Finite Fields and Their Applications最新文献_第4页

On two conjectures of the duals of AMDS BCH codes 关于AMDS BCH码对偶的两个猜想

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2026-03-01 Epub Date: 2025-12-01 DOI: 10.1016/j.ffa.2025.102765

Haode Yan

引用次数: 0

Kneser's theorem for codes and ℓ-divisible set families 码和可分集合族的Kneser定理

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2026-03-01 Epub Date: 2025-12-17 DOI: 10.1016/j.ffa.2025.102783

Chenying Lin , Gilles Zémor

{"title":"Kneser's theorem for codes and ℓ-divisible set families","authors":"Chenying Lin , Gilles Zémor","doi":"10.1016/j.ffa.2025.102783","DOIUrl":"10.1016/j.ffa.2025.102783","url":null,"abstract":"<div><div>A <em>k</em>-wise <em>ℓ</em>-divisible set family is a collection <span><math><mi>F</mi></math></span> of subsets of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span> such that any intersection of <em>k</em> sets in <span><math><mi>F</mi></math></span> has cardinality divisible by <em>ℓ</em>. If <span><math><mi>k</mi><mo>=</mo><mi>ℓ</mi><mo>=</mo><mn>2</mn></math></span>, it is well-known that <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow></msup></math></span>. We generalise this by proving that <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>⌊</mo><mi>n</mi><mo>/</mo><mi>p</mi><mo>⌋</mo></mrow></msup></math></span> if <span><math><mi>k</mi><mo>=</mo><mi>ℓ</mi><mo>=</mo><mi>p</mi></math></span>, for any prime number <em>p</em>.</div><div>For arbitrary values of <em>ℓ</em>, we prove that <span><math><mn>4</mn><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-wise <em>ℓ</em>-divisible set families <span><math><mi>F</mi></math></span> satisfy <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>⌊</mo><mi>n</mi><mo>/</mo><mi>ℓ</mi><mo>⌋</mo></mrow></msup></math></span> and that the only families achieving the upper bound are atomic, meaning that they consist of all the unions of disjoint subsets of size <em>ℓ</em>. This improves upon a recent result by Gishboliner, Sudakov and Timon, that arrived at the same conclusion for <em>k</em>-wise <em>ℓ</em>-divisible families, with values of <em>k</em> that behave exponentially in <em>ℓ</em>.</div><div>Our techniques rely heavily upon a coding-theory analogue of Kneser's Theorem from additive combinatorics.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102783"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145790234","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

Quantum (r,δ)-locally recoverable codes 量子(r,δ)-局部可恢复的代码

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2026-03-01 Epub Date: 2025-12-30 DOI: 10.1016/j.ffa.2025.102785

Carlos Galindo , Fernando Hernando , Helena Martín-Cruz , Ryutaroh Matsumoto

{"title":"Quantum (r,δ)-locally recoverable codes","authors":"Carlos Galindo , Fernando Hernando , Helena Martín-Cruz , Ryutaroh Matsumoto","doi":"10.1016/j.ffa.2025.102785","DOIUrl":"10.1016/j.ffa.2025.102785","url":null,"abstract":"<div><div>Classical <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-locally recoverable codes are designed for avoiding loss of information in large scale distributed and cloud storage systems. We introduce the quantum counterpart of those codes by defining quantum <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-locally recoverable codes which are quantum error-correcting codes capable of correcting <span><math><mi>δ</mi><mo>−</mo><mn>1</mn></math></span> qudit erasures from sets of at most <span><math><mi>r</mi><mo>+</mo><mi>δ</mi><mo>−</mo><mn>1</mn></math></span> qudits.</div><div>We give a necessary and sufficient condition for a quantum stabilizer code <span><math><mi>Q</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span> to be <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-locally recoverable. Our condition depends only on the puncturing and shortening at suitable sets of both the symplectic self-orthogonal code <em>C</em> used for constructing <span><math><mi>Q</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span> and its symplectic dual <span><math><msup><mrow><mi>C</mi></mrow><mrow><msub><mrow><mo>⊥</mo></mrow><mrow><mi>s</mi></mrow></msub></mrow></msup></math></span>. When <span><math><mi>Q</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span> comes from a Hermitian or Euclidean dual-containing code, and under an extra condition, we show that there is an equivalence between the classical and quantum concepts of <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-local recoverability. A Singleton-like bound is stated in this case and examples attaining the bound are given.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102785"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145883541","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

A family of optimal dual-containing and reversible linear codes over F4 F4上最优的双含可逆线性码族

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2026-03-01 Epub Date: 2025-12-11 DOI: 10.1016/j.ffa.2025.102777

Sujata Bansal, Pramod Kumar Kewat

{"title":"A family of optimal dual-containing and reversible linear codes over F4","authors":"Sujata Bansal, Pramod Kumar Kewat","doi":"10.1016/j.ffa.2025.102777","DOIUrl":"10.1016/j.ffa.2025.102777","url":null,"abstract":"<div><div>In this paper, we construct a family of optimal linear codes over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> with parameters <span><math><mo>[</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>e</mi></mrow></msup><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>e</mi></mrow></msup><mo>−</mo><mi>e</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>]</mo></math></span>, where <em>e</em> is a positive integer and <span><math><mi>e</mi><mo>≥</mo><mn>2</mn></math></span>. We determine the duals of these codes and establish that for <span><math><mi>e</mi><mo>≥</mo><mn>3</mn></math></span>, these codes are dual-containing. This property makes them suitable for the construction of CSS quantum error-correcting codes. Furthermore, we calculate the weight distribution of the duals of these codes and show that the duals are 3-weight codes. We derive the weight enumerator of these codes using the MacWilliams identities. Additionally, we establish that these codes are reversible for all <span><math><mi>e</mi><mo>≥</mo><mn>2</mn></math></span>. This ensures the symmetry in the code structure and facilitates them for the possible applications in DNA computing and bidirectional communication systems. The optimality, duality, and reversibility of this family of codes highlight the potential of these codes for various practical and theoretical applications in the error correction.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102777"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737312","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

Cycles and cuts in supersingular L-isogeny graphs 超奇异l -等构图中的环与切

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2026-03-01 Epub Date: 2025-12-11 DOI: 10.1016/j.ffa.2025.102768

Sarah Arpin , Ross Bowden , James Clements , Wissam Ghantous , Jason T. LeGrow , Krystal Maughan

{"title":"Cycles and cuts in supersingular L-isogeny graphs","authors":"Sarah Arpin , Ross Bowden , James Clements , Wissam Ghantous , Jason T. LeGrow , Krystal Maughan","doi":"10.1016/j.ffa.2025.102768","DOIUrl":"10.1016/j.ffa.2025.102768","url":null,"abstract":"<div><div>Supersingular elliptic curve isogeny graphs underlie isogeny-based cryptography. For isogenies of a single prime degree <em>ℓ</em>, their structure has been investigated graph-theoretically. We generalise the notion of <em>ℓ</em>-isogeny graphs to <em>L</em>-isogeny graphs (studied in the prime field case by Delfs and Galbraith), where <em>L</em> is a set of small primes dictating the allowed isogeny degrees in the graph. We analyse the graph-theoretic structure of <em>L</em>-isogeny graphs. Our approaches may be put into two categories: cycles and graph cuts.</div><div>On the topic of cycles, we provide: a count for the number of cycles in the <em>L</em>-isogeny graph with cyclic kernels using traces of Brandt matrices; an efficiently computable estimate based on this approach; and a third ideal-theoretic count for a certain subclass of <em>L</em>-isogeny cycles. We provide code to compute each of these three counts.</div><div>On the topic of graph cuts, we compare several algorithms to compute graph cuts which minimise a measure called the <em>edge expansion</em>, outlining a cryptographic motivation for doing so. Our results show that a <em>greedy neighbour</em> algorithm out-performs standard spectral algorithms for computing optimal graph cuts. We provide code and study explicit examples.</div><div>Furthermore, we describe several directions of active and future research.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102768"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737314","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

Multiplicative character sums over two classes of subsets of quadratic extensions of finite fields 有限域的二次扩展的两类子集上的乘法字符和

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2026-03-01 Epub Date: 2025-12-03 DOI: 10.1016/j.ffa.2025.102767

Kaimin Cheng , Arne Winterhof

{"title":"Multiplicative character sums over two classes of subsets of quadratic extensions of finite fields","authors":"Kaimin Cheng , Arne Winterhof","doi":"10.1016/j.ffa.2025.102767","DOIUrl":"10.1016/j.ffa.2025.102767","url":null,"abstract":"<div><div>Let <em>q</em> be a prime power and <em>r</em> a positive even integer. Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> be the finite field with <em>q</em> elements and <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span> be its extension field of degree <em>r</em>. Let <em>χ</em> be a nontrivial multiplicative character of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span> and <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> a polynomial over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span> with exactly one simple root in <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span>. In this paper, we improve estimates for character sums <span><math><munder><mo>∑</mo><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></munder><mi>χ</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>g</mi><mo>)</mo><mo>)</mo></math></span>, where <span><math><mi>G</mi></math></span> is either a subset of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span> of sparse elements, with respect to some fixed basis of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span> which contains a basis of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi><mo>/</mo><mn>2</mn></mrow></msup></mrow></msub></math></span>, or a subset avoiding affine hyperplanes in general position. While such sums have been previously studied, our approach yields sharper bounds by reducing them to sums over the subfield <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi><mo>/</mo><mn>2</mn></mrow></msup></mrow></msub></math></span> rather than sums over general linear spaces. These estimates can be used to prove the existence of primitive elements in <span><math><mi>G</mi></math></span> in the standard way.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102767"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145684471","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

Quantum MDS codes induced by the projective linear transformation 投影线性变换诱导的量子MDS码

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2026-03-01 Epub Date: 2025-12-01 DOI: 10.1016/j.ffa.2025.102764

Fengwei Li, Yuting Liu, Ruiyuan Jiang

引用次数: 0

Maps preserving a fixed rank-distance on matrices over finite fields 在有限域上矩阵上保持固定秩距的映射

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2026-03-01 Epub Date: 2025-11-25 DOI: 10.1016/j.ffa.2025.102759

A.M. Maksaev, N.Y. Medved, V.V. Promyslov

引用次数: 0

Two classes of NMDS codes from Roth-Lempel codes 从Roth-Lempel码中得到两类NMDS码