Designs, Codes and Cryptography最新文献

{"title":"On $$ell $$-rank additive intersection pairs (RAIP) of codes","authors":"Sanjit Bhowmick, Kuntal Deka, Sihem Mesnager","doi":"10.1007/s10623-026-01929-4","DOIUrl":"https://doi.org/10.1007/s10623-026-01929-4","url":null,"abstract":"","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"7 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148768981","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":"The weight spectrum of generalized Reed–Muller codes $$RM_q((m-2)(q-1)+3,m)$$","authors":"Somayyeh Golalizadeh, Claude Carlet, Nasrin Soltankhah","doi":"10.1007/s10623-026-01908-9","DOIUrl":"https://doi.org/10.1007/s10623-026-01908-9","url":null,"abstract":"","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"19 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148768982","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":"Constructions of binary self-orthogonal singly-even wide minimal linear codes with few weights","authors":"Kangquan Li, Hao Chen, Wengang Jin, Longjiang Qu","doi":"10.1007/s10623-026-01921-y","DOIUrl":"https://doi.org/10.1007/s10623-026-01921-y","url":null,"abstract":"","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"36 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148768980","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":"Subgroup perfect codes of simple N-groups","authors":"Zheng Gao Chen, Jing Jian Li, Hao Yu","doi":"10.1007/s10623-026-01927-6","DOIUrl":"https://doi.org/10.1007/s10623-026-01927-6","url":null,"abstract":"","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"9 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148768983","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":"Soft planes and groups","authors":"William M. Kantor","doi":"10.1007/s10623-026-01874-2","DOIUrl":"https://doi.org/10.1007/s10623-026-01874-2","url":null,"abstract":"Finite projective planes are constructed using groups that satisfy simple-looking conditions. The resulting projective planes include many known planes and possibly new ones, and are precisely those having a collineation group fixing a flag <inline-formula><alternatives><mml:math><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>∞</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mi>∞</mml:mi></mml:msub><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}$$(infty ,L_ infty )$$end{document}</tex-math></alternatives></inline-formula> and transitive on the flags (<italic>w</italic>, <italic>W</italic>) with <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>w</mml:mi><mml:mo>∉</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mi>∞</mml:mi></mml:msub></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}$$wnotin L_ infty $$end{document}</tex-math></alternatives></inline-formula> and <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>∞</mml:mi><mml:mo>∉</mml:mo><mml:mi>W</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}$$infty notin W$$end{document}</tex-math></alternatives></inline-formula>.","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"26 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687827","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":"Minimal 2n-covers of the hyperbolic quadric Q+(4n+1,q)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Q^+(4n+1,q)$$end{document}","authors":"Laure Schelfhout, Leo Storme","doi":"10.1007/s10623-026-01919-6","DOIUrl":"https://doi.org/10.1007/s10623-026-01919-6","url":null,"abstract":"We present constructions of small 2<italic>n</italic>-covers of the hyperbolic quadric <inline-formula><alternatives><mml:math><mml:mrow><mml:msup><mml:mi>Q</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></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}$$Q^+ (4n+1,q)$$end{document}</tex-math></alternatives></inline-formula>. We also present a lower bound on the size of a 2<italic>n</italic>-cover of <inline-formula><alternatives><mml:math><mml:mrow><mml:msup><mml:mi>Q</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></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}$$Q^+ (4n+1,q)$$end{document}</tex-math></alternatives></inline-formula>.","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"358 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687826","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 the decoding error weight of one or two deletion channels","authors":"Omer Sabary, Daniella Bar-Lev, Yotam Gershon, Alexander Yucovich, Eitan Yaakobi","doi":"10.1007/s10623-026-01873-3","DOIUrl":"https://doi.org/10.1007/s10623-026-01873-3","url":null,"abstract":"This paper tackles two problems that fall under the study of coding for insertions and deletions. These problems are motivated by several applications, among them is reconstructing strands in DNA-based storage systems. Under this paradigm, a word is transmitted over some fixed number of identical independent channels and the goal of the decoder is to output the transmitted word or some close approximation of it. The first part of the paper studies optimal decoding for a special case of the deletion channel, referred by the &lt;italic&gt;k&lt;/italic&gt;&lt;italic&gt;-deletion channel&lt;/italic&gt;, which deletes exactly &lt;italic&gt;k&lt;/italic&gt; symbols of the transmitted word uniformly at random. In this part, the goal is to understand how an optimal decoder operates in order to minimize the expected normalized distance. A full characterization of an efficient optimal decoder for this setup, referred to as &lt;italic&gt;the minimum expected distance (MED) decoder&lt;/italic&gt;, is given for a channel that deletes one or two symbols. For &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;k&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: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}$$k=1$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; it is shown that when the code is the entire space, the decoder is the &lt;italic&gt;lazy decoder&lt;/italic&gt; which simply returns the channel output. Similarly, for &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;k&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}$$k=2$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; it is shown that the decoder acts as the lazy decoder in almost all cases and when the longest run is significantly long (roughly &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:mn&gt;2&lt;/mml:mn&gt;&lt;mml:mo&gt;-&lt;/mml:mo&gt;&lt;mml:msqrt&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:msqrt&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;mml:mi&gt;n&lt;/mml:mi&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}$$(2-sqrt{2})n$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; when &lt;italic&gt;n&lt;/italic&gt; is the word length), it prolongs the longest run by one symbol. The second part of this paper studies the deletion channel that deletes a symbol with some fixed probability &lt;italic&gt;p&lt;/italic&gt;, while focusing on two instances of this channel. Since operating the MED decoder, in this case, is computational","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"9 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687828","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":"Tagged-FHE: strong syntax, stronger security, and standard model","authors":"Ziqing Wang, Mengdi Ouyang, Zhaosen Shi, Xinyan Wu, Fuchun Guo, Fagen Li","doi":"10.1007/s10623-026-01912-z","DOIUrl":"https://doi.org/10.1007/s10623-026-01912-z","url":null,"abstract":"Most existing fully homomorphic encryption (FHE) security models cannot resist attacks from a malicious server that can arbitrarily replace the ciphertexts and evaluation functions used during homomorphic evaluation and obtain the decryption results of the evaluated ciphertexts. To address this security issue of FHE caused by the breakdown of computational integrity, we proposed two new primitives, called tagged-FHE and &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msup&gt;&lt;mml:mtext&gt;tagged&lt;/mml:mtext&gt;&lt;mml:mo&gt;∗&lt;/mml:mo&gt;&lt;/mml:msup&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}$$hbox {tagged}^*$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;-FHE with corresponding security notions IND-TAG-CCA and IND-&lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msup&gt;&lt;mml:mtext&gt;TAG&lt;/mml:mtext&gt;&lt;mml:mo&gt;∗&lt;/mml:mo&gt;&lt;/mml:msup&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}$$hbox {TAG}^*$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;-CCA. Unlike the verifiable FHE, which also focuses on preserving computational integrity, the new primitives and security concepts still require the compactness of ciphertexts. The size of the output of the evaluation algorithm must be independent of the complexity of the function during evaluation. Tagged-FHE enables users to detect whether a server has replaced the input ciphertexts of evaluation. While &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msup&gt;&lt;mml:mtext&gt;tagged&lt;/mml:mtext&gt;&lt;mml:mo&gt;∗&lt;/mml:mo&gt;&lt;/mml:msup&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}$$hbox {tagged}^*$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;-FHE additionally enables users to detect whether a server has replaced the evaluation function. We present generic constructions for these two primitives with corresponding security notions. The IND-TAG-CCA tagged-FHE is achievable in the standard model using IND-CPA multi-key FHE and IND-CCA2 public-key encryption. The IND-&lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msup&gt;&lt;mml:mtext&gt;TAG&lt;/mml:mtext&gt;&lt;mml:mo&gt;∗&lt;/mml:mo&gt;&lt;/mml:msup&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}$$hbox {TAG}^*$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;-CCA &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msup&gt;&lt;mml:mtext&gt;tagged&lt;/mml:mtext&gt;&lt;mml:mo&gt;∗&lt;/mml:mo&gt;&lt;/mm","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"19 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148688051","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":"Constructions of duadic codes with minimum weights exceeding the square-root lower bound","authors":"Yuanting Zhang, Shixin Zhu","doi":"10.1007/s10623-026-01917-8","DOIUrl":"https://doi.org/10.1007/s10623-026-01917-8","url":null,"abstract":"Duadic codes are an interesting subclass of cyclic codes since they have large dimensions and their minimum distances may have a square-root lower bound. The objective of this paper is to construct several families of <inline-formula><alternatives><mml:math><mml:msup><mml:mi>q</mml:mi><mml:mn>2</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}$$q^2$$end{document}</tex-math></alternatives></inline-formula>-ary duadic codes of dimension near <inline-formula><alternatives><mml:math><mml:mfrac><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:mfrac></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}$$frac{n}{2}$$end{document}</tex-math></alternatives></inline-formula> with good lower bounds on their minimum distances. In this paper, we propose a construction for <inline-formula><alternatives><mml:math><mml:msup><mml:mi>q</mml:mi><mml:mn>2</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}$$q^2$$end{document}</tex-math></alternatives></inline-formula>-ary duadic codes whose splitting is given by <inline-formula><alternatives><mml:math><mml:msub><mml:mi>μ</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:msub></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}$$mu _{-q}$$end{document}</tex-math></alternatives></inline-formula>. Using this construction, we obtain four infinite families of <inline-formula><alternatives><mml:math><mml:msup><mml:mi>q</mml:mi><mml:mn>2</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}$$q^2$$end{document}</tex-math></alternatives></inline-formula>-ary duadic codes with lower bounds on their minimum distances exceeding the square-root lower bound. As byproducts, we derive several families of Hermitian self-dual codes and Hermitian self-orthogonal cyclic codes with lower bounds on their minimum distances also exceeding the square-root lower bound. These codes include distance-optimal codes and codes with the best known parameters.","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"19 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687802","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":"Generalized hamming weights of almost-MDS codes with respect to twisted Reed–Solomon codes","authors":"Ming Yang, Qin Yue, Fengwei Li","doi":"10.1007/s10623-026-01924-9","DOIUrl":"https://doi.org/10.1007/s10623-026-01924-9","url":null,"abstract":"Generalized Hamming weights are fundamental invariants of linear codes that provide a refined description of the support structure of subcodes. While the weight hierarchies of Maximum Distance Separable (MDS) codes and Near-MDS (NMDS) codes are well investigated, much less is known about Almost-MDS (AMDS) codes that are not NMDS. In this paper, we prove that an [<italic>n</italic>, <italic>k</italic>] linear code is AMDS but not NMDS if and only if its first two generalized Hamming weights are <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mi>k</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}$$n-k$$end{document}</tex-math></alternatives></inline-formula> and <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mi>k</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}$$n-k+1$$end{document}</tex-math></alternatives></inline-formula>, respectively. By introducing the concept of <italic>s</italic>-gap AMDS codes, we obtain that the unique missing entry in the AMDS hierarchy is <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mi>s</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}$$n-k+s$$end{document}</tex-math></alternatives></inline-formula>, <inline-formula><alternatives><mml:math><mml:mrow><mml:mn>2</mml:mn><mml:mo>≤</mml:mo><mml:mi>s</mml:mi><mml:mo>≤</mml:mo><mml:mi>k</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}$$2le sle k$$end{document}</tex-math></alternatives></inline-formula>. Moreover, we apply this framework to twisted Reed-Solomon codes, transforming the construction of <italic>s</italic>-gap AMDS codes into a problem of counting roots of specific polynomials over finite fields. Based on this transformation, we establish explicit construction criteria for three families of <italic>s</italic>-gap AMDS codes, along with several concrete illustrative examples.","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"35 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2026-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687823","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}