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 ∈ F_2^n, where the covering radius of some orthogonal array C ⊂ 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.

In this article, we show that the minimal vectors of the extremal even unimodular lattices in R^{32} define T-avoiding universally optimal spherical codes for suitable sets T. Moreover, these codes are minimal T-avoiding spherical designs and maximal Tavoiding codes for appropriate choices of T.

Mixed (asymmetric) orthogonal arrays (MOAs) generalize classical orthogonal arrays by allowing columns over different alphabets. However, their study requires very different structural tools than those used for symmetric orthogonal arrays (OAs), since several key features of the symmetric setting are no longer available in the mixed case, including Euclidean duality, a unique global index, and certain classical bounds. In this paper, we establish three structural results for mixed orthogonal arrays. First, we prove a Singleton-type upper bound and obtain a characterization of MDS and almost-MDS mixed orthogonal arrays. Second, we introduce a trace duality for 𝔽_q-linear MOAs over ∏_i=1^s𝔽_q^n_i and establish a correspondence with 𝔽_q-linear error-block codes that determines the strength of the MOA via the dual distance of the associated error-block code. Finally, we develop a structural theory of irredundant mixed orthogonal arrays (IrMOAs), motivated by their role in the construction of t-uniform and absolutely maximally entangled (AME) quantum states. In the extremal case t=⌊ s/2⌋, we prove that 𝔽_q-linear IrMOAs with minimum index 1 (yielding AME states of minimal support) are equivalent to 𝔽_q-linear error-block MDS codes.

We establish upper and lower universal bounds for potentials of weighted designs on the sphere Sn-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 Sn-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. (c) 2025 The Authors. Published by Elsevier GmbH. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

An orthogonal array (OA), denoted by OA(M, n, q, t), is an M & times;n matrix over an alphabet of size q such that every selection of t columns contains each possible t-tuple exactly lambda = M/qt times. An irredundant orthogonal array (IrOA) is an OA with the additional property that, in any selection of n-t columns, all resulting rows are distinct. IrOAs were first introduced by Goyeneche and & Zdot;yczkowski in 2014 to construct t-uniform quantum states without redundant information. Beyond their quantum applications, we focus on IrOAs as a combinatorial and coding theory problem. An OA is an IrOA if and only if its minimum Hamming distance is at least t+1. Using this characterization, we demonstrate that for any linear code, either the code itself or its Euclidean dual forms a linear IrOA, giving a huge source of IrOAs. In particular, the self-dual codes yield IrOAs. Moreover, we construct new families of linear IrOAs based on self-dual, Maximum Distance Separable (MDS), and MDS-self-dual codes. Finally, we establish bounds on the minimum distance and covering radius of IrOAs. (c) 2026 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Let μ be a measure on the Euclidean space ^d of unbounded total variation that is positive or translation bounded and has a pure point Fourier transform in the sense of distributions μ̂. We prove that the measure ν with the same support as μ̂ and masses equal to the squares of the masses of μ̂ is translation bounded. We also prove that if μ is as above and the restriction of its spectrum, i.e., of the support of μ̂, to each ball of fixed radius is a linearly independent set over , then the measure μ̂ is also translation bounded. These results imply certain conditions for a crystalline measure to be a Fourier quasicrystal.

Given an open set T⊂ [-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 α, β, γ (in our terminology (α,β)-avoiding γ-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.

This article is devoted to the study of discrete potentials on the sphere in $\mathbb{R}^n$ for sharp codes. We show that the potentials of most of the known sharp codes attain the universal lower bounds for polarization for spherical $\tau$-designs previously derived by the authors, where ``universal'' is meant in the sense of applying to a large class of potentials that includes absolutely monotone functions of inner products. We also extend our universal bounds to $T$-designs and the associated polynomial subspaces determined by the vanishing moments of spherical configurations and thus obtain the minima for the icosahedron, dodecahedron, and sharp codes coming from $E_8$ and the Leech lattice. For this purpose, we investigate quadrature formulas for certain subspaces of Gegenbauer polynomials $P^{(n)}_j$ which we call PULB subspaces, particularly those having basis $\{P_j^{(n)}\}_{j=0}^{2k+2}\setminus \{P_{2k}^{(n)}\}.$ Furthermore, for potentials with $h^{(\tau+1)}<0$ we prove that the strong sharp codes and the antipodal sharp codes attain the universal bounds and their minima occur at points of the codes. The same phenomenon is established for the $600$-cell when the potential $h$ satisfies $h^{(i)}\geq 0$, $i=1,\dots,15$, and $h^{(16)}\leq 0.$

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 explore our bounds for several previously studied weighted spherical codes.

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.

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.

In this article we investigate the N-point min-max and max-min polarization problems on the sphere for a large class of potentials in Rn. We derive universal lower and upper bounds on the polarization of spherical designs of fixed dimension, strength, and cardinality. The bounds are universal in the sense that they are a convex combination of potential function evaluations with nodes and weights independent of the class of potentials. As a consequence of our lower bounds, we obtain the Fazekas-Levenshtein bounds on the covering radius of spherical designs. Utilizing the existence of spherical designs, our polarization bounds are extended to general configurations. As examples we completely solve the min-max polarization problem for 120 points on S3 and show that the 600-cell is universally optimal for that problem. We also provide alternative methods for solving the max-min polarization problem when the number of points N does not exceed the dimension n and when N=n+1. We further show that the cross-polytope has the best max-min polarization constant among all spherical 2-designs of N=2n points for n=2,3,4; for n≥5, this statement is conditional on a well-known conjecture that the cross-polytope has the best covering radius. This max-min optimality is also established for all so-called centered codes.

The paper suggests an algorithm for finding Residue Number Systems (RNS) with six modules (6-tuples) with the Sum of Quotients SQ=2k for some positive integer k. It is shown that there are exactly thirteen such 6-tuples (m1,…,m6) with m1+m2≤10000 and m3+m4≤m5+m6≤10000 and investigated the three smallest among them, namely (5,2399,7,11,23,1691), (47,293,25,193,41,257), and (23,1433,13,29,681,821), having k=34, 36, and 40, respectively. The hypothesis is that such RNS allow efficient hardware implementations of non-modular operations — division, sign detection, comparison of numbers, and reverse conversion from the point of hardware resource usage and balancedness. The special case SQ=2k allows a significant simplification of these operations and increased efficiency of hardware implementations. Hardware modeling of circuits based on the above 6-tuples that implement magnitude comparison and reverse RNS to binary conversion (reverse conversion) is presented. The suggested magnitude comparison and reverse conversion devices, which operate with the six modules, are built using the methodology and values of the diagonal function, carry-save, and Kogge-Stone adders. The results of FPGA-based hardware modeling and the theoretical parameters of devices calculated using the unit-gate model are compared with state-of-the-art approaches. The unit-gate model showed that the use of the proposed circuits allows us to reduce the area by 17.82%–50.81% and the delay by 1.56%–97.76% for the implementation of the magnitude comparison, and to reduce the delay by 2.80%–95.03% for the implementation of reverse conversion. The FPGA synthesis also showed that the new design has reduced the area and delay by 46.96%–86.80% and 7.15%–47.65%, respectively, for the implementation of magnitude comparison and reduced the delay by 13.63%–81.53% for the implementation of reverse conversion. It is shown that a proposed technique to measure the RNS balance can adequately reflect differences in the RNS performance.

We derive upper and lower bounds on the sum of distances of a spherical code of size N in n dimensions when N=Θ(nα),0<α⩽2. The bounds are derived by specializing recent general, universal bounds on energy of spherical sets. We discuss asymptotic behavior of our bounds along with several examples of codes whose sum of distances closely follows the upper bound.

We enumerate all $$q$$ -ary additive (in particular, linear) block codes of length $$n$$ and cardinality $$N\ge q^2$$ with exactly two distances: $$d$$ and $$n$$ . For arbitrary codes of length $$n$$ with distances $$d$$ and $$n$$ , we obtain upper bounds on the cardinality via linear programming and using relationships to 2-distance sets on a Euclidean sphere.

We study the packing and covering properties of orthogonal arrays (OAs) as we propose new approaches for obtaining estimations on the minimum distance and covering radius of orthogonal arrays (designs) via examination of their distance distributions. First, we use some special representation of a linear system of equations which allows us to analyse extreme solutions (distance distributions). Second, we show how databases with (feasible) distance distributions can be used for obtaining sharp bounds for the minimum distance and covering radius of OAs. Correspondingly, new bounds are presented either in analytic form and as products of an ongoing project for computation and investigation of the possible distance distributions of OAs.

We prove that the inner products of spherical s-distance t-designs with t≥2s−2 (Delsarte codes) and s≥3 are rational with the only exception being the icosahedron. In other formulations, we prove that all sharp configurations have rational inner products and all spherical codes which attain the Levenshtein bound, have rational inner products, except for the icosahedron.

We consider q-ary (linear and nonlinear) block codes with exactly two distances: d and d+delta. We derive necessary conditions for existence of such codes (similar to the known conditions in the projective case). In the linear (but not necessary projective) case, we prove that under certain conditions the existence of such linear 2-weight code with delta > 1 implies the following equality of greatest common divisors: (d, q) = (delta, q). Upper bounds for the maximum cardinality of such codes are derived by linear programming and from few-distance spherical codes. Tables of lower and upper bounds for small q = 2, 3, 4 and q n < 50 are presented. (C) 2021 Elsevier B.V. All rights reserved.

Based on the Delsarte-Yudin linear programming approach, we extend Levenshtein's framework to obtain lower bounds for the minimum $h$-energy of spherical codes of prescribed dimension and cardinality, and upper bounds on the maximal cardinality of spherical codes of prescribed dimension and minimum separation. These bounds are universal in the sense that they hold for a large class of potentials $h$ and in the sense of Levenshtein. Moreover, codes attaining the bounds are universally optimal in the sense of Cohn-Kumar. Referring to Levenshtein bounds and the energy bounds of the authors as ``first level", our results can be considered as ``next level" universal bounds as they have the same general nature and imply necessary and sufficient conditions for their local and global optimality. For this purpose, we introduce the notion of Universal Lower Bound space (ULB-space), a space that satisfies certain quadrature and interpolation properties. While there are numerous cases for which our method applies, we will emphasize the model examples of $24$ points ($24$-cell) and $120$ points ($600$-cell) on $\mathbb{S}^3$. In particular, we provide a new proof that the $600$-cell is universally optimal, and in so doing, we derive optimality of the $600$-cell on a class larger than the absolutely monotone potentials considered by Cohn-Kumar.

We investigate spherical 4-distance 7-designs by studying their distance distributions. We compute these distance distributions and use their product (an integer) to derive certain divisibility conditions relating the dimension n and the cardinality M of our designs. It follows that n divides 12M and n+1 divides 4M 2 . This result provides a good base for computer experiments to support the folklore conjecture that the only spherical 4-distance 7-designs are the tight spherical 7-designs. We then proceed with a computer assisted proof of this conjecture in all dimensions n ≤ 1000.