Discrete Mathematics最新文献

{"title":"A survey of complex generalized weighing matrices and a construction of quantum error-correcting codes","authors":"","doi":"10.1016/j.disc.2024.114201","DOIUrl":"10.1016/j.disc.2024.114201","url":null,"abstract":"<div><p>Some combinatorial designs, such as Hadamard matrices, have been extensively researched and are familiar to readers across the spectrum of Science and Engineering. They arise in diverse fields such as cryptography, communication theory, and quantum computing. Objects like this also lend themselves to compelling mathematics problems, such as the Hadamard conjecture. However, complex generalized weighing matrices, which generalize Hadamard matrices, have not received anything like the same level of scrutiny. Motivated by an application to the construction of quantum error-correcting codes, which we outline in the latter sections of this paper, we survey the existing literature on complex generalized weighing matrices. We discuss and extend upon the known existence conditions and constructions, and compile known existence results for small parameters. Using these matrices we construct Hermitian self orthogonal codes over finite fields of square order, and consequently some interesting quantum codes are constructed to demonstrate the value of complex generalized weighing matrices.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003327/pdfft?md5=529c4d63c13ac71c4519138cdb73c99c&pid=1-s2.0-S0012365X24003327-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141978438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Terwilliger algebras of Odd graphs and Doubled Odd graphs","authors":"","doi":"10.1016/j.disc.2024.114216","DOIUrl":"10.1016/j.disc.2024.114216","url":null,"abstract":"<div><p>For an integer <span><math><mi>m</mi><mo>≥</mo><mn>1</mn></math></span>, let <span><math><mi>S</mi><mo>=</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn><mo>}</mo></math></span>. Denote by <span><math><mn>2</mn><mo>.</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> the Doubled Odd graph on <em>S</em> with vertex set <span><math><mi>X</mi><mo>:</mo><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>S</mi></mtd></mtr><mtr><mtd><mi>m</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>S</mi></mtd></mtr><mtr><mtd><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span>. By folding this graph, one can obtain a new graph called Odd graph <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> with vertex set <span><math><mi>X</mi><mo>:</mo><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>S</mi></mtd></mtr><mtr><mtd><mi>m</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span>. In this paper, we shall study the Terwilliger algebras of <span><math><mn>2</mn><mo>.</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. We first consider the case of <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. With respect to any fixed vertex <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>X</mi></math></span>, let <span><math><mi>A</mi><mo>:</mo><mo>=</mo><mi>A</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> denote the centralizer algebra of the stabilizer of <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> in the automorphism group of <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>, and <span><math><mi>T</mi><mo>:</mo><mo>=</mo><mi>T</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> the Terwilliger algebra of <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. For the algebras <span><math><mi>A</mi></math></span> and <span><math><mi>T</mi></math></span>: (i) we construct a basis of <span><math><mi>A</mi></math></span> by the stabilizer of <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> acting on <span><math><mi>X</mi><mo>×</mo><mi>X</mi></math></span>, compute its dimension and show that <span><math><mi>A</mi><mo>=</mo><mi>T</mi></math></span>; (ii) for <span><math><mi>m</mi><mo>≥</mo><mn>3</mn></math></span>, we give all the isomorphism classes of irreducible <span><math><mi>T</mi></math></span>-modu","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003479/pdfft?md5=8c465dc78658321c3a6c455f5d3877fe&pid=1-s2.0-S0012365X24003479-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141978439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"New methods for constructing AEAQEC codes","authors":"","doi":"10.1016/j.disc.2024.114202","DOIUrl":"10.1016/j.disc.2024.114202","url":null,"abstract":"<div><p>Recently, Liu and Liu gave the Singleton bound for pure asymmetric entanglement-assisted quantum error-correcting (AEAQEC) codes. They constructed three new families of AQEAEC codes by means of Vandermonde matrices, generalized Reed-Solomon (GRS) codes and cyclic codes. In this paper, we first exhibit the Singleton bound for any AEAQEC codes. Then we construct AEAQEC codes by two distinct constacyclic codes. By means of repeated-root cyclic codes, we construct new AEAQEC MDS codes. In addition, our methods allow for easily calculating the dimensions, <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>z</mi></mrow></msub></math></span>, <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>x</mi></mrow></msub></math></span> and the number <em>c</em> of pre-shared maximally entangled states of AEAQEC codes.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141978322","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}

{"title":"Ramsey numbers and a general Erdős-Rogers function","authors":"","doi":"10.1016/j.disc.2024.114203","DOIUrl":"10.1016/j.disc.2024.114203","url":null,"abstract":"<div><p>Given a graph <em>F</em>, let <span><math><mi>L</mi><mo>(</mo><mi>F</mi><mo>)</mo></math></span> be a fixed finite family of graphs consisting of a <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> and some bipartite graphs relying on an <em>s</em>-partite subgraph partitioning of edges of <em>F</em>. Define <span><math><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>-graph by an <span><math><mi>m</mi><mo>×</mo><mi>n</mi></math></span> bipartite graph with <span><math><mi>n</mi><mo>≥</mo><mi>m</mi></math></span> such that all vertices in the part of size <em>n</em> have degree <em>a</em> and all vertices in the part of size <em>m</em> have degree <span><math><mi>b</mi><mo>≥</mo><mi>a</mi></math></span>. In this paper, building upon the work of Janzer and Sudakov (2023⁺) and combining with the idea of Conlon, Mattheus, Mubayi and Verstraëte (2023⁺) we obtain that for each <span><math><mi>s</mi><mo>≥</mo><mn>2</mn></math></span>, if there exists an <span><math><mi>L</mi><mo>(</mo><mi>F</mi><mo>)</mo></math></span>-free <span><math><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>-graph, then there exists an <em>F</em>-free graph <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> with at least <span><math><mi>n</mi><msup><mrow><mi>a</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>s</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup><mo>−</mo><mn>1</mn></math></span> vertices in which every vertex subset of size <span><math><mi>m</mi><msup><mrow><mi>a</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mi>s</mi></mrow><mrow><mi>s</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>⁡</mo><mo>(</mo><mi>a</mi><mi>n</mi><mo>)</mo></math></span> contains a copy of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>. As applications, we obtain some upper bounds of general Erdős-Rogers functions for some special graphs of <em>F</em>. Moreover, we obtain the multicolor Ramsey numbers <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>;</mo><mi>t</mi><mo>)</mo><mo>=</mo><mover><mrow><mi>Ω</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msup><mrow><mi>t</mi></mrow><mrow><mfrac><mrow><mn>3</mn><mi>k</mi></mrow><mrow><mn>7</mn></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> and <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>;</mo><mi>t</mi><mo>)</mo><mo>=</mo><mover><mrow><mi>Ω</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msup><mrow><mi>t</mi></mrow><mrow><mfrac><mrow><mi>k</mi></mr","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141978323","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}

{"title":"A complete classification of edge-primitive graphs of valency 6","authors":"","doi":"10.1016/j.disc.2024.114205","DOIUrl":"10.1016/j.disc.2024.114205","url":null,"abstract":"<div><p>In 2020, the first author and Pan proved that every edge-primitive graph of valency 6 is 2-arc-transitive, and except the complete bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>6</mn><mo>,</mo><mn>6</mn></mrow></msub></math></span>, the automorphism group is almost simple, and they determined such graphs having a solvable edge stabilizer. The nonsolvable edge stabilizer case is settled in this work, which leads to a complete classification of edge-primitive graphs of valency 6.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141978182","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}

{"title":"Clustering of consecutive numbers in permutations avoiding a pattern of length three or avoiding a finite number of simple patterns","authors":"","doi":"10.1016/j.disc.2024.114199","DOIUrl":"10.1016/j.disc.2024.114199","url":null,"abstract":"<div><p>For <span><math><mi>η</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, let <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span> denote the set of permutations in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> that avoid the pattern <em>η</em>, and let <span><math><msubsup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span> denote the expectation with respect to the uniform probability measure on <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span>. For <span><math><mi>n</mi><mo>≥</mo><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>τ</mi><mo>∈</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span>, let <span><math><msubsup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>σ</mi><mo>)</mo></math></span> denote the number of occurrences of <em>k</em> consecutive numbers appearing in <em>k</em> consecutive positions in <span><math><mi>σ</mi><mo>∈</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span>, and let <span><math><msubsup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>;</mo><mi>τ</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>σ</mi><mo>)</mo></math></span> denote the number of such occurrences for which the order of the appearance of the <em>k</em> numbers is the pattern <em>τ</em>. We obtain explicit formulas for <span><math><msubsup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup><msubsup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>;</mo><mi>τ</mi><mo>)</mo></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup><msubsup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup></math></span>, for all <span><math><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></math></span>, all <span><math><mi>η</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> and all <span><math><mi>τ</mi><mo>∈</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span>. These exact formulas then yield asymptotic formulas as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span> with <em>k</em> fixed, and as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141963155","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}

{"title":"Average mixing in quantum walks of reversible Markov chains","authors":"","doi":"10.1016/j.disc.2024.114196","DOIUrl":"10.1016/j.disc.2024.114196","url":null,"abstract":"<div><p>The Szegedy quantum walk is a discrete time quantum walk model which defines a quantum analogue of any Markov chain. The long-term behavior of the quantum walk can be encoded in a matrix called the <em>average mixing matrix</em>, whose columns give the limiting probability distribution of the walk given an initial state. We define a version of the average mixing matrix of the Szegedy quantum walk which allows us to more readily compare the limiting behavior to that of the chain it quantizes. We prove a formula for our mixing matrix in terms of the spectral decomposition of the Markov chain and show a relationship with the mixing matrix of a continuous quantum walk on the chain. In particular, we prove that average uniform mixing in the continuous walk implies average uniform mixing in the Szegedy walk. We conclude by giving examples of Markov chains of arbitrarily large size which admit average uniform mixing in both the continuous and Szegedy quantum walk.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003273/pdfft?md5=837d04cd2734695aceae3a30d279780f&pid=1-s2.0-S0012365X24003273-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141953842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Hamiltonian decompositions of complete 3-uniform hypergraphs","authors":"","doi":"10.1016/j.disc.2024.114197","DOIUrl":"10.1016/j.disc.2024.114197","url":null,"abstract":"<div><p>Based on the definition of Hamiltonian cycles by Katona and Kierstead, we present a recursive construction of tight Hamiltonian decompositions of complete 3-uniform hypergraphs <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msubsup></math></span>, and complete multipartite 3-uniform hypergraph <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msubsup></math></span>, where <em>t</em> is the number of partite sets and <em>n</em> is the size of each partite set. For <span><math><mi>t</mi><mo>≡</mo><mn>4</mn><mo>,</mo><mn>8</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>12</mn><mo>)</mo></math></span>, we utilize a tight Hamiltonian decomposition of <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msubsup></math></span> to construct those of <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mi>t</mi></mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msubsup></math></span> for all positive integers <em>n</em>. By applying our method in conjunction with the current results in literature, we obtain tight Hamiltonian decompositions for infinitely many hypergraphs, namely complete hypergraphs <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msubsup></math></span> and complete multipartite hypergraphs <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msubsup></math></span> for any positive integer <em>n</em>, and <span><math><mi>t</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>,</mo><mn>5</mn><mo>⋅</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>,</mo><mn>7</mn><mo>⋅</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span>, and <span><math><mn>11</mn><mo>⋅</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span> when <span><math><mi>m</mi><mo>≥</mo><mn>2</mn></math></span>.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141963154","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}

{"title":"Z2Z4-ACP of codes and their applications to the noiseless two-user binary adder channel","authors":"","doi":"10.1016/j.disc.2024.114194","DOIUrl":"10.1016/j.disc.2024.114194","url":null,"abstract":"<div><p>Linear complementary pair (abbreviated to LCP) of codes were defined by Ngo et al. in 2015, and were proved that these pairs of codes can help to improve the security of the information processed by sensitive devices, especially against so-called side-channel attacks (SCA) and fault injection attacks (FIA). In this paper, we first generalize the LCP of codes over finite fields to the additive complementary pair (ACP) of codes in the ambient space with mixed binary and quaternary alphabets. Then we provide two characterizations for the <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-additive codes pair <span><math><mo>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> to be <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-ACP of codes. Meanwhile, we obtain a sufficient condition for the <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-additive codes pair <span><math><mo>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> to be <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-ACP of codes. Under suitable conditions, we derive a necessary and sufficient condition for the Gray map Φ image of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-ACP of codes <span><math><mo>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> to be LCP of codes over <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Finally, we exhibit an interesting application of a special class of the <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-ACP of codes in coding for the two-user binary adder channel.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141962685","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}

{"title":"On linear complementary pairs of algebraic geometry codes over finite fields","authors":"","doi":"10.1016/j.disc.2024.114193","DOIUrl":"10.1016/j.disc.2024.114193","url":null,"abstract":"<div><p>Linear complementary dual (LCD) codes and linear complementary pairs (LCP) of codes have been proposed for new applications as countermeasures against side-channel attacks (SCA) and fault injection attacks (FIA) in the context of direct sum masking (DSM). The countermeasure against FIA may lead to a vulnerability for SCA when the whole algorithm needs to be masked (in environments like smart cards). This led to a variant of the LCD and LCP problems, where several results were obtained intensively for LCD codes, but only partial results were derived for LCP codes. Given the gap between the thin results and their particular importance, this paper aims to reduce this by further studying the LCP of codes in special code families and, precisely, the characterization and construction mechanism of LCP codes of algebraic geometry codes over finite fields. Notably, we propose constructing explicit LCP of codes from elliptic curves. Besides, we also study the security parameters of the derived LCP of codes <span><math><mo>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> (notably for cyclic codes), which are given by the minimum distances <span><math><mi>d</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span> and <span><math><mi>d</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mo>⊥</mo></mrow></msup><mo>)</mo></math></span>. Further, we show that for LCP algebraic geometry codes <span><math><mo>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span>, the dual code <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⊥</mo></mrow></msup></math></span> is equivalent to <span><math><mi>D</mi></math></span> under some specific conditions we exhibit. Finally, we investigate whether MDS LCP of algebraic geometry codes exist (MDS codes are among the most important in coding theory due to their theoretical significance and practical interests). Construction schemes for obtaining LCD codes from any algebraic curve were given in 2018 by Mesnager, Tang and Qi in <span><span>[11]</span></span>. To our knowledge, it is the first time LCP of algebraic geometry codes has been studied.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141962684","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}