Designs, Codes and Cryptography最新文献

{"title":"Generalized hyperderivative Reed–Solomon codes","authors":"Mahir Bilen Can, Benjamin Horowitz","doi":"10.1007/s10623-026-01849-3","DOIUrl":"https://doi.org/10.1007/s10623-026-01849-3","url":null,"abstract":"This article introduces Generalized Hyperderivative Reed–Solomon codes (GHRS codes), which generalize NRT Reed–Solomon codes. Its main results are as follows: (1) every GHRS code is MDS, (2) the dual of a GHRS code is also an GHRS code, (3) determine subfamilies of GHRS codes whose members are low-density parity-check codes (LDPCs), and (4) determine a family of GHRS codes whose members are quasi-cyclic. We point out that there are GHRS codes having all of these properties.","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"19 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147702312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On group codes arising from Paley-type partial difference sets and skew–Hadamard difference sets","authors":"Vitor Araujo Garcia","doi":"10.1007/s10623-026-01846-6","DOIUrl":"https://doi.org/10.1007/s10623-026-01846-6","url":null,"abstract":"Paley-type partial difference sets and skew–Hadamard difference sets are classical objects in algebraic combinatorics, known for their rich connections with graph theory, coding theory, and group theory. In this paper, we explore new links between these combinatorial structures and group codes arising as ideals in finite group algebras. We construct such codes from difference sets and determine their dimensions in several cases. As an application of our links, we explicitly compute the full set of primitive central idempotents in certain abelian <inline-formula><alternatives><mml:math><mml:mi>p</mml:mi></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ p $$end{document}</tex-math></alternatives></inline-formula>-group algebras, by employing the classical sets of quadratic residues and non-residues modulo <inline-formula><alternatives><mml:math><mml:mi>p</mml:mi></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ p $$end{document}</tex-math></alternatives></inline-formula>, which are well-studied examples of difference and partial difference sets—we also obtain their dimensions and estimate their minimum weights.","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"23 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147702313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Binary code generated by the hyperbolic quadrics of W(2n-1,q),documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$W(2n-1,q),$$end{document}q even","authors":"Devjyoti Das, Bart De Bruyn, Binod Kumar Sahoo, N. S. Narasimha Sastry","doi":"10.1007/s10623-026-01803-3","DOIUrl":"https://doi.org/10.1007/s10623-026-01803-3","url":null,"abstract":"Consider a vector space of dimension 2&lt;italic&gt;n&lt;/italic&gt;, &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;mml:mo&gt;≥&lt;/mml:mo&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$n ge 2$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, defined over the finite field of order &lt;italic&gt;q&lt;/italic&gt;, that is equipped with a nondegenerate alternating bilinear form &lt;italic&gt;f&lt;/italic&gt;. Denote by &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;W&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;mml:mo&gt;-&lt;/mml:mo&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:mi&gt;q&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$W(2n-1,q)$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; the symplectic polar space associated with (&lt;italic&gt;V&lt;/italic&gt;, &lt;italic&gt;f&lt;/italic&gt;). For &lt;italic&gt;q&lt;/italic&gt; even, let &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mi mathvariant=\"script\"&gt;H&lt;/mml:mi&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathcal {H}}$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; denote the binary linear code spanned by those hyperbolic quadrics of &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mtext&gt;PG&lt;/mml:mtext&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;mml:mo&gt;-&lt;/mml:mo&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:mi&gt;q&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$textrm{PG}(2n-1,q)$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; with quadratic forms &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mi&gt;κ&lt;/mml:mi&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$kappa $$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; for which the associated symmetric bilinear form &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;f&lt;/mml:mi&gt;&lt;mml:mi&gt;κ&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} us","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"72 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147642130","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Further results on a family of bent functions from permutations","authors":"Daniele Bartoli, Marco Timpanella","doi":"10.1007/s10623-026-01815-z","DOIUrl":"https://doi.org/10.1007/s10623-026-01815-z","url":null,"abstract":"In 1997, Hou and Langevin introduced the idea of constructing bent functions as the composition of a Boolean function with a permutation. Recently, K. Li, C. Li, T. Helleseth, and L. Qu constructed several classes of bent functions from quadratic permutations and permutations with Niho exponents. In this paper, we further investigate one of these classes of bent functions, establishing more general sufficient conditions and discussing their relationship with the class of Maiorana–McFarland bent functions.","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"8 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147642165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Analysis of some classes of bent partitions and vectorial bent functions","authors":"Nurdagül Anbar, Fang-Wei Fu, Tekgül Kalaycı, Wilfried Meidl, Jiaxin Wang, Yadi Wei","doi":"10.1007/s10623-026-01835-9","DOIUrl":"https://doi.org/10.1007/s10623-026-01835-9","url":null,"abstract":"","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"81 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147617517","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A hybrid of lattice-reduction and Meet-LWE via near-collision on babai’s plane","authors":"Minki Hhan, Jiseung Kim, Changmin Lee, Yongha Son","doi":"10.1007/s10623-026-01813-1","DOIUrl":"https://doi.org/10.1007/s10623-026-01813-1","url":null,"abstract":"A cryptographic primitive based on the learning with errors (LWE) problem with variants is a promising candidate for the efficient quantum-resistant public key cryptosystem. As the parameters for such cryptosystems are chosen by the concrete attack cost for the corresponding LWE problem, improving the LWE solving algorithm has significant importance. In this paper, we present a new hybrid attack on the LWE problem. This new attack combines the primal lattice attack and an improved variant of the MitM attack, Meet-LWE, suggested by May (Crypto’21). This resolves the major open problem posed in the same paper. To this end, we develop several technical tools for hybrid attacks; a new property of Babai’s nearest plane algorithm with respect to projection, an approximate variant of Meet-LWE, and a locality-sensitive hashing-based near-collision finding algorithm. We also present a comprehensive analysis of the proposed attack, which involves the complicated arguments of both lattice and representation techniques. We finally estimate the concrete cost of our attack for sparse LWE keys. For some currently deployed parameter sets in fully homomorphic encryption libraries, our attack achieves up to 10 bits improvement than the previous primal hybrid attacks.","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"172 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147642128","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quasi-cyclic binary extended and expurgated Goppa codes and their parameters","authors":"Fengwei Li, Xue Jia, Huan Sun, Qin Yue","doi":"10.1007/s10623-025-01797-4","DOIUrl":"https://doi.org/10.1007/s10623-025-01797-4","url":null,"abstract":"In classic McEliece cryptosystems, separable quasi-cyclic binary Goppa codes play an important role in reducing the public key size. In this paper, we always assume that &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi mathvariant=\"double-struck\"&gt;F&lt;/mml:mi&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;q&lt;/mml:mi&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:msup&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathbb F_{q^2}$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; is a finite field with &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;q&lt;/mml:mi&gt;&lt;mml:mo&gt;=&lt;/mml:mo&gt;&lt;mml:msup&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;mml:mi&gt;s&lt;/mml:mi&gt;&lt;/mml:msup&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$q=2^{s}$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;G&lt;/mml:mi&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mi&gt;x&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;mml:mo&gt;=&lt;/mml:mo&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;x&lt;/mml:mi&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;q&lt;/mml:mi&gt;&lt;mml:mo&gt;+&lt;/mml:mo&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:msup&gt;&lt;mml:mo&gt;+&lt;/mml:mo&gt;&lt;mml:mi&gt;g&lt;/mml:mi&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;x&lt;/mml:mi&gt;&lt;mml:mi&gt;q&lt;/mml:mi&gt;&lt;/mml:msup&gt;&lt;mml:mo&gt;+&lt;/mml:mo&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;g&lt;/mml:mi&gt;&lt;mml:mi&gt;q&lt;/mml:mi&gt;&lt;/mml:msup&gt;&lt;mml:mi&gt;x&lt;/mml:mi&gt;&lt;mml:mo&gt;+&lt;/mml:mo&gt;&lt;mml:mi&gt;h&lt;/mml:mi&gt;&lt;mml:mo&gt;∈&lt;/mml:mo&gt;&lt;mml:msub&gt;&lt;mml:mi mathvariant=\"double-struck\"&gt;F&lt;/mml:mi&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;q&lt;/mml:mi&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:msup&gt;&lt;/mml:msub&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;[&lt;/mml:mo&gt;&lt;mml:mi&gt;x&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;]&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$G(x)=x^{q+1}+g x^{q}+g^q x +hin mathbb F_{q^{2 }}[x]$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; is a Goppa polynomial. We explicitly describe the complete irreducible factorizations of the polynomial &lt;italic&gt;G&lt;/italic&gt;(&lt;italic&gt;x&lt;/italic&gt;) over &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi mathvariant=\"double-struck\"&gt;F&lt;/mml:mi&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;q&lt;/mml:mi&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:msup&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathbb F_{q^2}$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. Let &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;L&lt;/mml:mi&gt;&lt;mml:mo&gt;=&lt;/mml:mo&gt;&lt;mml:m","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"126 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147642125","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Weightwise almost perfectly balanced functions, construction from a permutation group action view","authors":"Deepak Kumar Dalai, Krishna Mallick, Pierrick Méaux","doi":"10.1007/s10623-026-01818-w","DOIUrl":"https://doi.org/10.1007/s10623-026-01818-w","url":null,"abstract":"The construction of Boolean functions with good cryptographic properties over subsets of vectors with fixed Hamming weight is significant for lightweight stream ciphers like FLIP. This article introduces a general construction for a class of Weightwise Almost Perfectly Balanced (WAPB) Boolean functions, based on the action of a cyclic permutation group on &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msubsup&gt;&lt;mml:mi mathvariant=\"double-struck\"&gt;F&lt;/mml:mi&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;/mml:msubsup&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathbb {F}_2^n$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. This class generalizes the Weightwise Perfectly Balanced (WPB) &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;mml:mo&gt;=&lt;/mml:mo&gt;&lt;mml:msup&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;mml:mi&gt;m&lt;/mml:mi&gt;&lt;/mml:msup&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$n = 2^m$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;-variable Boolean function construction by Liu and Mesnager to any &lt;italic&gt;n&lt;/italic&gt;. We establish theoretical bounds on the nonlinearity and weightwise nonlinearity of the resulting functions. Particularly, we explore two significant permutation groups, &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;⟨&lt;/mml:mo&gt;&lt;mml:mi&gt;ψ&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;⟩&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$langle psi rangle $$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;⟨&lt;/mml:mo&gt;&lt;mml:mi&gt;σ&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;⟩&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$langle sigma rangle $$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, where &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mi&gt;ψ&lt;/mml:mi&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$psi $$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; is a distinct binary-cycle permutation and &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mi&gt;σ&lt;/m","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"18 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147642126","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"LRC codes over characteristic 2","authors":"F. Galluccio","doi":"10.1007/s10623-026-01845-7","DOIUrl":"https://doi.org/10.1007/s10623-026-01845-7","url":null,"abstract":"In this work the construction of LRC codes given in Chara et al. (Finite Fields Appl 94:102359, 2024) is completed, in the case of even characteristic. A general construction is presented, that enables us to obtain linear LRC codes of large length <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>n</mml:mi><mml:mo>≈</mml:mo><mml:msup><mml:mi>q</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$n approx q^4$$end{document}</tex-math></alternatives></inline-formula>, dimension and distance of order <inline-formula><alternatives><mml:math><mml:msup><mml:mi>q</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$q^4$$end{document}</tex-math></alternatives></inline-formula>, and locality <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mi>q</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$r =q-1$$end{document}</tex-math></alternatives></inline-formula>. In addition, the cases <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$q = 4$$end{document}</tex-math></alternatives></inline-formula> and <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>8</mml:mn></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$q=8$$end{document}</tex-math></alternatives></inline-formula> are studied.","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"276 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147642120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Perfect codes in Cayley graphs of abelian groups","authors":"Peter J. Cameron, Roro Sihui Yap, Sanming Zhou","doi":"10.1007/s10623-026-01821-1","DOIUrl":"https://doi.org/10.1007/s10623-026-01821-1","url":null,"abstract":"A perfect code in a graph <inline-formula><alternatives><mml:math><mml:mrow><mml:mi mathvariant=\"normal\">Γ</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>V</mml:mi><mml:mo>,</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Gamma = (V, E)$$end{document}</tex-math></alternatives></inline-formula> is a subset <italic>C</italic> of <italic>V</italic> such that no two vertices in <italic>C</italic> are adjacent and every vertex in <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>V</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\"></mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$V setminus C$$end{document}</tex-math></alternatives></inline-formula> is adjacent to exactly one vertex in <italic>C</italic>. A total perfect code in <inline-formula><alternatives><mml:math><mml:mi mathvariant=\"normal\">Γ</mml:mi></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Gamma $$end{document}</tex-math></alternatives></inline-formula> is a subset <italic>C</italic> of <italic>V</italic> such that every vertex of <inline-formula><alternatives><mml:math><mml:mi mathvariant=\"normal\">Γ</mml:mi></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Gamma $$end{document}</tex-math></alternatives></inline-formula> is adjacent to exactly one vertex in <italic>C</italic>. In this paper we prove several results on perfect codes and total perfect codes in Cayley graphs of finite abelian groups.","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"33 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147642123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}