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In our previous work with Grifo and Hà, we showed the stable Harbourne-Huneke containment and Chudnovsky's conjecture for the defining ideal of sufficiently many general points in P N. In this paper, we establish the conjectures for all remaining cases, and hence, give the affirmative answer to Harbourne-Huneke containment and Chudnovsky's conjecture for the general points in P N for all N. Our new technique is to develop the Cremona reduction process that provides effective lower bounds for the Waldschmidt constant of the defining ideals of generic points in projective spaces. [ABSTRACT FROM AUTHOR]

Let N k (m , n) denote the number of partitions of n with Garvan k -rank m. It is well-known that Andrews–Garvan–Dyson's crank and Dyson's rank are the k -rank for k = 1 and k = 2 , respectively. In this paper, we prove that the sequences (N k (m , n)) | m | ≤ n − k − 71 are log-concave for all sufficiently large integers n and each integer k. In particular, we partially solve the log-concavity conjecture for Andrews–Garvan–Dyson's crank and Dyson's rank, which was independently proposed by Bringmann–Jennings-Shaffer–Mahlburg and Ji–Zang recently. [ABSTRACT FROM AUTHOR]

A simple graph G with maximum degree Δ is overfull if | E (G) | > Δ ⌊ | V (G) | / 2 ⌋. The core of G , denoted G Δ , is the subgraph of G induced by its vertices of degree Δ. Clearly, the chromatic index of G equals Δ + 1 if G is overfull. Conversely, Hilton and Zhao in 1996 conjectured that if G is a simple connected graph with Δ ≥ 3 and Δ (G Δ) ≤ 2 , then χ ′ (G) = Δ + 1 implies that G is overfull or G = P ⁎ , where P ⁎ is obtained from the Petersen graph by deleting a vertex. Cariolaro and Cariolaro settled the base case Δ = 3 in 2003, and Cranston and Rabern proved the next case, Δ = 4 , in 2019. In this paper, we give a proof of this conjecture for all Δ ≥ 4. [ABSTRACT FROM AUTHOR]

Let r 1 , ... , r t be positive integers and let R 2 (r 1 , ... , r t) be the set of positive odd integers that can be represented as p + 2 k 1 r 1 + ⋯ + 2 k t r t , where p is a prime and k 1 , ... , k t are positive integers. It is easy to see that if r 1 − 1 + ⋯ + r t − 1 < 1 , then the set R 2 (r 1 , ... , r t) has asymptotic density zero. In this paper, we prove that if r 1 − 1 + ⋯ + r t − 1 ≥ 1 , then the set R 2 (r 1 , ... , r t) has a positive lower asymptotic density. Several conjectures and open problems are posed for further research. [ABSTRACT FROM AUTHOR]

In this paper, the fractional problem of the Ambrosetti-Prodi type involving the critical Sobolev exponent is taken into account in a bounded domain of R N { A α u = u + 2 α ⁎ − 1 + λ u − s ¯ φ 1 , u > 0 , in Ω , u = 0 , on ∂ Ω , where A α is the spectral fractional operator, λ and s ¯ are real numbers, Ω ⊂ R N is bounded, 2 α ⁎ = 2 N N − 2 α is a critical exponent, 0 < α < 1 , φ 1 is the first eigenfunction of −Δ with zero Dirichlet boundary condition. We will construct bubbling solutions when the parameter is large enough, and the location of the bubbling point is near the boundary of the domain. [ABSTRACT FROM AUTHOR]

Given a finite group G and an irreducible complex character χ of G , the codegree of χ is defined as the integer cod (χ) = | G : ker (χ) | / χ (1). It was conjectured by G. Qian in [16] that, for every element g of G , there exists an irreducible character χ of G such that cod (χ) is a multiple of the order of g ; the conjecture has been verified under the assumption that G is solvable ([16]) or almost-simple ([13]). In this paper, we prove that Qian's conjecture is true for every finite group whose Fitting subgroup is trivial, and we show that the analysis of the full conjecture can be reduced to groups having a solvable socle. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove a conjecture made by T. Fukaya in 2008. This conjecture concerns the exact value of the Z 2 -cup-length of the Grassmann manifold G ˜ n , 3 of oriented 3-planes in R n. Along the way, we calculate the heights of the Stiefel–Whitney classes of the canonical vector bundle over G ˜ n , 3. [ABSTRACT FROM AUTHOR]

In this paper, we prove that every graph with average degree at least s + t + 2 has a vertex partition into two parts, such that one part has average degree at least s , and the other part has average degree at least t. This solves a conjecture of Csóka, Lo, Norin, Wu and Yepremyan. [ABSTRACT FROM AUTHOR]

The famous List Colouring Conjecture from the 1970s states that for every graph G the chromatic index of G is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds asymptotically. Our main result is a local generalization of Kahn's theorem. More precisely, we show that, for a graph G with sufficiently large maximum degree Δ and minimum degree δ ≥ ln 25 ⁡ Δ , the following holds: for every assignment L of lists of colours to the edges of G , such that | L (e) | ≥ (1 + o (1)) ⋅ max ⁡ { deg ⁡ (u) , deg ⁡ (v) } for each edge e = u v , there is an L -edge-colouring of G. Furthermore, Kahn showed that the List Colouring Conjecture holds asymptotically for linear, k -uniform hypergraphs, and recently Molloy generalized Kahn's original result to correspondence colouring as well as its hypergraph generalization. We prove local versions of all of these generalizations by showing a weighted version that simultaneously implies all of our results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate sums of four squares of integers whose prime factorizations are restricted, making progress towards a conjecture of Sun that states that two of the integers may be restricted to the forms 2 a 3 b and 2 c 5 d. We obtain an ineffective generalization of results of Gauss and Legendre on sums of three squares and an effective generalization of Lagrange's four-square theorem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the multi-species nonlinear Schrödinger systems in R N : { − Δ u j + V j (x) u j = μ j u j 3 + ∑ i = 1 ; i ≠ j d β i , j u i 2 u j in R N , u j (x) > 0 in R N , u j (x) → 0 as | x | → + ∞ , j = 1 , 2 , ⋯ , d , where N = 2 , 3 , μ j > 0 are constants, β i , j = β j , i ≠ 0 are coupling parameters, d ≥ 2 and V j (x) are potentials. By Ljapunov-Schmidt reduction arguments, we construct infinitely many nonradial positive solutions of the above system under some mild assumptions on potentials V j (x) and coupling parameters { β i , j } , without any symmetric assumptions on the limit case of the above system. Our result, giving a positive answer to the conjecture in Pistoia and Viara [50] and extending the results in [50,52] , reveals new phenomenon in the case of N = 2 and d = 2 and is almost optimal for the coupling parameters { β i , j }. [ABSTRACT FROM AUTHOR]

We prove the conjecture that exotic and block-exotic fusion systems coincide holds for all fusion systems on exceptional p -groups of maximal nilpotency class, where p ≥ 5. This is done by considering a family of exotic fusion systems discovered by Parker and Stroth. Together with a previous result by the author, which we also generalise in this paper, and a result by Grazian and Parker this implies the conjecture for fusion systems on such groups. Considering small primes, there are no exotic fusion systems on 2-groups of maximal class and for p = 3 , we prove block-exoticity of two exotic fusion systems described by Diaz–Ruiz–Viruel. [ABSTRACT FROM AUTHOR]

A combinatorial property that characterizes Cohen-Macaulay binomial edge ideals has long been elusive. A recent conjecture ties the Cohen-Macaulayness of a binomial edge ideal J G to special disconnecting sets of vertices of its underlying graph G , called cut sets. More precisely, the conjecture states that J G is Cohen-Macaulay if and only if J G is unmixed and the collection of the cut sets of G is an accessible set system. In this paper we prove the conjecture theoretically for all graphs with up to 12 vertices and develop an algorithm that allows to computationally check the conjecture for all graphs with up to 15 vertices and all blocks with whiskers where the block has at most 11 vertices. This significantly extends previous computational results. [ABSTRACT FROM AUTHOR]

This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that for all k and Δ, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of the (k × k) -wall or the line graph of a subdivision of the (k × k) -wall as an induced subgraph. We prove two theorems supporting this conjecture, as follows. 1. For t ≥ 2 , a t-theta is a graph consisting of two nonadjacent vertices and three internally vertex-disjoint paths between them, each of length at least t. A t-pyramid is a graph consisting of a vertex v , a triangle B disjoint from v and three paths starting at v and vertex-disjoint otherwise, each joining v to a vertex of B , and each of length at least t. We prove that for all k , t and Δ, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a t -theta, or a t -pyramid, or the line graph of a subdivision of the (k × k) -wall as an induced subgraph. This affirmatively answers a question of Pilipczuk et al. asking whether every graph of bounded maximum degree and sufficiently large treewidth contains either a theta or a triangle as an induced subgraph (where a theta means a t -theta for some t ≥ 2). 2. A subcubic subdivided caterpillar is a tree of maximum degree at most three whose all vertices of degree three lie on a path. We prove that for every Δ and subcubic subdivided caterpillar T , every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of T or the line graph of a subdivision of T as an induced subgraph. [ABSTRACT FROM AUTHOR]

The minimal degree of a permutation group G is the minimum number of points not fixed by non-identity elements of G. Lower bounds on the minimal degree have strong structural consequences on G. Babai conjectured that if a primitive coherent configuration with n vertices is not a Cameron scheme, then its automorphism group has minimal degree ≥ c n for some constant c > 0. In 2014, Babai proved the desired lower bound on the minimal degree of the automorphism groups of strongly regular graphs, thus confirming the conjecture for primitive coherent configurations of rank 3. In this paper, we extend Babai's result to primitive coherent configurations of rank 4, confirming the conjecture in this special case. The proofs combine structural and spectral methods. Recently (March 2022) Sean Eberhard published a class of counterexamples of rank 28 to Babai's conjecture and suggested to replace "Cameron schemes" in the conjecture with a more general class he calls "Cameron sandwiches". Naturally, our result also confirms the rank 4 case of Eberhard's version of the conjecture. [ABSTRACT FROM AUTHOR]

Applying a recent extension (2015) of a structure theorem of Robertson, Seymour and Thomas from 1993, in this paper we establish Robertson's magic-tree conjecture from 1997. [ABSTRACT FROM AUTHOR]

In the paper, we give an affirmative answer to the conjecture in [1]. We prove that a Shamsuddin derivation D is simple if and only if Aut (K [ x , y 1 , ... , y n ]) D = { i d }. In addition, we calculate the isotropy groups of the Shamsuddin derivations D = ∂ x + ∑ j = 1 r (a (x) y j + b j (x)) ∂ j of K [ x , y 1 , ... , y r ]. We also prove that Im D is a Mathieu-Zhao subspace if and only if a (x) ∈ K. [ABSTRACT FROM AUTHOR]

Can we find a self-similar set on the line with positive Lebesgue measure and empty interior? Currently, we do not have the answer for this question for deterministic self-similar sets. In this paper we answer this question negatively for random self-similar sets which are defined with the construction introduced in the paper by Jordan et al. (2007) [6]. For the same type of random self-similar sets we prove the Palis-Takens conjecture which asserts that at least typically the algebraic difference of dynamically defined Cantor sets is either large in the sense that it contains an interval or small in the sense that it is a set of zero Lebesgue measure. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to present an answer to Bressoud's conjecture for the case j = 0 , resulting in a complete solution to Bressoud's conjecture. The case for j = 1 has been recently resolved by Kim. Using the connection established in our previous paper between the ordinary partition function B 0 and the overpartition function B ¯ 1 , we found that the proof of Bressoud's conjecture for the case j = 0 is equivalent to establishing an overpartition analogue of the conjecture for the case j = 1. By generalizing Kim's method, we obtain the desired overpartition analogue of Bressoud's conjecture for the case j = 1 , which eventually enables us to confirm Bressoud's conjecture for the case j = 0. [ABSTRACT FROM AUTHOR]

Many conjectures and open problems in graph theory can either be reduced to cubic graphs or are directly stated for them. Furthermore, it is known that for a lot of problems, a counterexample must be a snark, that is, a bridgeless cubic graph which is not 3-edge-colourable. In this paper we deal with the fact that the family of potential counterexamples to many interesting conjectures can be narrowed even further to the family S ≥ 5 of bridgeless cubic graphs whose edge set cannot be covered with four perfect matchings. The Cycle Double Cover Conjecture, the Shortest Cycle Cover Conjecture and the Fan–Raspaud Conjecture are examples of statements for which S ≥ 5 is crucial. In this paper we study parameters which have the potential to further refine S ≥ 5 and thus enlarge the set of cubic graphs for which the mentioned conjectures can be verified. We show that S ≥ 5 can be naturally decomposed into subsets with increasing complexity, thereby producing a natural scale for proving these conjectures. More precisely, we consider the following parameters and questions: given a bridgeless cubic graph, (i) how many perfect matchings need to be added, (ii) how many copies of the same perfect matching need to be added, and (iii) how many 2-factors need to be added so that the resulting regular graph is Class I? We present new results for these parameters and we also establish some strong relations between these problems and some long-standing conjectures. [ABSTRACT FROM AUTHOR]

We reduce the chromatic Iwasawa main conjecture for elliptic curves to the conjectured existence of certain Beilinson–Flach classes in the supersingular case. This generalizes the method of Wan in which a p = 0 was assumed; the main innovation of this paper consists in the new 2-dimensional technique to overcome this condition. We also derive the 3-part of the leading term formula in the Birch and Swinnerton-Dyer conjecture for analytic rank 0 or 1 from the main conjecture. Another consequence is that among those elliptic curves with a prescribed number of solutions modulo any fixed prime, infinitely many would satisfy the full Birch and Swinnerton-Dyer conjecture. [ABSTRACT FROM AUTHOR]

Let F i denote the i th Fibonacci number, and define ∏ i = 1 n 1 + x F i + 1 = ∑ k c n (k) x k . The paper is concerned primarily with the coefficients c n (k). In particular, for any r ≥ 0 the generating function ∑ n ≥ 0 (∑ k c n (k) r) x n is rational. The coefficients c n (k) can be displayed in an array called the Fibonacci triangle poset F with some interesting further properties, including an encoding of a certain dense linear order on the nonnegative integers. Some generalizations are briefly considered, but there remain many open questions. [ABSTRACT FROM AUTHOR]

The mirror P=W conjecture, formulated by Harder-Katzarkov-Przyjalkowski [27] , predicts a correspondence between weight and perverse filtrations in the context of mirror symmetry. In this paper, we revisit this conjecture through the lens of mirror symmetry for a Fano pair (X , D) , where X is a smooth Fano variety and D is a simple normal crossing divisor. We introduce its mirror object as a multi-potential analogue of a Landau-Ginzburg (LG) model, which we call the hybrid LG model. This model is expected to capture the mirrors of all irreducible components of D. We study the topological aspects, particularly the perverse filtration, and the Hodge theory of hybrid LG models, building upon the work of Katzarkov-Kontsevich-Pantev [32]. As an application, we discover an interesting upper bound on the multiplicativity of the perverse filtration for a hybrid LG model. Additionally, we propose a relative version of the homological mirror symmetry conjecture and explain how the mirror P=W conjecture naturally emerges from it. [ABSTRACT FROM AUTHOR]

Let G = (Z / n Z) ⊕ (Z / n Z). Let s ≤ k (G) be the smallest integer ℓ such that every sequence of ℓ terms from G , with repetition allowed, has a nonempty zero-sum subsequence with length at most k. It is known that s ≤ 2 n − 1 − k (G) = 2 n − 1 + k for k ∈ [ 0 , n − 1 ]. The structure of extremal sequences that show this bound is tight was determined for k ∈ { 0 , 1 , n − 1 } , and for various special cases when k ∈ [ 2 , n − 2 ]. For the remaining values k ∈ [ 2 , n − 2 ] , the characterization of extremal sequences of length 2 n − 2 + k avoiding a nonempty zero-sum of length at most 2 n − 1 − k remained open in general. It is conjectured that they must all have the form e 1 [ n − 1 ] ⋅ e 2 [ n − 1 ] ⋅ (e 1 + e 2) [ k ] for some basis (e 1 , e 2) for G. Here x [ n ] denotes a sequence consisting of the term x repeated n times. In this paper, we establish this conjecture for all k ∈ [ 2 , n − 2 ] when n is prime, which in view of other recent work, implies the conjectured structure for all abelian groups of rank two. [ABSTRACT FROM AUTHOR]

In this paper we study L p -norm spherical copulas for arbitrary p ∈ [ 1 , ∞ ] and arbitrary dimensions. The study is motivated by a conjecture that these distributions lead to a sharp bound for the value of a certain generalized mean difference. We fully characterize conditions for existence and uniqueness of L p -norm spherical copulas. Explicit formulas for their densities and correlation coefficients are derived and the distribution of the radial part is determined. Moreover, statistical inference and efficient simulation are considered. [ABSTRACT FROM AUTHOR]

We prove two conjectures involving permutation polynomials in a paper of Dmytrenko, Lazebnik, Williford, in a low degree regime, using the theory of algebraic curves over finite fields. More precisely, we prove that Conjecture A holds whenever q ≥ max ⁡ { (2 k − 1) 2 + 1 , 1.823 (4 k 2 − 14 k + 12) } , whereas Conjecture B holds if q ≥ 2.233 (9 k 2 − 21 k + 12). Although one of these conjectures was already proved by Hou without any restriction on the degree of the polynomials, we consider the proof contained in this paper is more direct and less computational. [ABSTRACT FROM AUTHOR]

Let q = p e , where p is a prime and e is a positive integer. The family of graphs D (k , q) , defined for any positive integer k and prime power q , were introduced by Lazebnik and Ustimenko in 1995. To this day, the connected components of the graphs D (k , q) , provide the best known general lower bound for the size of a graph of given order and given girth. Furthermore, Ustimenko conjectured that the second largest eigenvalue of D (k , q) is always less than or equal to 2 q. If true, this would imply that for a fixed q and k growing, D (k , q) would define a family of expanders that are nearly Ramanujan. In this paper we prove the smallest open case of the conjecture, showing that for all odd prime powers q , the second largest eigenvalue of D (5 , q) is less than or equal to 2 q. [ABSTRACT FROM AUTHOR]

A family F of sets is union-closed if the union of any two sets from F belongs to F. The union-closed sets conjecture states that if F is a finite union-closed family of finite sets, then there is an element that belongs to at least half of the sets in F. The conjecture has several equivalent formulations in terms of other combinatorial structures such as lattices and graphs. In its whole generality the conjecture remains wide open, but it was verified for various important classes of lattices, such as lower semimodular lattices, and graphs, such as chordal bipartite graphs. In the present paper we develop a Boolean approach to the conjecture and verify it for several classes of Boolean functions, such as submodular functions and double Horn functions. [ABSTRACT FROM AUTHOR]

In this paper we study the following eigenvalue problem { − Δ v = λ C (α) (p α − ε) | x | α u ε p α − ε − 1 v in Ω , u = 0 on ∂ Ω , where Ω ⊂ R N is a smooth bounded domain containing the origin, C (α) = (N + α) (N − 2) , N ≥ 3 , p α = N + 2 + 2 α N − 2 , α > 0 , ε > 0 is a small parameter and u ε is a single peaked solution of Hénon equation { − Δ u = C (α) | x | α u p α − ε in Ω , u > 0 in Ω , u = 0 on ∂ Ω , which established by Gladiali and Grossi (2012) [21]. By using various local Pohozaev identities and blow-up analysis, we prove some asymptotic behavior of the eigenvalues λ ε , i and corresponding eigenfunctions v ε , i , i = 2 , ⋯ , ∑ 1 ≤ k < 2 + α 2 (N + 2 k − 2) (N + k − 3) ! (N − 2) ! k ! + 2 when α is not an even integer. As a consequence, if 0 < α < 2 , we have that the Morse index of the single peaked solutions is N + 1 , which gives an affirmative answer to a conjecture raised by Gladiali and Grossi. [ABSTRACT FROM AUTHOR]

The reconfiguration graph C k (G) for the k -colourings of a graph G has a vertex for each proper k -colouring of G , and two vertices of C k (G) are adjacent precisely when those k -colourings differ on a single vertex of G. Much work has focused on bounding the maximum value of diam C k (G) over all n -vertex graphs G. We consider the analogous problems for list colourings and for correspondence colourings. We conjecture that if L is a list-assignment for a graph G with | L (v) | ≥ d (v) + 2 for all v ∈ V (G) , then diam C L (G) ≤ n (G) + μ (G). We also conjecture that if (L , H) is a correspondence cover for a graph G with | L (v) | ≥ d (v) + 2 for all v ∈ V (G) , then diam C (L , H) (G) ≤ n (G) + τ (G). (Here μ (G) and τ (G) denote the matching number and vertex cover number of G.) For every graph G , we give constructions showing that both conjectures are best possible, which also hints towards an exact form of Cereceda's Conjecture for regular graphs. Our first main result proves the upper bounds (for the list and correspondence versions, respectively) diam C L (G) ≤ n (G) + 2 μ (G) and diam C (L , H) (G) ≤ n (G) + 2 τ (G). Our second main result proves that both conjectured bounds hold, whenever all v satisfy | L (v) | ≥ 2 d (v) + 1. We conclude by proving one or both conjectures for various classes of graphs such as complete bipartite graphs, subcubic graphs, cactuses, and graphs with bounded maximum average degree. The full paper can also be found at arxiv.org/abs/2204.07928. [ABSTRACT FROM AUTHOR]

The famous open Vizing conjecture claims that the domination number of the Cartesian product graph of two graphs G and H is at least the product of the domination numbers of G and H. Recently Gaar, Krenn, Margulies and Wiegele used the graph class G of all graphs with n G vertices and domination number k G and reformulated Vizing's conjecture as the problem that for all graph classes G and H the Vizing polynomial is sum-of-squares (SOS) modulo the Vizing ideal. By solving semidefinite programs (SDPs) and clever guessing they derived SOS-certificates for some values of k G , n G , k H , and n H. In this paper, we consider their approach for k G = k H = 1. For this case we are able to derive the unique reduced Gröbner basis of the Vizing ideal. Based on this, we deduce the minimum degree (n G + n H − 1) / 2 of an SOS-certificate for Vizing's conjecture, which is the first result of this kind. Furthermore, we present a method to find certificates for graph classes G and H with n G + n H − 1 = d for general d , which is again based on solving SDPs, but does not depend on guessing and depends on much smaller SDPs. We implement our new method in SageMath and give new SOS-certificates for all graph classes G and H with k G = k H = 1 and n G + n H ≤ 15. [ABSTRACT FROM AUTHOR]

A singular k -tuple of a tensor T of format (n 1 , ... , n k) is essentially a complex critical point of the distance function from T constrained to the cone of tensors of format (n 1 , ... , n k) of rank at most one. A generic tensor has finitely many complex singular k -tuples, and their number depends only on the tensor format. Furthermore, if we fix the first k − 1 dimensions n i , then the number of singular k -tuples of a generic tensor becomes a monotone non-decreasing function in one integer variable n k , that stabilizes when (n 1 , ... , n k) reaches a boundary format. In this paper, we study the linear span of singular k -tuples of a generic tensor. Its dimension also depends only on the tensor format. In particular, we concentrate on special order three tensors and order- k tensors of format (2 , ... , 2 , n). As a consequence, if again we fix the first k − 1 dimensions n i and let n k increase, we show that in these special formats, the dimension of the linear span stabilizes as well, but at some concise non-sub-boundary format. We conjecture that this phenomenon holds for an arbitrary format with k > 3. Finally, we provide equations for the linear span of singular triples of a generic order three tensor T of some special non-sub-boundary format. From these equations, we conclude that T belongs to the linear span of its singular triples, and we conjecture that this is the case for every tensor format. [ABSTRACT FROM AUTHOR]

The new identifiable case appeared in Angelini et al. (2018) , together with the analysis on simultaneous identifiability of pairs of ternary forms recently developed in Beorchia and Galuppi (2022) , suggested the following conjecture towards a complete classification of all simultaneous Waring identifiable cases: for any d ≥ 2 , the general polynomial vectors consisting of d − 1 ternary forms of degree d and a ternary form of degree d + 1 , with rank d 2 + d + 2 2 , are identifiable over C. In this paper, by means of a computer-aided procedure inspired to the one described in Angelini et al. (2018) , we obtain that the case d = 4 contradicts the previous conjecture, admitting at least 36 complex simultaneous Waring decompositions (of length 11) instead of 1. [ABSTRACT FROM AUTHOR]

In 1968, Golomb and Welch conjectured that there is no perfect Lee codes with radius r ≥ 2 and dimension n ≥ 3. A diameter perfect code is a natural generalization of the perfect code. In 2011, Etzion (2011) [5] proposed the following problem: Are there diameter perfect Lee (DPL, for short) codes with distance greater than four besides the D P L (3 , 6) code? Later, Horak and AlBdaiwi (2012) [12] conjectured that there are no D P L (n , d) codes for dimension n ≥ 3 and distance d > 4 except for (n , d) = (3 , 6). In this paper, we give a counterexample to this conjecture. Moreover, we prove that for n ≥ 3 , there is a linear D P L (n , 6) code if and only if n = 3 , 11. [ABSTRACT FROM AUTHOR]