Publications
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Peter Boyvalenkov and Danila Cherkashin. The kissing number in 48 dimensions for codes with certain forbidden distances is 52 416 000. Results in Mathematics, 80(1):art. 3, 2025.
arXiv JournalAbstractWe prove that the kissing number in 48 dimensions among antipodal spherical codes with certain forbidden inner products is 52 416 000. Constructions of attaining codes as kissing configurations of minimum vectors in even unimodular extremal lattices are well known since the 1970's. We also prove that corresponding spherical 11-designs with the same cardinality are minimal. We use appropriate modifications of the linear programming bounds for spherical codes and designs introduced by Delsarte, Goethals and Seidel in 1977.
MetricsImpact Factor 1.1 (2023), Q1 (Mathematics)
Scientific Journal Rankings 0.689 (2024), Q1 (Mathematics, miscellaneous)
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Sergiy Borodachov, Peter Boyvalenkov, Peter Dragnev, Douglas Hardin, Edward Saff, and Maya Stoyanova. Bounds on energy and potentials of discrete measures on the sphere. arXiv preprint arXiv:2412.07442, 2024.
arXivAbstractWe establish upper and lower universal bounds for potentials of weighted designs on the sphere \(\mathbb{S}^{n-1}\) that depend only on quadrature nodes and weights derived from the design structure. Our bounds hold for a large class of potentials that includes absolutely monotone functions. The classes of spherical designs attaining these bounds are characterized. Additionally, we study the problem of constrained energy minimization for Borel probability measures on \(\mathbb{S}^{n-1}\) and apply it to optimal distribution of charge supported at a given number of points on the sphere. In particular, our results apply to \(p\)-frame energy.
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Sergiy Borodachov, Peter Boyvalenkov, Peter Dragnev, Douglas Hardin, Edward Saff, and Maya Stoyanova. Bounds on discrete potentials of spherical \((k, k)\)-designs. arXiv preprint arXiv:2411.00290, 2024.
arXivAbstractWe derive universal lower and upper bounds for max-min and min-max problems (also known as polarization) for the potential of spherical \((k,k)\)-designs and provide certain examples, including unit-norm tight frames, that attain these bounds. The universality is understood in the sense that the bounds hold for all spherical \((k,k)\)-designs and for a large class of potential functions, and the bounds involve certain nodes and weights that are independent of the potential. When the potential function is \(h(t)=t^{2k}\), we prove an optimality property of the spherical \((k,k)\)-designs in the class of all spherical codes of the same cardinality both for max-min and min-max potential problems.
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Sergiy Borodachov, Peter Boyvalenkov, Peter Dragnev, Douglas Hardin, Edward Saff, and Maya Stoyanova. Energy bounds for weighted spherical codes and designs via linear programming. Analysis and Mathematical Physics, 15(1):art. 19, 2025.
arXiv JournalAbstractUniversal bounds for the potential energy of weighted spherical codes are obtained by linear programming. The universality is in the sense of Cohn-Kumar -- every attaining code is optimal with respect to a large class of potential functions (absolutely monotone), in the sense of Levenshtein -- there is a bound for every weighted code, and in the sense of parameters (nodes and weights) -- they are independent of the potential function. We derive a necessary condition for optimality (in the linear programming framework) of our lower bounds which is also shown to be sufficient when the potential is strictly absolutely monotone. Bounds are also obtained for the weighted energy of weighted spherical designs. We explore our bounds for several previously studied weighted spherical codes.
MetricsImpact Factor 1.4 (2023), Q1 (Mathematics)
Scientific Journal Rankings 0.846 (2024), Q1 (Analysis)
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Sergiy Borodachov, Peter Boyvalenkov, Peter Dragnev, Douglas Hardin, Edward Saff, and Maya Stoyanova. Linear programming lower bounds for energy of weighted spherical codes. Proceedings of Workshop on Coding and Cryptograprhy, Perugia 2024, 2024, pp. 77-86.
ProceedingsAbstractUniversal lower bounds for potential energy of weighted spherical codes are obtained by linear programming. The universality is in the sense of Cohn-Kumar – every attaining code (if any) is optimal with respect to a large class of potential functions, in the sense of Levenshtein – there is a bound for every weighted code, and in the sense of parameters (nodes and weights) which do not depend on the potential function.
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Peter Boyvalenkov and Peter Dragnev. Energy of codes with forbidden distances in 48 dimensions. arXiv preprint arXiv:2412.07577, 2024.
arXivAbstractWe prove the universal optimality of four remarkable spherical 11-designs in 48 dimensions either among all antipodal codes, or all spherical 3-designs, whose inner-products avoid the set \(T_1=(−1/3,−1/6) \cup (1/6,1/3)\). We also prove the universal optimality of these configurations among all codes whose distance-avoiding set is \(T_2=(−1/2,−1/3) \cup (1/3,1/2)\).
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Peter Boyvalenkov, Danila Cherkashin and Peter Dragnev. Universal optimality of \(T\)-avoiding spherical codes and designs. arXiv preprint arXiv:2501.13906, 2024.
arXivAbstractGiven an open set (a union of open intervals), \(T\subset[−1,1]\) we introduce the concepts of \(T\)-avoiding spherical codes and designs, that is, spherical codes that have no inner products in the set T. We show that certain codes found in the minimal vectors of the Leech lattices, as well as the minimal vectors of the Barnes--Wall lattice and codes derived from strongly regular graphs, are universally optimal in the restricted class of \(T\)-avoiding codes. We also extend a result of Delsarte--Goethals--Seidel about codes with three inner products \(\alpha,\beta,\gamma\) (in our terminology \((\alpha,\beta)\)-avoiding \(\gamma\)-codes). Parallel to the notion of tight spherical designs, we also derive that these codes are minimal (tight) \(T\)-avoiding spherical designs of fixed dimension and strength. In some cases, we also find that codes under consideration have maximal cardinality in their \(T\)-avoiding class for given dimension and minimum distance.
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Danila Cherkashin. On set systems without singleton intersections. Discrete Mathematics Letters, 14:85–88, 2024.
arXiv JournalAbstractConsider a family \(\mathcal{F}\) of \(k\)-subsets of an ambient \((k^2-k+1)\)-set such that no pair of \(k\)-subsets in \(\mathcal{F}\) intersects in exactly one element. In this short note we show that the maximal size of such \(\mathcal{F}\) is \(\binom{k^2-k-1}{k-2}\) for every \(k > 1\).
MetricsImpact Factor 1 (2023), Q1 (Mathematics)
Scientific Journal Rankings 0.361 (2024), Q3 (Discrete Mathematics and Combinatorics)
Open Access
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Thomas Honold, Michael Kiermaier, and Ivan Landjev. New results on arcs in projective Hjelmslev planes over small chain rings. arXiv preprint arXiv:2409.02099, 2024.
arXivAbstractWe present various new constructions and bounds for arcs in projective Hjelmslev planes over finite chain rings of nilpotency index 2. For the chain rings of cardinality at most 25 we give updated tables with the best known upper and lower bounds for the maximum size of such arcs.
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Ivan Landjev and Konstantin Vorob’ev. On binary codes with distances \(d\) and \(d + 2\). arXiv preprint arXiv:2402.13420, 2024.
arXivAbstractWe consider the problem of finding \(A_2(n,\{d_1,d_2\})\) defined as the maximal size of a binary (non-linear) code of length \(n\) with two distances \(d_1\) and \(d_2\). Binary codes with distances \(d\) and \(d+2\) of size \(\sim\frac{n^2}{\frac{d}{2}(\frac{d}{2}+1)}\) can be obtained from \(2\)-packings of an \(n\)-element set by blocks of cardinality \(\frac{d}{2}+1\). This value is far from the upper bound \(A_2(n,\{d_1,d_2\})\le1+{n\choose2}\) proved recently by Barg et al.
In this paper we prove that for every fixed \(d\) (\(d\) even) there exists an integer \(N(d)\) such that for every \(n\ge N(d)\) it holds \(A_2(n,\{d,d+2\})=D(n,\frac{d}{2}+1,2)\), or, in other words, optimal codes are isomorphic to constant weight codes. We prove also estimates on \(N(d)\) for \(d=4\) and \(d=6\). -
Assia Rousseva. On affine blocking sets. Mathematics and Education in Mathematics, 53:9–17, 2024.
JournalAbstractWe survey the known results on the minimal size of a blocking set in the finiteaffine geometries \(AG(n, q)\).
MetricsScientific Journal Rankings 0.111 (2024), Q4 (Mathematics, miscellaneous)
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Ivan Landjev, Sascha Kurz and Assia Rousseva. Quadratic Sets and \((3\, \mbox{mod}\, 5)\)-Arcs in \(PG(r, 5)\). International Conference on Computer Science and Education in Computer Science, Springer Series LNICST, volume 609, 88–96, 2025.
JournalAbstractTo goal of this contribution is to provide a characterization of the \((3\, \mbox{mod}\, 5)\)-arcs in \(PG(r, 5)\), \(r \geq 4\). Such arcs are either lifted or quadratic and by a construction described by Kurz-Landjev-Pavese-Rousseva.
MetricsScientific Journal Rankings 0.158 (2024), Q4 (Computer Networks and Communications)