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An asymptotic property of quaternary additive codes 四元加法码的渐近特性
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-06-12 DOI: 10.1007/s10623-024-01438-2
Jürgen Bierbrauer, Stefano Marcugini, Fernanda Pambianco
{"title":"An asymptotic property of quaternary additive codes","authors":"Jürgen Bierbrauer, Stefano Marcugini, Fernanda Pambianco","doi":"10.1007/s10623-024-01438-2","DOIUrl":"https://doi.org/10.1007/s10623-024-01438-2","url":null,"abstract":"<p>Let <span>(n_k(s))</span> be the maximal length <i>n</i> such that a quaternary additive <span>([n,k,n-s]_4)</span>-code exists. We solve a natural asymptotic problem by determining the lim sup <span>(lambda _k)</span> of <span>(n_k(s)/s)</span> for <i>s</i> going to infinity, and the smallest value of <i>s</i> such that <span>(n_k(s)/s=lambda _k.)</span> Our new family of quaternary additive codes has parameters <span>([4^k-1,k,4^k-4^{k-1}]_4=[2^{2k}-1,k,3cdot 2^{2k-2}]_4)</span> (where <span>(k=l/2)</span> and <i>l</i> is an odd integer). These are constant-weight codes. The binary codes obtained by concatenation with inner code <span>([3,2,2]_2)</span> meet the Griesmer bound with equality. The proof is in terms of multisets of lines in <span>(PG(l-1,2))</span>.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141309080","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}
引用次数: 0
Circular external difference families: construction and non-existence 循环外部差异族：构建与不存在
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-06-12 DOI: 10.1007/s10623-024-01443-5
Huawei Wu, Jing Yang, Keqin Feng
{"title":"Circular external difference families: construction and non-existence","authors":"Huawei Wu, Jing Yang, Keqin Feng","doi":"10.1007/s10623-024-01443-5","DOIUrl":"https://doi.org/10.1007/s10623-024-01443-5","url":null,"abstract":"<p>The circular external difference family and its strong version are of great significance both in theory and in applications. In this paper, we apply the classical cyclotomic construction to the circular external differnece family and exhibit several concrete examples, in particular constructing an infinite family. Furthermore, we prove that all strong circular external differnece families are constructed by patching together several strong external difference families consisting of two subsets, thereby solving the open problem raised by Veitch and Stinson. We also present a new result on the non-existence of a certain type of strong external difference families.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141315778","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}
引用次数: 0
On prefer-one sequences 关于优选序列
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-06-10 DOI: 10.1007/s10623-024-01425-7
Yupeng Jiang, Ming Li, Ying Gao, Dongdai Lin
{"title":"On prefer-one sequences","authors":"Yupeng Jiang, Ming Li, Ying Gao, Dongdai Lin","doi":"10.1007/s10623-024-01425-7","DOIUrl":"https://doi.org/10.1007/s10623-024-01425-7","url":null,"abstract":"<p>We study the sequences generated by prefer-one rule with different initial vectors. Firstly, we give upper bounds of their periods and for initial vectors with Hamming weight one, we prove that the generated sequences are modified de Bruijn sequences. Moreover, for two of them, we give the truth tables of their feedback functions. We also investigate the feedback functions of prefer-one de Bruijn sequences. For order <i>n</i> prefer-one de Bruijn sequence, we give linear and quadratic terms in its feedback function and prove that the number of degree <span>(n-2)</span> terms has the same parity as <i>n</i>. The statistical result for small <i>n</i> shows that about half of all terms occur in the feedback functions.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141299115","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}
引用次数: 0
External codes for multiple unicast networks via interference alignment 通过干扰对齐实现多个单播网络的外部编码
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-06-10 DOI: 10.1007/s10623-024-01439-1
F. R. Kschischang, F. Manganiello, A. Ravagnani, K. Savary
{"title":"External codes for multiple unicast networks via interference alignment","authors":"F. R. Kschischang, F. Manganiello, A. Ravagnani, K. Savary","doi":"10.1007/s10623-024-01439-1","DOIUrl":"https://doi.org/10.1007/s10623-024-01439-1","url":null,"abstract":"<p>We introduce a formal framework to study the multiple unicast problem for a coded network in which the network code is linear over a finite field and fixed. We show that the problem corresponds to an interference alignment problem over a finite field. In this context, we establish an outer bound for the achievable rate region and provide examples of networks where the bound is sharp. We finally give evidence of the crucial role played by the field characteristic in the problem.\u0000</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141299090","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}
引用次数: 0
On the classification of skew Hadamard matrices of order $$varvec{36}$$ and related structures 关于阶为 $$varvec{36}$ 的倾斜哈达玛矩阵的分类及相关结构
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-06-07 DOI: 10.1007/s10623-024-01427-5
Makoto Araya, Masaaki Harada, Hadi Kharaghani, Ali Mohammadian, Behruz Tayfeh-Rezaie
{"title":"On the classification of skew Hadamard matrices of order $$varvec{36}$$ and related structures","authors":"Makoto Araya, Masaaki Harada, Hadi Kharaghani, Ali Mohammadian, Behruz Tayfeh-Rezaie","doi":"10.1007/s10623-024-01427-5","DOIUrl":"https://doi.org/10.1007/s10623-024-01427-5","url":null,"abstract":"<p>Two skew Hadamard matrices are considered <span>SH</span>-equivalent if they are similar by a signed permutation matrix. This paper determines the number of <span>SH</span>-inequivalent skew Hadamard matrices of order 36 for some types. We also study ternary self-dual codes and association schemes constructed from the skew Hadamard matrices of order 36.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141287241","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}
引用次数: 0
Indicator functions, v-numbers and Gorenstein rings in the theory of projective Reed–Muller-type codes 投影里德-穆勒型编码理论中的指示函数、v数和戈伦斯坦环
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-06-05 DOI: 10.1007/s10623-024-01437-3
Manuel González-Sarabia, Humberto Muñoz-George, Jorge A. Ordaz, Eduardo Sáenz-de-Cabezón, Rafael H. Villarreal
{"title":"Indicator functions, v-numbers and Gorenstein rings in the theory of projective Reed–Muller-type codes","authors":"Manuel González-Sarabia, Humberto Muñoz-George, Jorge A. Ordaz, Eduardo Sáenz-de-Cabezón, Rafael H. Villarreal","doi":"10.1007/s10623-024-01437-3","DOIUrl":"https://doi.org/10.1007/s10623-024-01437-3","url":null,"abstract":"<p>For projective Reed–Muller-type codes we give a global duality criterion in terms of the v-number and the Hilbert function of a vanishing ideal. As an application, we provide a global duality theorem for projective Reed–Muller-type codes over Gorenstein vanishing ideals, generalizing the known case where the vanishing ideal is a complete intersection. We classify self dual Reed–Muller-type codes over Gorenstein ideals using the regularity and a parity check matrix. For projective evaluation codes, we give a duality theorem inspired by that of affine evaluation codes. We show how to compute the regularity index of the <i>r</i>-th generalized Hamming weight function in terms of the standard indicator functions of the set of evaluation points.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141264905","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}
引用次数: 0
On the number of rational points of Artin–Schreier’s curves and hypersurfaces 论阿尔廷-施莱尔曲线和超曲面的有理点数
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-06-05 DOI: 10.1007/s10623-024-01431-9
F. E. Brochero Martínez, Daniela Oliveira
{"title":"On the number of rational points of Artin–Schreier’s curves and hypersurfaces","authors":"F. E. Brochero Martínez, Daniela Oliveira","doi":"10.1007/s10623-024-01431-9","DOIUrl":"https://doi.org/10.1007/s10623-024-01431-9","url":null,"abstract":"<p>Let <span>(mathbb {F}_{q^n})</span> represent the finite field with <span>(q^n)</span> elements. In this paper, our focus is on determining the number of <span>(mathbb {F}_{q^n})</span>-rational points for two specific objects: an affine Artin–Schreier curve given by the equation <span>(y^q-y = x(x^{q^i}-x)-lambda )</span>, and an Artin–Schreier hypersurface given by the equation <span>(y^q-y=sum _{j=1}^r a_jx_j(x_j^{q^{i_j}}-x_j)-lambda )</span>. Additionally, we establish that the Weil bound is only achieved in these cases when the trace of the element <span>(lambda in mathbb {F}_{q^n})</span> over the subfield <span>(mathbb {F}_q)</span> is equal to zero.\u0000</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141264885","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}
引用次数: 0
A note on approximate Hadamard matrices 关于近似哈达玛矩阵的说明
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-06-04 DOI: 10.1007/s10623-024-01430-w
Stefan Steinerberger
{"title":"A note on approximate Hadamard matrices","authors":"Stefan Steinerberger","doi":"10.1007/s10623-024-01430-w","DOIUrl":"https://doi.org/10.1007/s10623-024-01430-w","url":null,"abstract":"<p>A Hadamard matrix is a scaled orthogonal matrix with <span>(pm 1)</span> entries. Such matrices exist in certain dimensions: the Hadamard conjecture is that such a matrix always exists when <i>n</i> is a multiple of 4. A conjecture attributed to Ryser is that no circulant Hadamard matrices exist when <span>(n &gt; 4)</span>. Recently, Dong and Rudelson proved the existence of <i>approximate</i> Hadamard matrices in all dimensions: there exist universal <span>(0&lt; c&lt; C &lt; infty )</span> so that for all <span>(n ge 1)</span>, there is a matrix <span>(A in left{ -1,1right} ^{n times n})</span> satisfying, for all <span>(x in mathbb {R}^n)</span>, </p><span>$$begin{aligned} c sqrt{n} Vert xVert _2 le Vert AxVert _2 le C sqrt{n} Vert xVert _2. end{aligned}$$</span><p>We observe that, as a consequence of the existence of flat Littlewood polynomials, <i>circulant</i> approximate Hadamard matrices exist for all <span>(n ge 1)</span>.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141246578","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}
引用次数: 0
ZLR: a fast online authenticated encryption scheme achieving full security ZLR：实现全面安全的快速在线验证加密方案
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-05-30 DOI: 10.1007/s10623-024-01434-6
Wonseok Choi, Seongha Hwang, Byeonghak Lee, Jooyoung Lee
{"title":"ZLR: a fast online authenticated encryption scheme achieving full security","authors":"Wonseok Choi, Seongha Hwang, Byeonghak Lee, Jooyoung Lee","doi":"10.1007/s10623-024-01434-6","DOIUrl":"https://doi.org/10.1007/s10623-024-01434-6","url":null,"abstract":"<p>Online authenticated encryption has been considered of practical relevance in light-weight environments due to low latency and constant memory usage. In this paper, we propose a new tweakable block cipher-based online authenticated encryption scheme, dubbed <span>ZLR</span>, and its domain separation variant, dubbed <span>DS-ZLR</span>. <span>ZLR</span> and <span>DS-ZLR</span> follow the Encrypt-Mix-Encrypt paradigm. However, in contrast to existing schemes using the same paradigm such as <span>ELmE</span> and <span>CoLM</span>, <span>ZLR</span> and <span>DS-ZLR</span> enjoy <i>n</i>-bit security by using larger internal states with an efficient <span>ZHash</span>-like hashing algorithm. In this way, 2<i>n</i>-bit blocks are processed with only a single primitive call for hashing and two primitive calls for encryption and decryption, when they are based on an <i>n</i>-bit tweakable block cipher using <i>n</i>-bit (resp. 2<i>n</i>-bit) tweaks for <span>ZLR</span> (resp. <span>DS-ZLR</span>). Furthermore, they support pipelined computation as well as online nonce-misuse resistance. To the best of our knowledge, <span>ZLR</span> and <span>DS-ZLR</span> are the first pipelineable tweakable block cipher-based online authenticated encryption schemes of rate-2/3 that provide <i>n</i>-bit security with online nonce-misuse resistance.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141177536","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}
引用次数: 0
Constructions of t-strongly multimedia IPP codes with length $$t+1$$ 长度为 $$t+1$$ 的 t 强多媒体 IPP 编码的构造
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-05-28 DOI: 10.1007/s10623-024-01422-w
Jing Jiang, Fenggui Pei, Cailin Wen, Minquan Cheng, Henk D. L. Hollmann
{"title":"Constructions of t-strongly multimedia IPP codes with length $$t+1$$","authors":"Jing Jiang, Fenggui Pei, Cailin Wen, Minquan Cheng, Henk D. L. Hollmann","doi":"10.1007/s10623-024-01422-w","DOIUrl":"https://doi.org/10.1007/s10623-024-01422-w","url":null,"abstract":"<p>Strongly multimedia identifiable parent property code (<i>t</i>-SMIPPC) was introduced for protecting multimedia contents from illegally redistributing under the averaging collusion attack. Such a code has efficient algorithms for tracing colluders. However, there are few results about the existence of such codes up to now. In this paper, we focus on <i>t</i>-SMIPPCs with length <span>(t+1)</span> where <span>(t ge 2)</span> is an integer. We first improve the lower bound on the size of such codes. For the case <span>(t=2)</span>, i.e., 2-SMIPPCs with length 3, we further investigate combinatorial properties of the codes. Based on these properties, optimal <i>q</i>-ary 2-SMIPPCs with length 3 are constructed for <span>(qequiv 0,1,2,5 pmod 6)</span>.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141165113","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}
引用次数: 0
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