Pub Date : 2025-12-30DOI: 10.1016/j.ffa.2025.102785
Carlos Galindo , Fernando Hernando , Helena Martín-Cruz , Ryutaroh Matsumoto
Classical -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 -locally recoverable codes which are quantum error-correcting codes capable of correcting qudit erasures from sets of at most qudits.
We give a necessary and sufficient condition for a quantum stabilizer code to be -locally recoverable. Our condition depends only on the puncturing and shortening at suitable sets of both the symplectic self-orthogonal code C used for constructing and its symplectic dual . When 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 -local recoverability. A Singleton-like bound is stated in this case and examples attaining the bound are given.
{"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":"2025-12-30","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}
Pub Date : 2025-12-17DOI: 10.1016/j.ffa.2025.102783
Chenying Lin , Gilles Zémor
A k-wise ℓ-divisible set family is a collection of subsets of such that any intersection of k sets in has cardinality divisible by ℓ. If , it is well-known that . We generalise this by proving that if , for any prime number p.
For arbitrary values of ℓ, we prove that -wise ℓ-divisible set families satisfy and that the only families achieving the upper bound are atomic, meaning that they consist of all the unions of disjoint subsets of size ℓ. This improves upon a recent result by Gishboliner, Sudakov and Timon, that arrived at the same conclusion for k-wise ℓ-divisible families, with values of k that behave exponentially in ℓ.
Our techniques rely heavily upon a coding-theory analogue of Kneser's Theorem from additive combinatorics.
一个向k可整除的集合族是{1，…，n}的子集的集合F，使得F中k个集合的任何交集都具有可被r整除的基数。若k= n =2，则已知|F|≤2⌊n/2⌋。我们通过证明|F|≤2⌊n/p⌋，如果k= r =p，对于任意素数p，我们证明了4个2 ~ 2可分集合族F满足|F|≤2⌊n/p⌋，并且唯一达到上限的族是原子族，这意味着它们由大小为r的不相交子集的所有并组成。这改进了Gishboliner， Sudakov和Timon最近的一个结果，他们对k-可分族得出了相同的结论，其中k的值在r中表现为指数。我们的技术在很大程度上依赖于可加组合学中克尼泽定理的编码理论类比。
{"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":"2025-12-17","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}
Pub Date : 2025-12-17DOI: 10.1016/j.ffa.2025.102782
Juanjo Rué , Christoph Spiegel
We study an analogue of the Ramsey multiplicity problem for additive structures, in particular establishing the minimum number of monochromatic 3-APs in 3-colorings of as well as obtaining the first non-trivial lower bound for the minimum number of monochromatic 4-APs in 2-colorings of . The former parallels results by Cumings et al. [8] in extremal graph theory and the latter improves upon results of Saad and Wolf [42]. The lower bounds are notably obtained by extending the flag algebra calculus of Razborov [39] to additive structures in vector spaces over finite fields.
我们研究了可加性结构Ramsey多重性问题的一个类似问题，特别是建立了F3n的3-着色中单色3- ap的最小数目，以及F5n的2-着色中单色4- ap的最小数目的第一个非平凡下界。前者与Cumings et al.[8]在极值图论中的结果相似，后者改进了Saad和Wolf[8]的结果。将Razborov[39]的标志代数演算推广到有限域上向量空间的加性结构，得到了下界。
{"title":"The Rado multiplicity problem in vector spaces over finite fields","authors":"Juanjo Rué , Christoph Spiegel","doi":"10.1016/j.ffa.2025.102782","DOIUrl":"10.1016/j.ffa.2025.102782","url":null,"abstract":"<div><div>We study an analogue of the Ramsey multiplicity problem for additive structures, in particular establishing the minimum number of monochromatic 3-APs in 3-colorings of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> as well as obtaining the first non-trivial lower bound for the minimum number of monochromatic 4-APs in 2-colorings of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>. The former parallels results by Cumings et al. <span><span>[8]</span></span> in extremal graph theory and the latter improves upon results of Saad and Wolf <span><span>[42]</span></span>. The lower bounds are notably obtained by extending the flag algebra calculus of Razborov <span><span>[39]</span></span> to additive structures in vector spaces over finite fields.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102782"},"PeriodicalIF":1.2,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145790235","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 : 2025-12-16DOI: 10.1016/j.ffa.2025.102780
H. Navarro, Luz A. Pérez
In this paper, we present two methods for constructing curves of Kummer type with many rational points over finite fields. The first method is based on binomials, while the second employs reciprocal polynomials. The latter is an extension of the method introduced by Gupta et al. (2023) [19] over quadratic finite fields, to non-prime finite fields. As a result, we found 63 new records and 37 new entries for the online table of curves with many points found at manYPoints.
{"title":"New curves of Kummer type with many rational points over finite fields","authors":"H. Navarro, Luz A. Pérez","doi":"10.1016/j.ffa.2025.102780","DOIUrl":"10.1016/j.ffa.2025.102780","url":null,"abstract":"<div><div>In this paper, we present two methods for constructing curves of Kummer type with many rational points over finite fields. The first method is based on binomials, while the second employs reciprocal polynomials. The latter is an extension of the method introduced by Gupta et al. (2023) <span><span>[19]</span></span> over quadratic finite fields, to non-prime finite fields. As a result, we found 63 new records and 37 new entries for the online table of curves with many points found at <span><span>manYPoints</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102780"},"PeriodicalIF":1.2,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145789554","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 : 2025-12-15DOI: 10.1016/j.ffa.2025.102784
Mariusz Kwiatkowski, Mark Pankov, Adam Tyc
Consider the point-line geometry whose maximal singular subspaces correspond to q-ary simplex codes of dimension k. Maximal cliques in the collinearity graph of this geometry contain no more than elements and maximal singular subspaces of are n-cliques of this graph. If , then and there is a one-to-one correspondence between -cliques of the collinearity graph and symmetric -designs. For the case when we construct a class of n-cliques distinct from maximal singular subspaces. In the case when , some of these cliques are normal rational curves.
{"title":"One class of maximal cliques in the collinearity graphs of geometries related to simplex codes","authors":"Mariusz Kwiatkowski, Mark Pankov, Adam Tyc","doi":"10.1016/j.ffa.2025.102784","DOIUrl":"10.1016/j.ffa.2025.102784","url":null,"abstract":"<div><div>Consider the point-line geometry <span><math><mi>S</mi><mo>(</mo><mi>k</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> whose maximal singular subspaces correspond to <em>q</em>-ary simplex codes of dimension <em>k</em>. Maximal cliques in the collinearity graph of this geometry contain no more than <span><math><mi>n</mi><mo>=</mo><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> elements and maximal singular subspaces of <span><math><mi>S</mi><mo>(</mo><mi>k</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> are <em>n</em>-cliques of this graph. If <span><math><mi>q</mi><mo>=</mo><mn>2</mn></math></span>, then <span><math><mi>n</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>−</mo><mn>1</mn></math></span> and there is a one-to-one correspondence between <span><math><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-cliques of the collinearity graph and symmetric <span><math><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>)</mo></math></span>-designs. For the case when <span><math><mi>q</mi><mo>≥</mo><mn>5</mn></math></span> we construct a class of <em>n</em>-cliques distinct from maximal singular subspaces. In the case when <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span>, some of these cliques are normal rational curves.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102784"},"PeriodicalIF":1.2,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145789556","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 : 2025-12-15DOI: 10.1016/j.ffa.2025.102778
Yajing Zhou , Rongquan Feng
Let be the ring of residue classes modulo n, and let be the group of its units. In 2017, Mollahajiaghaei presented a formula for the number of solutions of the congruence . This paper considers the addition of squares and cubes over . Specifically, when n is a prime number such that , we correct the formula given by Mollahajiaghaei.
{"title":"On the addition of squares and cubes of units modulo n","authors":"Yajing Zhou , Rongquan Feng","doi":"10.1016/j.ffa.2025.102778","DOIUrl":"10.1016/j.ffa.2025.102778","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the ring of residue classes modulo <em>n</em>, and let <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span> be the group of its units. In 2017, Mollahajiaghaei presented a formula for the number of solutions <span><math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msup></math></span> of the congruence <span><math><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>≡</mo><mi>c</mi><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>n</mi><mo>)</mo></math></span>. This paper considers the addition of squares and cubes over <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. Specifically, when <em>n</em> is a prime number such that <span><math><mi>n</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math></span>, we correct the formula given by Mollahajiaghaei.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102778"},"PeriodicalIF":1.2,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145789555","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 : 2025-12-12DOI: 10.1016/j.ffa.2025.102779
Zhonghao Liang, Qunying Liao
Since near maximum distance separable (NMDS) codes have good algebraic properties and excellent error-correcting capabilities, they have been widely used in various fields such as communication systems, data storage, quantum codes, and so on. In this paper, basing on the generator matrix of Roth-Lempel codes, we present two classes of NMDS codes which generalize Han's, Zheng's and Zhou's constructions in 2023 and 2025, respectively. And we also completely determine their weight distributions.
{"title":"Two classes of NMDS codes from Roth-Lempel codes","authors":"Zhonghao Liang, Qunying Liao","doi":"10.1016/j.ffa.2025.102779","DOIUrl":"10.1016/j.ffa.2025.102779","url":null,"abstract":"<div><div>Since near maximum distance separable (NMDS) codes have good algebraic properties and excellent error-correcting capabilities, they have been widely used in various fields such as communication systems, data storage, quantum codes, and so on. In this paper, basing on the generator matrix of Roth-Lempel codes, we present two classes of NMDS codes which generalize Han's, Zheng's and Zhou's constructions in 2023 and 2025, respectively. And we also completely determine their weight distributions.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102779"},"PeriodicalIF":1.2,"publicationDate":"2025-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737315","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 : 2025-12-11DOI: 10.1016/j.ffa.2025.102781
Tetsushi Ito , Ren Tatematsu , Takahiro Tsushima
The van der Geer–van der Vlugt curves are Artin–Schreier coverings of the affine line defined by linearized polynomials over finite fields. We provide several criteria for them to be maximal or minimal, i.e. attaining the upper or lower bound in the Hasse–Weil inequalities. As applications, we identify several maximal (or minimal) curves within this family. Our proofs are based on an explicit formula for the L-polynomials, recently obtained by Takeuchi and the third author.
van der Geer-van der Vlugt曲线是有限域上由线性化多项式定义的仿射线的Artin-Schreier覆盖。我们给出了它们最大或最小的几个标准，即达到Hasse-Weil不等式的上界或下界。作为应用，我们在这个族中确定了几个最大（或最小）曲线。我们的证明是基于最近由Takeuchi和第三作者获得的l多项式的显式公式。
{"title":"Criteria of maximality and minimality of van der Geer–van der Vlugt curves","authors":"Tetsushi Ito , Ren Tatematsu , Takahiro Tsushima","doi":"10.1016/j.ffa.2025.102781","DOIUrl":"10.1016/j.ffa.2025.102781","url":null,"abstract":"<div><div>The van der Geer–van der Vlugt curves are Artin–Schreier coverings of the affine line defined by linearized polynomials over finite fields. We provide several criteria for them to be maximal or minimal, i.e. attaining the upper or lower bound in the Hasse–Weil inequalities. As applications, we identify several maximal (or minimal) curves within this family. Our proofs are based on an explicit formula for the <em>L</em>-polynomials, recently obtained by Takeuchi and the third author.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102781"},"PeriodicalIF":1.2,"publicationDate":"2025-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737313","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 : 2025-12-11DOI: 10.1016/j.ffa.2025.102777
Sujata Bansal, Pramod Kumar Kewat
In this paper, we construct a family of optimal linear codes over with parameters , where e is a positive integer and . We determine the duals of these codes and establish that for , 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 . 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.
{"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":"2025-12-11","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}
Pub Date : 2025-12-11DOI: 10.1016/j.ffa.2025.102768
Sarah Arpin , Ross Bowden , James Clements , Wissam Ghantous , Jason T. LeGrow , Krystal Maughan
Supersingular elliptic curve isogeny graphs underlie isogeny-based cryptography. For isogenies of a single prime degree ℓ, their structure has been investigated graph-theoretically. We generalise the notion of ℓ-isogeny graphs to L-isogeny graphs (studied in the prime field case by Delfs and Galbraith), where L is a set of small primes dictating the allowed isogeny degrees in the graph. We analyse the graph-theoretic structure of L-isogeny graphs. Our approaches may be put into two categories: cycles and graph cuts.
On the topic of cycles, we provide: a count for the number of cycles in the L-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 L-isogeny cycles. We provide code to compute each of these three counts.
On the topic of graph cuts, we compare several algorithms to compute graph cuts which minimise a measure called the edge expansion, outlining a cryptographic motivation for doing so. Our results show that a greedy neighbour algorithm out-performs standard spectral algorithms for computing optimal graph cuts. We provide code and study explicit examples.
Furthermore, we describe several directions of active and future research.
{"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":"2025-12-11","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}
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