Publications

• Danila Cherkashin. On branching points in the Gilbert–Steiner problem. To appear in Journal of Convex Analysis, 2027.

  arXiv

  Abstract

  The Gilbert–Steiner problem is a generalization of the Steiner tree problem and specific optimal mass transportation, which allows the use additional (branching) point in a transport plan. A specific feature of the problem is that the cost of transporting a mass \(m\) along a segment of length \(l\) is equal to \(l \times m^p\) for a fixed \(0 < p < 1\) and segments may end at points not belonging to the supports of given measures (branching points). Main result of this paper determines all pairs of \( (p,d) \) for which the Gilbert–Steiner problem in \(\mathbb{R}^d\) admits only branching points of degree 3. Namely, it happens if and only if \(d = 2\) or \(p < 1/2\).

  Metrics

  Impact Factor 0.8 (2025), Q2 (Mathematics)

  Scientific Journal Rankings 0.479 (2025), Q2 (Analysis)

• Peter Boyvalenkov, Ferruh Özbudak, and Maya Stoyanova. New bounds on the covering radius of orthogonal arrays of even strength. Journal of Algebraic Combinatorics, 64, art. 26, 2026.

  arXivJournal

  Abstract

  We obtain new linear programming (LP) and constructive bounds for the covering radius of binary orthogonal arrays of strength \(2k\). Our LP bounds develop in two alternative scenarios. First, if a point \(y \in F_2^n\), where the covering radius of some orthogonal array \(C \subset F_2^n\) of strength \(2k\) is realized, is such that the farthest point of \(C\) to \(y\) is not antipodal to \(y\) we obtain a bound which is better than the Tietäväinen (or Fazekas–Levenshtein) bound for non-tight arrays (i.e., the cardinality strictly exceeds the Rao lower bound). Second, if all points where the covering radius is realized are such that their antipodes are in \(C\), we obtain a bound which depends on the cardinality of \(C\) and is again better whenever the orthogonal array is not tight. We further describe three infinite families of binary orthogonal arrays related to the duals of BCH, Melas, and Zetterberg codes. For these families, we derive lower bounds on the covering radius by applying techniques from algebraic curves over finite fields, while the improved linear programming methods developed in this paper provide upper bounds, leading in some cases to fairly close estimates.

  Metrics

  Impact Factor 0.9 (2025), Q2 (Mathematics)

  Scientific Journal Rankings 0.841 (2025), Q1 (Algebra and Number Theory; Discrete Mathematics and Combinatorics)

• Ivan Landjev, Assia Rousseva, and Leo Storme. On the reducibility of minihypers. Designs, Codes and Cryptography, 94, art. 156, 2026.

  Journal

  Abstract

  In this paper, we prove a new reducibility theorem for minihypers which uses a more general notion for reducibility. This result can be used in the characterization of arcs, minihypers and linear codes with optimal parameters.

  Metrics

  Impact Factor 1.2 (2025), Q2 (Mathematics, Applied)

  Scientific Journal Rankings 0.712 (2025), Q1 (Discrete Mathematics and Combinatorics)

• Peter Boyvalenkov, Danila Cherkashin, and Peter Dragnev. Universal optimality of \(T\)-avoiding spherical codes and designs. Constructive Approximation, 64, pp. 145—182, 2026.

  arXivJournal

  Abstract

  Given an open set \(T\subset [-1,1)\), 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 lattice, 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.

  Metrics

  Impact Factor 1.4 (2025), Q1 (Mathematics)

  Scientific Journal Rankings 1.285 (2025), Q1 (Analysis)

• Ivan Landjev and Assia Rousseva. On the maximal length of near-MDS codes. Mathematics and Education in Mathematics, 55, pp. 405—413, 2026.

  Proceedings

  Abstract

  This paper summarizes the known results about near-MDS codes proved in the past thirty years. The main focus is put on the problem of determining the value of the function \(m'(k,q)\) defined as the maximal length of a near-MDS code of dimension \(k\) over the field with \(q\) elements. For dimensions \(k>q+2\) we improve the upper bound \(m'(k,q)\le 2q+k-2\) which follows from the nonexistence of maximal arcs over fields of odd characteristic. In analogy with the main conjecture for MDS codes we formulate some conjectures on the exact value of \(m'(k,q)\).

  Metrics

  Scientific Journal Rankings 0.146 (2025), Q4

• Ivan Landjev, Emiliyan Rogachev, and Assia Rousseva. Characterization of some minihypers in \(\mathrm{PG}(r,3)\) and the nonexistence of some ternary Griesmer codes. Designs, Codes and Cryptography, 94, art. 22, 2026.

  Journal

  Abstract

  We consider the problem of finding the minimal length \(n_q(k,d)\) of a linear code over \(\mathbb F_q\) of fixed dimension \(k\) and fixed minimum distance \(d\). For ternary codes of dimension 6 this problem is solved for all but 70 values of \(d\). In this paper, we resolve three of the undecided cases: \(d=344\), 345, 346. The problem is tackled by associating with the linear codes in question certain minihypers with bounded point multiplicity. In this paper we make use of the characterization of the minihypers with parameters \((66,21)\), \((67,21)\) and \((68,21)\) in \(\mathrm{PG}(4,3)\) to rule out the existence of the minihypers with parameters \((210,68)\), \((209,68)\) and \((207,67)\) in \(\mathrm{PG}(5,3)\). This violates the existence of the hypothetical ternary codes with parameters \([518,6,344]\), \([519,6,345]\), \([521,6,346]\), and implies the three exact values; \(n_3(6,344)=519,\quad n_3(6,345)=520,\quad n_3(6,346)=522.\) The proof is based on geometric arguments and is entirely computer-free.

  Metrics

  Impact Factor 1.2 (2025), Q2 (Mathematics, Applied)

  Scientific Journal Rankings 0.712 (2025), Q1 (Discrete Mathematics and Combinatorics)

• Sergiy Borodachov, Peter Boyvalenkov, Peter Dragnev, Douglas Hardin, Edward Saff, and Maya Stoyanova. Bounds on energy and potentials of discrete measures on the sphere. Expositiones Mathematicae, 44(1), art. 125712, 2026.

  arXivJournal

  Abstract

  We 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.

  Metrics

  Impact Factor 0.8 (2025), Q2 (Mathematics)

  Scientific Journal Rankings 0.664 (2025), Q1 (Mathematics, miscellaneous)

• Peter Boyvalenkov and Peter Dragnev. Energy of Codes with Forbidden Distances in 48 Dimensions. In Recent Advances in Approximation and Potential Theory: In Honor of Edward Saff, Applied and Numerical Harmonic Analysis, pp. 157—172, Birkhäuser, 2026.

  arXivChapter

  Abstract

  We 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 \(T_2=(-1/2,-1/3)\cup(1/3,1/2)\).

• Peter Boyvalenkov, Danila Cherkashin, and Peter Dragnev. Linear programming for \(T\)-avoiding spherical codes. Workshop on Coding and Cryptography (WCC 2026), Paris, 2026.

  Proceedings

  Abstract

  This note summarizes recent results by the authors on \(T\)-avoiding spherical codes and designs, i.e. codes and designs whose inner products avoid a certain set \(T\subset[-1,1)\). We discuss the methods used to obtain these results and list open questions in this field.

• Ivan Landjev and Assia Rousseva. On the Divisibility of Linear Codes and Arcs. Workshop on Coding and Cryptography (WCC 2026), Paris, 2026.

  Proceedings

  Abstract

  We prove several divisibility results for codes over non-prime fields. They can be viewed as a generalization of Ward’s classical divisibility theorem for Griesmer codes over prime fields.

• Sascha Kurz, Ivan Landjev, and Assia Rousseva. Optimal Codes and Arcs for the Generalized Hamming Weights. Workshop on Coding and Cryptography (WCC 2026), Paris, 2026.

  arXivProceedings

  Abstract

  This text contains some notes on the generalized Griesmer bound, i.e. the Griesmer bound with respect to the generalized Hamming weight. We give a geometric characterization of minihypers with certain parameters. This implies that the associated codes are obtained by the classical Solomon–Stiffler construction. We present tables with the parameters of optimal binary codes for dimensions at most seven.

• Sergiy Borodachov, Peter Boyvalenkov, Peter Dragnev, Douglas Hardin, Edward Saff, and Maya Stoyanova. Bounds on discrete potentials of spherical \((k,k)\)-⁠designs. Designs, Codes and Cryptography, 93, pp. 3875—3902, 2025.

  arXivJournal

  Abstract

  We 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 polarization problems.

  Metrics

  Impact Factor 1.2 (2025), Q2 (Mathematics, Applied)

  Scientific Journal Rankings 0.712 (2025), Q1 (Discrete Mathematics and Combinatorics)

• 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.

  arXivJournal

  Abstract

  Universal 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 demonstrate our bounds for several previously studied weighted spherical codes.

  Metrics

  Impact Factor 1.3 (2025), Q1 (Mathematics)

  Scientific Journal Rankings 0.716 (2025), Q1 (Analysis)

• 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.

  arXivJournal

  Abstract

  We 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.

  Metrics

  Impact Factor 1.1 (2025), Q2 (Mathematics)

  Scientific Journal Rankings 0.682 (2025), Q1 (Mathematics, miscellaneous)

• Ivan Landjev and Konstantin Vorob’ev. On binary codes with distances \(d\) and \(d+2\). Designs, Codes and Cryptography, 93, pp. 4313—4322, 2025.

  arXivJournal

  Abstract

  We 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 \(\sim\frac{n^2}{\frac d2(\frac d2+1)}\) can be obtained from 2-packings of an \(n\)-element set by blocks of cardinality \(\frac d2+1\). This value is far from the upper bound \(A_2(n,\{d_1,d_2\})\le 1+{n\choose2}\) 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 \(A_2(n,\{d,d+2\})=D(n,\frac d2+1,2)\), or, in other words, optimal codes are isomorphic to constant weight codes. We prove also estimates on \(N(d)\) for \(d=4\) and \(d=6\).

  Metrics

  Impact Factor 1.2 (2025), Q2 (Mathematics, Applied)

  Scientific Journal Rankings 0.712 (2025), Q1 (Discrete Mathematics and Combinatorics)

• Ivan Landjev, Sascha Kurz, and Assia Rousseva. Quadratic Sets and \((3 \bmod 5)\)-Arcs in \(\mathrm{PG}(r,5)\). In Computer Science and Education in Computer Science, Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, 609, pp. 88—96, Springer, 2025.

  Chapter

  Abstract

  The goal of this contribution is to provide a characterization of the \((3 \bmod 5)\)-arcs in \(\mathrm{PG}(r,5)\) for \(r\ge4\). Such arcs are either lifted or quadratic and described by a construction of Kurz–Landjev–Pavese–Rousseva.

  Metrics

  Scientific Journal Rankings 0.156 (2025), Q4

• Ivan Landjev, Assia Rousseva, and Konstantin Vorob’ev. A New Reducibility Result for Minihypers in Finite Projective Geometries. Annual of Sofia University St. Kliment Ohridski, Faculty of Mathematics and Informatics, 112, pp. 93—103, 2025.

  arXivJournal

  Abstract

  In this paper, we prove a new reducibility result for minihypers in projective geometries over finite fields. It is further used to characterize the minihypers with parameters \((70,22)\) in \(\mathrm{PG}(4,3)\). The latter can be used to attack the existence problem for some hypothetical ternary Griesmer codes of dimension 6.

  Metrics

  Scientific Journal Rankings 0.154 (2025), Q4

• Ivan Landjev and Konstantin Vorob’ev. An Upper Bound on the Size of a Binary Code with \(s\) Distances. Comptes Rendus de l’Académie Bulgare des Sciences, 78(5), pp. 656—662, 2025.

  arXivJournal

  Abstract

  Let \(C\) be a binary code of length \(n\) with distances \(0<d_1<\cdots<d_s\le n\). In this note we prove a general upper bound on the size of \(C\) without any restriction on the distances \(d_i\). The bound is asymptotically optimal.

  Metrics

  Impact Factor 0.4 (2025), Q4 (Multidisciplinary sciences)

  Scientific Journal Rankings 0.179 (2025), Q3 (Multidisciplinary)

• Danila Cherkashin. On set systems without singleton intersections. Discrete Mathematics Letters, 14, pp. 85—88, 2024.

  arXivJournal

  Abstract

  Consider 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 \(k>1\).

  Metrics

  Impact Factor 0.8 (2024), Q2 (Mathematics)

  Scientific Journal Rankings 0.361 (2024), Q3 (Discrete Mathematics and Combinatorics)

• Assia Rousseva. On Affine Blocking Sets. Mathematics and Education in Mathematics, 53, pp. 9—17, 2024.

  Proceedings

  Abstract

  We survey the known results on the minimal size of a blocking set in the finite affine geometries \(\mathrm{AG}(n,q)\).

  Metrics

  Scientific Journal Rankings 0.111 (2024), Q4

• Assia Rousseva, Ivan Landjev, and Emiliyan Rogachev. Characterization of some non-canonical minihypers in \(\mathrm{PG}(r,3)\) and the main problem of coding theory. Workshop on Coding and Cryptography (WCC 2024), Perugia, pp. 342—350, 2024.

  Proceedings

  Abstract

  In this paper we characterize certain minihypers in \(\mathrm{PG}(r,3)\), \(r\le 5\). Based on the obtained results we rule out the existence of certain hypothetical Griesmer codes with \(q=3\), \(k=6\). As a by-product we prove several general results on minihypers that turn out to be important in the investigation of the main problem of coding theory.

• Sergiy Borodachov, Peter Boyvalenkov, Peter Dragnev, Douglas Hardin, Edward Saff, and Maya Stoyanova. Linear Programming Lower Bounds for Energy of Weighted Spherical Codes. Workshop on Coding and Cryptography (WCC 2024), Perugia, pp. 77—86, 2024.

  Proceedings

  Abstract

  Universal 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.

• Danila Cherkashin and Pavel Prozorov. On uniform eventowns. arXiv preprint arXiv:2608.06248, 2026.

  arXiv

  Abstract

  Suppose that \(n=2m\), \(k=2t\) and \(n>10k^7\). We show that if a family \(\mathcal F\) of \(k\)-subsets of an \(n\)-set has only even pairwise intersections then \(|\mathcal F|\le\binom mt\). Moreover, every extremal family has an atomic structure. This result was previously proved by Frankl and Tokushige for \(n>n_{FT}(k)\), where \(n_{FT}(k)\) is at least exponential. The main technique is Delsarte linear programming.

• Danila Cherkashin and Yakov Shubin. Short proofs of three combinatorial results in the Johnson scheme. arXiv preprint arXiv:2605.30092, 2026.

  arXiv

  Abstract

  In this note, we give short proofs of three theorems concerning extremal problems in the Johnson scheme, or, in other terminology, on \((n,k,L)\)-systems. The main result is a proof of the Aljohani–Bamberg–Cameron conjecture which claims that if \(n>n_0(k)\) and there are an \((n,k,L)\)-system and an \((n,k,\{0,\dots,k-1\}\setminus L)\)-system whose sizes have product \(\binom nk\), then they are a \(t\)-intersecting family and a Steiner system \(S(t,k,n)\) for some \(t\).

• Ivan Landjev, Emiliyan Rogachev, and Assia Rousseva. A Chain Ring Analogue of the Erdős–Ko–Rado Theorem. arXiv preprint arXiv:2602.12219, 2026.

  arXiv

  Abstract

  In this paper, we prove an analogue of the Erdős–Ko–Rado theorem intersecting families of subspaces in projective Hjelmslev geometries over finite chain rings of nilpotency index 2. We give an example of maximal families that are not canonically intersectng.

• Sergiy Borodachov, Peter Boyvalenkov, Peter Dragnev, Douglas Hardin, Edward Saff, and Maya Stoyanova. Universal polar dual pairs of spherical codes found in \(E_8\) and \(\Lambda_{24}\). arXiv preprint arXiv:2512.25037, 2025.

  arXiv

  Abstract

  We identify universal polar dual pairs of spherical codes \(C\) and \(D\) such that for a large class of potential functions \(h\) the minima of the discrete \(h\)-potential of \(C\) on the sphere occur at the points of \(D\) and vice versa. Moreover, the minimal values of their normalized potentials are equal. These codes arise from the known sharp codes embedded in the even unimodular extremal lattices \(E_8\) and \(\Lambda_{24}\) (Leech lattice). This embedding allows us to use the lattices’ properties to find new universal polar dual pairs. In the process we extensively utilize the interplay between the binary Golay codes and the Leech lattice. As a byproduct of our analysis, we identify a new universally optimal (in the sense of energy) code in the projective space \(\mathbb{RP}^{21}\) with 1408 points (lines). Furthermore, we extend the Delsarte–Goethals–Seidel definition of derived codes from their seminal 1977 paper and generalize their Theorem 8.2 to show that if a \(\tau\)-design is enclosed in \(k\le\tau\) parallel hyperplanes, then each of the hyperplane’s sub-code is a \((\tau+1-k)\)-design in the ambient subspace.

• 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.

  arXiv

  Abstract

  We 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.