Finite Fields and Their Applications最新文献

{"title":"Linear codes with few weights over finite fields","authors":"","doi":"10.1016/j.ffa.2024.102509","DOIUrl":"10.1016/j.ffa.2024.102509","url":null,"abstract":"<div><div>Linear codes with a few weights have wide applications in digital signatures, authentication codes, secret sharing protocols and some other fields. Using definition sets to construct linear codes is an effective method. In this paper, we investigate a new defining set and obtain linear codes with four weights, five weights and six weights over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, where <em>p</em> is an odd prime number. The parameters and weight distribution of the constructed linear code are completely determined by accurately calculating the exponential sum over the finite field.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328070","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 explicit values of the UBCT, the LBCT and the DBCT of the inverse function","authors":"","doi":"10.1016/j.ffa.2024.102508","DOIUrl":"10.1016/j.ffa.2024.102508","url":null,"abstract":"<div><p>Substitution boxes (S-boxes) play a significant role in ensuring the resistance of block ciphers against various attacks. The Upper Boomerang Connectivity Table (UBCT), the Lower Boomerang Connectivity Table (LBCT) and the Double Boomerang Connectivity Table (DBCT) of a given S-box are crucial tools to analyze its security concerning specific attacks. However, there are currently no research results on determining these tables of a function. The inverse function is crucial for constructing S-boxes of block ciphers with good cryptographic properties in symmetric cryptography. Therefore, extensive research has been conducted on the inverse function, exploring various properties related to standard attacks. Thanks to the recent advances in boomerang cryptanalysis, particularly the introduction of concepts such as UBCT, LBCT, and DBCT, this paper aims to further investigate the properties of the inverse 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>n</mi></mrow></msup><mo>−</mo><mn>2</mn></mrow></msup></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> for arbitrary <em>n</em>. As a consequence, by carrying out certain finer manipulations of solving specific equations 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>, we give all entries of the UBCT, LBCT of <span><math><mi>F</mi><mo>(</mo><mi>x</mi><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> for arbitrary <em>n</em>. Besides, based on the results of the UBCT and LBCT for the inverse function, we determine that <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is hard when <em>n</em> is odd. Furthermore, we completely compute all entries of the DBCT of <span><math><mi>F</mi><mo>(</mo><mi>x</mi><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> for arbitrary <em>n</em>. Additionally, we provide the precise number of elements with a given entry by means of the values of some Kloosterman sums. Further, we determine the double boomerang uniformity of <span><math><mi>F</mi><mo>(</mo><mi>x</mi><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> for arbitrary <em>n</em>. Our in-depth analysis of the DBCT of <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> contributes to a better evaluation of the S-box's resistance against boomerang attacks.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142240735","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":"Oriented supersingular elliptic curves and Eichler orders of prime level","authors":"","doi":"10.1016/j.ffa.2024.102501","DOIUrl":"10.1016/j.ffa.2024.102501","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; be a prime and &lt;em&gt;E&lt;/em&gt; be a supersingular elliptic curve defined over &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. Let &lt;em&gt;c&lt;/em&gt; be a prime with &lt;span&gt;&lt;math&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;em&gt;G&lt;/em&gt; be a subgroup of &lt;span&gt;&lt;math&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of order &lt;em&gt;c&lt;/em&gt;. The pair &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is called a supersingular elliptic curve with level-&lt;em&gt;c&lt;/em&gt; structure, and the endomorphism ring &lt;span&gt;&lt;math&gt;&lt;mtext&gt;End&lt;/mtext&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is isomorphic to an Eichler order with level &lt;em&gt;c&lt;/em&gt;. We construct two kinds of Eichler orders &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; with level &lt;em&gt;c&lt;/em&gt;. Interestingly, we prove that each &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; or &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; can represent a primitive reduced binary quadratic form with discriminant &lt;span&gt;&lt;math&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; or &lt;span&gt;&lt;math&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; respectively. If a curve &lt;em&gt;E&lt;/em&gt; is &lt;span&gt;&lt;math&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-oriented or &lt;span&gt;&lt;math&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-oriented, then we prove that &lt;span&gt;&lt;math&gt;&lt;mtext&gt;End&lt;/mtext&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is isomorphic to &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; or &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; respectively. Due to the fact that &lt;span&gt;&lt;math&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-oriented isogenies between &lt;span&gt;&lt;math&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142240734","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 duality of cyclic codes of length ps over Fpm[u]〈u3〉","authors":"","doi":"10.1016/j.ffa.2024.102500","DOIUrl":"10.1016/j.ffa.2024.102500","url":null,"abstract":"<div><p>In this paper, we determine the dual codes of cyclic codes of length <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> over <span><math><msub><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup></mrow></msub><mo>[</mo><mi>u</mi><mo>]</mo></mrow><mrow><mo>〈</mo><msup><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>〉</mo></mrow></mfrac></math></span>, where <em>p</em> is a prime number and <span><math><msup><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>=</mo><mn>0</mn></math></span>. Also, we improve and give correction of the results stated by B. Kim and J. Lee (2020) in <span><span>[11]</span></span>. Finally, we provide some examples of optimal and near-MDS cyclic codes of length <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> over <span><math><msub><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> and compute dual of them.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142136897","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":"Denniston partial difference sets exist in the odd prime case","authors":"","doi":"10.1016/j.ffa.2024.102499","DOIUrl":"10.1016/j.ffa.2024.102499","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Denniston constructed partial difference sets (PDSs) with the parameters &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; in elementary abelian groups of order &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; for all &lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. These correspond to maximal arcs in Desarguesian projective planes of even order. In this paper, we show that - although maximal arcs do not exist in Desarguesian projective planes of odd order - PDSs with the Denniston parameters &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1071579724001382/pdfft?md5=03b1e738d3c4bc750b4b0f4af02289e1&pid=1-s2.0-S1071579724001382-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142129238","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":"The resultant method in higher dimensions","authors":"","doi":"10.1016/j.ffa.2024.102493","DOIUrl":"10.1016/j.ffa.2024.102493","url":null,"abstract":"<div><p>Stability results play an important role in Galois geometries. The famous resultant method, developed by Szőnyi and Weiner <span><span>[12]</span></span>, <span><span>[11]</span></span>, became very fruitful and resulted in many stability theorems in the last two decades. This method is based on some bivariate polynomials associated to point sets. In this paper we generalize the method for the multidimensional case and show some new applications. We build up the multivariate polynomial machinery and apply it for <em>Rédei polynomials</em>. We can prove a high dimensional analogue of the result of Szőnyi-Weiner <span><span>[9]</span></span>, concerning the number of hyperplanes being skew to a point set of the space. We prove general results on “partial blocking sets”, using the tools we have developed.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142097573","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":"Achromatic colorings of polarity graphs","authors":"","doi":"10.1016/j.ffa.2024.102497","DOIUrl":"10.1016/j.ffa.2024.102497","url":null,"abstract":"<div><p>A complete partition of a graph <em>G</em> is a partition of the vertex set such that there is at least one edge between any two parts. The largest <em>r</em> such that <em>G</em> has a complete partition into <em>r</em> parts, each of which is an independent set, is the achromatic number of <em>G</em>. We determine the achromatic number of polarity graphs of biaffine planes coming from generalized polygons. Our colorings of a family of unitary polarity graphs are used to solve a problem of Axenovich and Martin on complete partitions of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-free graphs. Furthermore, these colorings prove that there are sequences of graphs which are optimally complete and have unbounded degree, a problem that had been studied for the sequence of hypercubes independently by Roichman, and Ahlswede, Bezrukov, Blokhuis, Metsch, and Moorhouse.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142097574","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 note on (2,2)-isogenies via theta coordinates","authors":"","doi":"10.1016/j.ffa.2024.102496","DOIUrl":"10.1016/j.ffa.2024.102496","url":null,"abstract":"<div><p>In this paper, we revisit the algorithm for computing chains of <span><math><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>-isogenies between products of elliptic curves via theta coordinates proposed by Dartois et al. For each fundamental block of this algorithm, we provide an explicit inversion-free version. Besides, we exploit the technique of <em>x</em>-only ladder to speed up the computation of gluing isogeny. Finally, we present a mixed optimal strategy, which combines the inversion-elimination tool with the original methods together to execute a chain of <span><math><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>-isogenies.</p><p>We make a cost analysis and present a concrete comparison between ours and the previously known methods for inversion elimination. Furthermore, we implement the mixed optimal strategy for benchmark. The results show that when computing <span><math><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>-isogeny chains with lengths of 126, 208 and 632, compared to Dartois, Maino, Pope and Robert's latest implementation, utilizing our techniques can reduce 9.7%, 9.5% and 9.6% multiplications over the base field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, respectively. Therefore, even for the updated version that employs their inversion-free algorithms, our tools still possess an advantage.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142097567","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 a recent extension of a family of biprojective APN functions","authors":"","doi":"10.1016/j.ffa.2024.102494","DOIUrl":"10.1016/j.ffa.2024.102494","url":null,"abstract":"<div><p>APN functions play a big role as primitives in symmetric cryptography as building blocks that yield optimal resistance to differential attacks. In this note, we consider a recent extension, done by Calderini et al. (2023), of a biprojective APN family introduced by Göloğlu (2022) defined on <span><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></span>. We show that this generalization yields functions equivalent to Göloğlu's original family if <span><math><mn>3</mn><mo>∤</mo><mi>m</mi></math></span>. If <span><math><mn>3</mn><mo>|</mo><mi>m</mi></math></span> we show exactly how many inequivalent APN functions this new family contains. We also show that the family has the minimal image set size for an APN function and determine its Walsh spectrum, hereby settling some open problems. In our proofs, we leverage a group theoretic technique recently developed by Göloğlu and the author in conjunction with a group action on the set of projective polynomials.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142083363","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":"Four new families of NMDS codes with dimension 4 and their applications","authors":"","doi":"10.1016/j.ffa.2024.102495","DOIUrl":"10.1016/j.ffa.2024.102495","url":null,"abstract":"<div><p>For an <span><math><msub><mrow><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>d</mi><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> linear code <span><math><mi>C</mi></math></span>, the singleton defect of <span><math><mi>C</mi></math></span> is defined by <span><math><mi>S</mi><mo>(</mo><mi>C</mi><mo>)</mo><mo>=</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>d</mi></math></span>. When <span><math><mi>S</mi><mo>(</mo><mi>C</mi><mo>)</mo><mo>=</mo><mi>S</mi><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>⊥</mo></mrow></msup><mo>)</mo><mo>=</mo><mn>1</mn></math></span>, the code <span><math><mi>C</mi></math></span> is called a near maximum distance separable (NMDS) code, where <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⊥</mo></mrow></msup></math></span> is the dual code of <span><math><mi>C</mi></math></span>. NMDS codes have important applications in finite projective geometries, designs and secret sharing schemes. In this paper, we present four new constructions of infinite families of NMDS codes with dimension 4 and completely determine their weight enumerators. As an application, we also determine the locality of the dual codes of these NMDS codes and obtain four families of distance-optimal and dimension-optimal locally recoverable codes.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142075831","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}