (<span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>) with nonlinearity greater than 0 and differential uniformity not exceeding <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msup></math></span>, we explicitly construct functions which are CCZ-equivalent but EA-inequivalent to <em>F</em>. Also for every non-planar quadratic function on <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> <span><math><mo>(</mo><mi>p</mi><mo>></mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>≥</mo><mn>4</mn><mo>)</mo></math></span> with <span><math><mo>|</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>|</mo><mo>≤</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> and differential uniformity not exceeding <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msup></math></span>, we explicitly construct functions which are CCZ-equivalent but EA-inequivalent to <em>F</em>. As an application, for a proper divisor <em>m</em> of <em>n</em>, we present many examples
{"title":"On the functions which are CCZ-equivalent but not EA-equivalent to quadratic functions over Fpn","authors":"Jaeseong Jeong , Namhun Koo , Soonhak Kwon","doi":"10.1016/j.ffa.2025.102574","DOIUrl":"10.1016/j.ffa.2025.102574","url":null,"abstract":"<div><div>For a given function <em>F</em> from <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> to itself, determining whether there exists a function which is CCZ-equivalent but EA-inequivalent to <em>F</em> is a very important and interesting problem. For example, Kölsch <span><span>[33]</span></span> showed that there is no function which is CCZ-equivalent but EA-inequivalent to the inverse function. On the other hand, for the cases of Gold function <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>i</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></msup></math></span> and <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mrow><mi>Tr</mi></mrow><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>9</mn></mrow></msup><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>, Budaghyan, Carlet and Pott (respectively, Budaghyan, Carlet and Leander) <span><span>[12]</span></span>, <span><span>[14]</span></span> found functions which are CCZ-equivalent but EA-inequivalent to <em>F</em>. In this paper, when a given function <em>F</em> has a component function which has a linear structure, we present functions which are CCZ-equivalent to <em>F</em>, and if suitable conditions are satisfied, the constructed functions are shown to be EA-inequivalent to <em>F</em>. As a consequence, for every quadratic function <em>F</em> on <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> (<span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>) with nonlinearity greater than 0 and differential uniformity not exceeding <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msup></math></span>, we explicitly construct functions which are CCZ-equivalent but EA-inequivalent to <em>F</em>. Also for every non-planar quadratic function on <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> <span><math><mo>(</mo><mi>p</mi><mo>></mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>≥</mo><mn>4</mn><mo>)</mo></math></span> with <span><math><mo>|</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>|</mo><mo>≤</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> and differential uniformity not exceeding <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msup></math></span>, we explicitly construct functions which are CCZ-equivalent but EA-inequivalent to <em>F</em>. As an application, for a proper divisor <em>m</em> of <em>n</em>, we present many examples","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102574"},"PeriodicalIF":1.2,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139997","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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