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Optimal combinatorial neural codes via symmetric designs
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-12-18 DOI: 10.1007/s10623-024-01534-3
Xingyu Zheng, Shukai Wang, Cuiling Fan
{"title":"Optimal combinatorial neural codes via symmetric designs","authors":"Xingyu Zheng, Shukai Wang, Cuiling Fan","doi":"10.1007/s10623-024-01534-3","DOIUrl":"https://doi.org/10.1007/s10623-024-01534-3","url":null,"abstract":"<p>Combinatorial neural (CN) codes are binary codes introduced firstly by Curto et al. for asymmetric channel, and then are further studied by Cotardo and Ravagnani under the metric <span>(delta _r)</span> (called asymmetric discrepancy) which measures the differentiation of codewords in CN codes. When <span>(r&gt;1)</span>, CN codes are different from the usual error-correcting codes in symmetric channel (<span>(r=1)</span>). In this paper, we focus on the optimality of some CN codes with <span>(r&gt;1)</span>. An upper bound for the size of CN codes with <span>(delta _r=r+1)</span> is deduced, by discussing the relationship between such CN codes and error-detecting codes for asymmetric channels, which is shown to be tight in this case. We also propose an improved Plotkin bound for CN codes. Notably, by applying symmetric designs related with Hadamard matrices, we not only generalize one former construction of optimal CN codes by bent functions obtained by Zhang et al. (IEEE Trans Inf Theory 69:5440–5448, 2023), but also obtain seven classes of new optimal CN codes meeting the improved Plotkin bound.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"8 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142841501","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
Relating code equivalence to other isomorphism problems 将代码等价与其他同构问题联系起来
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-12-16 DOI: 10.1007/s10623-024-01542-3
Huck Bennett, Kaung Myat Htay Win
{"title":"Relating code equivalence to other isomorphism problems","authors":"Huck Bennett, Kaung Myat Htay Win","doi":"10.1007/s10623-024-01542-3","DOIUrl":"https://doi.org/10.1007/s10623-024-01542-3","url":null,"abstract":"<p>We study the complexity of the <i>Code Equivalence Problem</i> on linear error-correcting codes by relating its variants to isomorphism problems on other discrete structures—graphs, lattices, and matroids. Our main results are a fine-grained reduction from the Graph Isomorphism Problem to the Linear Code Equivalence Problem over any field <span>(mathbb {F})</span>, and a reduction from the Linear Code Equivalence Problem over any field <span>(mathbb {F}_p)</span> of prime, polynomially bounded order <i>p</i> to the Lattice Isomorphism Problem. Both of these reductions are simple and natural. We also give reductions between variants of the Code Equivalence Problem, and study the relationship between isomorphism problems on codes and linear matroids.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"2021 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142832039","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
Hulls of projective Reed–Muller codes 射影里德-穆勒码的船体
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-12-14 DOI: 10.1007/s10623-024-01543-2
Nathan Kaplan, Jon-Lark Kim
{"title":"Hulls of projective Reed–Muller codes","authors":"Nathan Kaplan, Jon-Lark Kim","doi":"10.1007/s10623-024-01543-2","DOIUrl":"https://doi.org/10.1007/s10623-024-01543-2","url":null,"abstract":"<p>Projective Reed–Muller codes are constructed from the family of projective hypersurfaces of a fixed degree over a finite field <span>(mathbb {F}_q)</span>. We consider the relationship between projective Reed–Muller codes and their duals. We determine when these codes are self-dual, when they are self-orthogonal, and when they are LCD. We then show that when <i>q</i> is sufficiently large, the dimension of the hull of a projective Reed–Muller code is 1 less than the dimension of the code. We determine the dimension of the hull for a wider range of parameters and describe how this leads to a new proof of a recent result of Ruano and San-José.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"63 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142823245","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 set systems with strongly restricted intersections
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-12-05 DOI: 10.1007/s10623-024-01535-2
Xin Wei, Xiande Zhang, Gennian Ge
{"title":"On set systems with strongly restricted intersections","authors":"Xin Wei, Xiande Zhang, Gennian Ge","doi":"10.1007/s10623-024-01535-2","DOIUrl":"https://doi.org/10.1007/s10623-024-01535-2","url":null,"abstract":"<p>Set systems with strongly restricted intersections, called <span>(alpha )</span>-intersecting families for a vector <span>(alpha )</span>, were introduced recently as a generalization of several well-studied intersecting families including the classical oddtown and eventown. Given a binary vector <span>(alpha =(a_1, ldots , a_k))</span>, a collection <span>({mathcal {F}})</span> of subsets over an <i>n</i> element set is an <span>(alpha )</span>-intersecting family modulo 2 if for each <span>(i=1,2,ldots ,k)</span>, all <i>i</i>-wise intersections of distinct members in <span>({mathcal {F}})</span> have sizes with the same parity as <span>(a_i)</span>. Let <span>(f_alpha (n))</span> denote the maximum size of such a family. In this paper, we study the asymptotic behavior of <span>(f_alpha (n))</span> when <i>n</i> goes to infinity. We show that if <i>t</i> is the maximum integer such that <span>(a_t=1)</span> and <span>(2tle k)</span>, then <span>(f_alpha (n)sim (t! n)^{1/t})</span>. More importantly, we show that for any constant <span>(c&gt;0)</span>, as the length <i>k</i> goes larger, <span>(f_alpha (n))</span> is upper bounded by <span>(O (n^c))</span> for almost all <span>(alpha )</span>. Equivalently, no matter what <i>k</i> is, there are only finitely many <span>(alpha )</span> satisfying <span>(f_alpha (n)=Omega (n^c))</span>. This answers an open problem raised by Johnston and O’Neill in 2023. All of our results can be generalized to modulo <i>p</i> setting for any prime <i>p</i> smoothly.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"37 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776806","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 3-dimensional MRD codes of type $$langle X^{q^t},X+delta X^{q^{2t}},G(X) rangle $$
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-12-05 DOI: 10.1007/s10623-024-01528-1
Daniele Bartoli, Francesco Ghiandoni
{"title":"On 3-dimensional MRD codes of type $$langle X^{q^t},X+delta X^{q^{2t}},G(X) rangle $$","authors":"Daniele Bartoli, Francesco Ghiandoni","doi":"10.1007/s10623-024-01528-1","DOIUrl":"https://doi.org/10.1007/s10623-024-01528-1","url":null,"abstract":"<p>In this work we present results on the classification of <span>(mathbb {F}_{q^n})</span>-linear MRD codes of dimension three. In particular, using connections with certain algebraic varieties over finite fields, we provide non-existence results for MRD codes <span>(mathcal {C}=langle X^{q^t}, F(X), G(X) rangle subseteq mathcal {L}_{n,q})</span> of exceptional type, i.e. such that <span>(mathcal {C})</span> is MRD over infinitely many extensions of the base field. These results partially address a conjecture of Bartoli, Zini and Zullo in 2023.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"20 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776813","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
Derivative descendants of cyclic codes and constacyclic codes
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-12-04 DOI: 10.1007/s10623-024-01536-1
Li Xu, Cuiling Fan, Chunming Tang, Zhengchun Zhou
{"title":"Derivative descendants of cyclic codes and constacyclic codes","authors":"Li Xu, Cuiling Fan, Chunming Tang, Zhengchun Zhou","doi":"10.1007/s10623-024-01536-1","DOIUrl":"https://doi.org/10.1007/s10623-024-01536-1","url":null,"abstract":"<p>Cyclic codes, as a special type of constacyclic codes, have been extensively studied due to their favorable theoretical and mathematical properties. Very recently, by using the derivative of the Mattson-Solomon polynomials, Huang and Zhang (IEEE Trans Inf Theor 70(4):2395–2410, 2024) studied the cyclic derivative descendants (DDs) and linear DDs of binary extended cyclic codes and proposed the corresponding derivative decoding methods. One objective of this paper is to generalize these conclusions to <i>q</i>-ary extended cyclic codes with group algebra theory. It demonstrates that the cyclic DDs of a <i>q</i>-ary extended cyclic code are the same codes and its linear DDs are equivalent codes. In addition, we show that the relevant results can be generalized to <i>q</i>-ary constacyclic codes and the linear codes generated by Plotkin construction. Our conclusions reveal that the soft-decision decoding method proposed by Huang and Zhang for binary cyclic codes is also applicable to <i>q</i>-ary cyclic codes, <i>q</i>-ary constacyclic codes and the linear codes generated by Plotkin construction.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"67 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776828","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
Codes over $$mathbb {F}_4$$ and $$mathbb {F}_2 times mathbb {F}_2$$ and theta series of the corresponding lattices in quadratic fields
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-12-04 DOI: 10.1007/s10623-024-01537-0
Josline Freed
{"title":"Codes over $$mathbb {F}_4$$ and $$mathbb {F}_2 times mathbb {F}_2$$ and theta series of the corresponding lattices in quadratic fields","authors":"Josline Freed","doi":"10.1007/s10623-024-01537-0","DOIUrl":"https://doi.org/10.1007/s10623-024-01537-0","url":null,"abstract":"<p>Using codes defined over <span>(mathbb {F}_4)</span> and <span>(mathbb {F}_2 times mathbb {F}_2)</span>, we simultaneously define the theta series of corresponding lattices for both real and imaginary quadratic fields <span>(mathbb {Q}(sqrt{d}))</span> with <span>(d equiv 1mod 4)</span> a square-free integer. For such a code, we use its weight enumerator to prove which term in the code’s corresponding theta series is the first to depend on the choice of <i>d</i>. For a given choice of real or imaginary quadratic field, we find conditions on the length of the code relative to the choice of quadratic field. When these conditions are satisfied, the generated theta series is unique to the code’s symmetric weight enumerator. We show that whilst these conditions ensure all non-equivalent codes will produce distinct theta series, for other codes that do not satisfy this condition, the length of the code and choice of quadratic field is not always enough to determine if the corresponding theta series will be unique.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"29 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776825","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 pair of orthogonal orthomorphisms of finite nilpotent groups
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-12-04 DOI: 10.1007/s10623-024-01540-5
Shikang Yu, Tao Feng, Menglong Zhang
{"title":"A pair of orthogonal orthomorphisms of finite nilpotent groups","authors":"Shikang Yu, Tao Feng, Menglong Zhang","doi":"10.1007/s10623-024-01540-5","DOIUrl":"https://doi.org/10.1007/s10623-024-01540-5","url":null,"abstract":"<p>A bijection <span>(theta :Grightarrow G)</span> of a finite group <i>G</i> is an orthomorphism of <i>G</i> if the mapping <span>(xmapsto x^{-1}theta (x))</span> is also a bijection. Two orthomorphisms <span>(theta )</span> and <span>(phi )</span> of a finite group <i>G</i> are orthogonal if the mapping <span>(xmapsto theta (x)^{-1}phi (x))</span> is also bijective. We show that there is a pair of orthogonal orthomorphisms of a finite nilpotent group <i>G</i> if and only if the Sylow 2-subgroup of <i>G</i> is either trivial or noncyclic with the definite exceptions of <span>(Gcong G')</span> where <span>(G'in {D_8,Q_8,{mathbb {Z}}_3,{mathbb {Z}}_9})</span> and except possibly for <span>(Gcong Q_8times {mathbb {Z}}_9)</span> or <span>(Gcong SD_{2^n}times {mathbb {Z}}_3)</span> for any <span>(ngeqslant 4)</span>. This result yields the existence of difference matrices over finite nilpotent groups with four rows.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"9 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776826","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
Factorization and irreducibility of composed products
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-12-04 DOI: 10.1007/s10623-024-01529-0
Lukas Kölsch, Lucas Krompholz, Gohar Kyureghyan
{"title":"Factorization and irreducibility of composed products","authors":"Lukas Kölsch, Lucas Krompholz, Gohar Kyureghyan","doi":"10.1007/s10623-024-01529-0","DOIUrl":"https://doi.org/10.1007/s10623-024-01529-0","url":null,"abstract":"<p>Brawley and Carlitz introduced diamond products of elements of finite fields and associated composed products of polynomials in 1987. Composed products yield a method to construct irreducible polynomials of large composite degrees from irreducible polynomials of lower degrees. We show that the composed product of two irreducible polynomials of degrees <i>m</i> and <i>n</i> is again irreducible if and only if <i>m</i> and <i>n</i> are coprime and the involved diamond product satisfies a special cancellation property, the so-called conjugate cancellation. This completes the characterization of irreducible composed products, considered in several previous papers. More generally, we give precise criteria when a diamond product satisfies conjugate cancellation. For diamond products defined via bivariate polynomials, we prove simple criteria that characterize when conjugate cancellation holds. We also provide efficient algorithms to check these criteria. We achieve stronger results as well as more efficient algorithms in the case that the polynomials are bilinear. Lastly, we consider possible constructions of normal elements using composed products and the methods we developed.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"3 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776830","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 translation hyperovals in semifield planes
IF 1.6 2区 数学
Designs, Codes and Cryptography Pub Date : 2024-12-04 DOI: 10.1007/s10623-024-01533-4
Kevin Allen, John Sheekey
{"title":"On translation hyperovals in semifield planes","authors":"Kevin Allen, John Sheekey","doi":"10.1007/s10623-024-01533-4","DOIUrl":"https://doi.org/10.1007/s10623-024-01533-4","url":null,"abstract":"<p>In this paper we demonstrate the first example of a finite translation plane which does not contain a translation hyperoval, disproving a conjecture of Cherowitzo. The counterexample is a semifield plane, specifically a Generalised Twisted Field plane, of order 64. We also relate this non-existence to the covering radius of two associated rank-metric codes, and the non-existence of scattered subspaces of maximum dimension with respect to the associated spread.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"83 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776804","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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