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Let FI(X, K) be the finitary incidence algebra of a non-connected partially ordered set X over a field K of characteristic different from 2. For the case where every multiplicative automorphism of FI(X, K) is inner, we present necessary and sufficient conditions for two involutions on FI(X, K) to be equivalent. [ABSTRACT FROM AUTHOR]

A Hadamard matrix is a scaled orthogonal matrix with ± 1 entries. Such matrices exist in certain dimensions: the Hadamard conjecture is that such a matrix always exists when n is a multiple of 4. A conjecture attributed to Ryser is that no circulant Hadamard matrices exist when n > 4 . Recently, Dong and Rudelson proved the existence of approximate Hadamard matrices in all dimensions: there exist universal 0 < c < C < ∞ so that for all n ≥ 1 , there is a matrix A ∈ - 1 , 1 n × n satisfying, for all x ∈ R n , c n ‖ x ‖ 2 ≤ ‖ A x ‖ 2 ≤ C n ‖ x ‖ 2. We observe that, as a consequence of the existence of flat Littlewood polynomials, circulant approximate Hadamard matrices exist for all n ≥ 1 . [ABSTRACT FROM AUTHOR]

Two skew Hadamard matrices are considered SH-equivalent if they are similar by a signed permutation matrix. This paper determines the number of SH-inequivalent skew Hadamard matrices of order 36 for some types. We also study ternary self-dual codes and association schemes constructed from the skew Hadamard matrices of order 36. [ABSTRACT FROM AUTHOR]

Balanced generalized weight matrices are used to construct optimal constant weight codes that are monomially inequivalent to codes derived from the classical simplex codes. What's more, these codes can be assumed to be generated entirely by ω -shifts of a single codeword where ω is a primitive element of a Galois field. Additional constant weight codes are derived by projecting onto subgroups of the alphabet sets. These too are shown to be optimal. [ABSTRACT FROM AUTHOR]

Denote by $ \omega _n $ ω n the convex polytope of $ n\times n $ n × n doubly substochastic matrices and by $ \Omega _n $ Ω n the convex polytope of $ n\times n $ n × n doubly stochastic matrices. We generalize the Bruhat order for permutation matrices and doubly stochastic matrices to the subpermutation matrices and doubly substochastic matrices, respectively. Moreover we study the Bruhat shadow of a subpermutation matrix and the Bruhat faces of $ \omega _n $ ω n . The relations between the Bruhat faces in $ \omega _n $ ω n and those in $ \Omega _n $ Ω n are also given. [ABSTRACT FROM AUTHOR]

A quaternionic Hadamard matrix (QHM) of order n is an n × n matrix H with non-zero entries in the quaternions such that H H ∗ = n I n , where I n and H ∗ denote the identity matrix and the conjugate-transpose of H, respectively. A QHM is dephased if all the entries in its first row and first column are 1, and it is non-commutative if its entries generate a non-commutative group. The aim of our work is to provide new constructions of infinitely many (non-commutative dephased) QHMs; such matrices are used by Farkas et al. (IEEE Trans Inform Theory 69(6):3814–3824, 2023) to produce mutually unbiased measurements. [ABSTRACT FROM AUTHOR]

It is shown that a normalized complex Hadamard matrix of order 6 having three distinct columns each containing at least one - 1 entry, necessarily belongs to the transposed Fourier family, or to the family of 2-circulant complex Hadamard matrices. The proofs rely on solving polynomial systems of equations by Gröbner basis techniques, and make use of a structure theorem concerning regular Hadamard matrices. As a consequence, members of these two families can be easily recognized in practice. In particular, one can identify complex Hadamard matrices appearing in known triplets of pairwise mutually unbiased bases in dimension 6. [ABSTRACT FROM AUTHOR]

We consider distributed approval voting schemes. Each voter i ∈ I has α i cards that (s)he distributes among the candidates a ∈ A as a measure of approval. One (or several) candidate(s) who received the maximum number of cards is (are) elected. We provide polynomial algorithms to recognize voting forms and voting correspondences generated by such voting schemes in cases when either the number of candidates or the number of voters is equal to 2. We prove that for two voters, if α 2 ≥ α 1 - 2 ≥ 0 then the unique voting correspondence has distinct rows. We also characterize voting forms with distinct rows. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of binary sequences with two-valued non-zero periodic autocorrelation sum and good periodic crosscorrelation sum as well as balanced properties. We make use of the sequences obtained in (No, J. et al., IEEE Trans. Inform. Theory 44(3), 1278-1282 2001) and adopt the extraction method similar to (Lüke, H. IEEE Trans. Inform. Theory 43(1) 1997). The new sequences are proven to be balanced or almost balanced. Based on these correlation and balanced properties, an important application is to construct Hadamard matrices of order p + 1 for p ≡ 3 ( mod 4) and 2 p + 2 for p ≡ 1 ( mod 4). Some examples are shown to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The so-called matrix properties of finitely complete categories are a special type of exactness properties that can be encoded as extended matrices of integers. The relation of entailment of matrix properties gives an interesting preorder on the set of such matrices, which can be investigated independently of the category-theoretic considerations. In this paper, we conduct such investigation and obtain several results dealing with binary matrices, i.e., when the only integer entries in the matrix are 0 or 1. Central to these results is a complete description of the preorder in the case of a special type of binary matrices, which we call diagonal matrices, and which include matrices that define Mal'tsev, majority and arithmetical categories. [ABSTRACT FROM AUTHOR]

In a recent survey, Schmidt compiled equivalences between generalized bent functions, group invariant Butson Hadamard matrices, and abelian splitting relative difference sets. We establish a broader network of equivalences by considering Butson matrices that are cocyclic rather than strictly group invariant. This result has several applications; for example, to the construction of Boolean functions whose expansions are generalized partially bent functions, including cases where no bent function can exist. [ABSTRACT FROM AUTHOR]

Let S be a finite set, each element of which receives a color. A rainbow t-set of S is a t-subset of S in which different elements receive different colors. Let S t denote the set of all rainbow t-sets of S , let S ≤ t represent the union of S i for i = 0 , … , t , and let 2 S stand for the set of all rainbow subsets of S . The rainbow inclusion matrix W S is the 2 S × 2 S (0, 1) matrix whose (T, K)-entry is one if and only if T ⊆ K . We write W t , k S and W ≤ t , k S for the S t × S k submatrix and the S ≤ t × S k submatrix of W S , respectively, and so on. We determine the diagonal forms and the ranks of W t , k S and W ≤ t , k S . We further calculate the singular values of W t , k S and construct accordingly a complete system of (0 , ± 1) eigenvectors for them when the numbers of elements receiving any two given colors are the same. Let D t , k S denote the integral lattice orthogonal to the rows of W ≤ t , k S and let D ¯ t , k S denote the orthogonal lattice of D t , k S . We make use of Frankl rank to present a (0 , ± 1) basis of D t , k S and a (0, 1) basis of D ¯ t , k S . For any commutative ring R, those nonzero functions f ∈ R 2 S satisfying W t , ≥ 0 S f = 0 are called null t-designs over R, while those satisfying W ≤ t , ≥ 0 S f = 0 are called null (≤ t) -designs over R. We report some observations on the distributions of the support sizes of null designs as well as the structure of null designs with extremal support sizes. [ABSTRACT FROM AUTHOR]

Let A = [aij]m×n be an m × n matrix of zeros and ones. The matrix A is said to be a Ferrers matrix if it has decreasing row sums and it is row and column dense with nonzero (1,1)-entry. We characterize all linear maps perserving the set of n × 1 Ferrers vectors over the binary Boolean semiring and over the Boolean ring ℤ 2 . Also, we have achieved the number of these linear maps in each case. [ABSTRACT FROM AUTHOR]

This work brings together several types of combinatorial designs: difference matrices, difference covering arrays and difference schemes by defining the concept of covering scheme of strength t over an abelian additive group. Connections of covering schemes with orthogonal arrays and covering arrays are also established. We show general results of covering schemes of strength t using a method based on the factorization of a group and some refinements for particular classes. We apply the previous results to investigate covering schemes having three, four and five factors. Finally, a reformulation of covering schemes in terms of graph theory is established. [ABSTRACT FROM AUTHOR]

A subspace code is a nonempty set of subspaces of a vector space F q n . Linear codes with complementary duals, or LCD codes, are linear codes whose intersection with their duals is trivial. In this paper, we introduce a notion of LCD subspace codes. We show that the minimum distance decoding problem for an LCD subspace code reduces to a problem that is simpler than for a general subspace code. Further, we show that under some conditions equitable partitions of association schemes yield such LCD subspace codes and as an illustration of the method give some examples from distance-regular graphs. We also give constructions from mutually unbiased weighing matrices, that include constructions from mutually unbiased Hadamard matrices. [ABSTRACT FROM AUTHOR]

In this article, we consider the resistance matrix of a connected graph. A connected graph is said to be resistance regular if all the row(column) sums of its resistance matrix are equal. We establish some necessary and sufficient conditions for a simple connected graph to be a resistance regular graph. Also, we find some relationship between the Laplacian matrix and the resistance matrix in the case of weighted graphs where all edge weights are positive definite matrices of given order. [ABSTRACT FROM AUTHOR]

In this note, we report the classification of all symmetric 2-(36, 15, 6) designs that admit an automorphism of order 2 and their incidence matrices generate an extremal ternary self-dual code. It is shown that up to isomorphism, there exists only one such design, having a full automorphism group of order 24, and the ternary code spanned by its incidence matrix is equivalent to the Pless symmetry code. [ABSTRACT FROM AUTHOR]

In this this paper all binary codes of length 552 which admit the sporadic simple group Co 3 as an imprimitive transitive permutation group are determined. Our aim is to understand the results also by using theoretical arguments and to discuss the combinatorial properties of the codes as well as their relation to some special properties of the Leech lattice group Co 3 . For all codes (with two exceptions) we obtain the weight enumerators and in many interesting cases the classification of codewords under the action of the group of code automorphisms Co 3 . The exceptional codes are both self-dual and have minimum weight 12. [ABSTRACT FROM AUTHOR]

In 2013, Nebe and Villar gave a series of ternary self-dual codes of length 2 (p + 1) for a prime p congruent to 5 modulo 8. As a consequence, the third ternary extremal self-dual code of length 60 was found. We show that these ternary self-dual codes contain codewords which form a Hadamard matrix of order 2 (p + 1) when p is congruent to 5 modulo 24. In addition, we show that the ternary self-dual codes found by Nebe and Villar are generated by the rows of the Hadamard matrices. We also demonstrate that the third ternary extremal self-dual code of length 60 contains at least two inequivalent Hadamard matrices. [ABSTRACT FROM AUTHOR]

Pseudorandom sequences, sometimes shortened as sequences, have played a key role in the applications of digital communications, cryptography and computer science. This research field is an example of scientific research directly born from the real world applications. Specifically, the research on sequences stems from the application of the sequences generated by maximal length linear feedback shift registers to detect returning signals from Explorer 1, the satellite launched on January 31, 1958 by US, shortly after Sputnik, launched by Soviet Union on October 4, 1957 which is the first satellite in the human being civilization. With more than seven decades of the developments of theory and practice of sequences, this field has evolved to acquire a wide range of the tools and methodologies from extremely deep mathematic fields (comparing with other engineering subjects), such as algebraic geometry, number theory, combinatorics, representation theory, harmonic analysis, to just mention a few. In this survey, we present the current status of the research in sequence design along three different directions, i.e., the sequences with 2-level autocorrelation, the sequence sets with low ambiguity, and Golay complementary sequence sets and complete complementary codes. [ABSTRACT FROM AUTHOR]

Boolean functions naturally induce binary sequences of length 2 m and a large number of such orthogonal sequences is required in the design of code-division multiple-access (CDMA) systems. In this paper, Boolean functions are used to construct nonlinear phase orthogonal sequence sets for CDMA communications. For even m, employing carefully designed an m-variable Boolean function with five-valued Walsh spectra, one can get 16 different orthogonal sequence sets with sequence length 2 m . These sequence sets are assigned to a lattice of regular hexagonal cells, and we can ensure the orthogonality of adjacent cells. Moreover, the cross-correlation values between the sequences in a given cell and the sequences in non-neighbouring cells belong to { 0 , ± 2 m 2 , ± 2 m 2 + 1 } . On the other hand, the cardinality of the sequences sets is 2 m - 3 thus implying a trade-off between the quality of communication and the number of users assigned to each cell. This method can be improved so that the number of users is increased to 2 m - 2 in one half of the network while preserving the orthogonality between adjacent cells and the same level of low cross-correlation values to the non-neighbouring cells. [ABSTRACT FROM AUTHOR]

In this paper we study Butson Hadamard matrices, and codes over finite rings coming from these matrices in logarithmic form, called BH-codes. We introduce a new morphism of Butson Hadamard matrices through a generalized Gray map on the matrices in logarithmic form, which is comparable to the morphism given in a recent note of Ó Catháin and Swartz. That is, we show how, if given a Butson Hadamard matrix over the k th roots of unity, we can construct a larger Butson matrix over the ℓ th roots of unity for any ℓ dividing k, provided that any prime p dividing k also divides ℓ . We prove that a Z p s -additive code with p a prime number is isomorphic as a group to a BH-code over Z p s and the image of this BH-code under the Gray map is a BH-code over Z p (binary Hadamard code for p = 2 ). Further, we investigate the inherent propelinear structure of these codes (and their images) when the Butson matrix is cocyclic. Some structural properties of these codes are studied and examples are provided. [ABSTRACT FROM AUTHOR]

We present a construction of partial spread bent functions using subspaces generated by linear recurring sequences (LRS). We first show that the kernels of the linear mappings defined by two LRS have a trivial intersection if and only if their feedback polynomials are relatively prime. Then, we characterize the appropriate parameters for a family of pairwise coprime polynomials to generate a partial spread required for the support of a bent function, showing that such families exist if and only if the degrees of the underlying polynomials are either 1 or 2. We then count the resulting sets of polynomials and prove that, for degree 1, our LRS construction coincides with the Desarguesian partial spread. Finally, we perform a computer search of all PS - and PS + bent functions of n = 8 variables generated by our construction and compute their 2-ranks. The results show that many of these functions defined by polynomials of degree d = 2 are not EA-equivalent to any Maiorana–McFarland or Desarguesian partial spread function. [ABSTRACT FROM AUTHOR]

It is known that the minimum number of distinct eigenvalues c(T) of a symmetric matrix whose graph is a given tree T is at least the diameter d(T) of that tree. However, the disparity c(T) − d(T) can be positive. Using branch duplication and rooted seeds, the notion of the 'most complex seed' is introduced, and an explicit upper bound on the disparity is given for any tree of a given diameter. [ABSTRACT FROM AUTHOR]

The existence of a quasi-symmetric 2-(41, 9, 9) design with intersection numbers x = 1 , y = 3 is a long-standing open question. Using linear codes and properties of subdesigns, we prove that a cyclic quasi-symmetric 2-(41, 9, 9) design does not exist, and if p < 41 is a prime number being the order of an automorphism of a quasi-symmetric 2-(41, 9, 9) design, then p ≤ 5 . The derived design with respect to a point of a quasi-symmetric 2-(41, 9, 9) design with block intersection numbers 1 and 3 is a quasi-symmetric 1-(40, 8, 9) design with block intersection numbers 0 and 2. The incidence matrix of the latter generates a binary doubly even code of length 40. Using the database of binary doubly even self-dual codes of length 40 classified by Betsumiya et al. (Electron J Combin 19(P18):12, 2012), we prove that there is no quasi-symmetric 2-(41, 9, 9) design with an automorphism ϕ of order 5 with exactly one fixed point such that the binary code of the derived design is contained in a doubly-even self-dual [40, 20] code invariant under ϕ . [ABSTRACT FROM AUTHOR]

Balanced weighing matrices with parameters (1 + 18 ⋅ 9 m + 1 − 1 8 , 9 m + 1 , 4 ⋅ 9 m) , for each nonzero integer m are constructed. This is the first infinite class not belonging to those with classical parameters. It is shown that any balanced weighing matrix is equivalent to a five-class association scheme. [ABSTRACT FROM AUTHOR]

It is proved that a code L(q) which is monomially equivalent to the Pless symmetry code C(q) of length 2 q + 2 contains the (0,1)-incidence matrix of a Hadamard 3- (2 q + 2 , q + 1 , (q - 1) / 2) design D(q) associated with a Paley–Hadamard matrix of type II. Similarly, any ternary extended quadratic residue code contains the incidence matrix of a Hadamard 3-design associated with a Paley–Hadamard matrix of type I. If q = 5 , 11 , 17 , 23 , then the full permutation automorphism group of L(q) coincides with the full automorphism group of D(q), and a similar result holds for the ternary extended quadratic residue codes of lengths 24 and 48. All Hadamard matrices of order 36 formed by codewords of the Pless symmetry code C(17) are enumerated and classified up to equivalence. There are two equivalence classes of such matrices: the Paley–Hadamard matrix H of type I with a full automorphism group of order 19584, and a second regular Hadamard matrix H ′ such that the symmetric 2-(36, 15, 6) design D associated with H ′ has trivial full automorphism group, and the incidence matrix of D spans a ternary code equivalent to C(17). [ABSTRACT FROM AUTHOR]

In this paper, we introduce a switching for 2-designs, which defines a type of trade. We illustrate this method by applying it to some symmetric (64, 28, 12) designs, showing that the switching introduced in this paper in some cases can be applied directly to orbit matrices. In that way we obtain six new symmetric (64, 28, 12) designs. Further, we show that this type of switching (of trades) can be applied to any symmetric design related to a Bush-type Hadamard matrix and construct symmetric designs with parameters (36, 15, 6) leading to new Bush-type Hadamard matrices of order 36, and symmetric (100, 45, 20) designs yielding Bush-type Hadamard matrices of order 100. [ABSTRACT FROM AUTHOR]

Let m and n be two positive integers. We use [m] for the set { 1 , ... , m } . For any integer i, i m designates the minimum nonnegative integer that is congruent to i modulo m, and S m , i n stands for the set of those x ∈ [ m ] [ n ] satisfying ∑ t = 1 n x (t) is congruent to i modulo m. An enumeration of S m , i n as x 1 , ... , x m n - 1 is called a tight balanced cyclic sequence if ∑ p ∈ [ m n - 1 ] x p + 1 (t) - x p (t) m = m n n holds for each t ∈ [ n ] , where the subscript m n - 1 + 1 should be understood as 1. Assuming that m n is a multiple of n, is there always a tight balanced cyclic sequence over S m , i n ? We provide a positive answer to this question when n is a power of m. [ABSTRACT FROM AUTHOR]

Many problems in combinatorial linear algebra require upper bounds on the number of solutions to an underdetermined system of linear equations $Ax = b$ , where the coordinates of the vector x are restricted to take values in some small subset (e.g. $\{\pm 1\}$) of the underlying field. The classical ways of bounding this quantity are to use either a rank bound observation due to Odlyzko or a vector anti-concentration inequality due to Halász. The former gives a stronger conclusion except when the number of equations is significantly smaller than the number of variables; even in such situations, the hypotheses of Halász's inequality are quite hard to verify in practice. In this paper, using a novel approach to the anti-concentration problem for vector sums, we obtain new Halász-type inequalities that beat the Odlyzko bound even in settings where the number of equations is comparable to the number of variables. In addition to being stronger, our inequalities have hypotheses that are considerably easier to verify. We present two applications of our inequalities to combinatorial (random) matrix theory: (i) we obtain the first non-trivial upper bound on the number of $n\times n$ Hadamard matrices and (ii) we improve a recent bound of Deneanu and Vu on the probability of normality of a random $\{\pm 1\}$ matrix. [ABSTRACT FROM AUTHOR]

For various quasi-synchronous (QS) CDMA systems, to reduce or eliminate the multiple access interference and multipath interference, it is required to design a set of spreading sequences which are mutually orthogonal within a designed shift zone. In this article, we demonstrate that a concept of irregular spatial assignment, with flexibility to assign different number of users to different cells, can be used to provide the maximal number of orthogonal sequences in any three adjacent cells in networks with a regular tessellation of hexagonal cells. We first consider p-phase spreading sequences of length p m (thus nonbinary p-valued sequences) suitable for synchronous (S)-CDMA applications, for p > 3 , and give an efficient design method for reaching the maximal cardinality achievable (being p m ). A simple solution for a flexible assignment of our orthogonal sets of spreading sequences to the cells in hexagonal networks is given. To address QS-CDMA applications as well, an efficient method to combine these orthogonal sequences with Zadeoff–Chu sequences is proposed for the purpose of designing sets of zero correlation zone (ZCZ) sequences (within a certain shift zone) with optimal parameters, thus reaching the Tang–Fan–Matsufuji bound. A similar design framework, based on the use of some special classes of Boolean functions, is then employed for the binary case to provide the maximum cardinality of pairwise orthogonal sequences of length 2 m through this irregular spatial assignment. This improves upon the best known results achieved in Zhang et al. (IEEE Trans Inf Theory 62:3757–3767, 2016), which assigns 2 m - 2 orthogonal sequences (users) per cell, by doubling the number of users in one third of the network. [ABSTRACT FROM AUTHOR]

The two-circulant core (TCC) construction for Hadamard matrices uses two sequences with almost perfect autocorrelation to construct a Hadamard matrix. A research problem of K. Horadam asks whether such matrices are cocyclic. Using techniques from the theory of permutation groups, we prove that the order of a cocyclic TCC matrix coincides with the order of a Hadamard matrix of Paley type, of Sylvester type or certain multiples of these orders. We show that there exist cocyclic TCC Hadamard matrices at all allowable orders ⩽ 1000 with at most one exception. Of the four families of TCC matrices known in the literature, we establish that two are cocyclic, prove that one is not cocyclic, and leave one undecided. The undecided family consists of matrices of 2-power order; we show that these are inequivalent to the Sylvester matrices. As a generalisation of the TCC construction, we introduce quadruple-circulant core (QCC) Hadamard matrices; our results give a complete description of the orders that admit cocyclic QCC Hadamard matrices. [ABSTRACT FROM AUTHOR]

We present a method for determining the number of decimation classes of density δ binary vectors indexed by a finite abelian group G of size ℓ and exponent ℓ ∗ such that δ is relatively prime to ℓ ∗ . This method is the first which is not based on exhaustive vector generation and exploits the subgroup lattice of Z ℓ ∗ × . Instead, our method is based on our newly developed theory of multipliers for arbitrary subsets of finite abelian groups, our results on orbits under the action of the multiplier group, and finding the number of solutions of a potentially highly symmetric subset sum problem. Implementing our method on vectors indexed by Z ℓ of odd length ℓ and density (ℓ + 1) / 2 greatly increased the number of ℓ for which the number of decimation classes of such vectors is known. Additionally, our newly developed theory provides information on the number of distinct translates fixed by each member of the multiplier group as well as sufficient conditions for each member of the multiplier group to be translate fixing. [ABSTRACT FROM AUTHOR]

A quasi-orthogonal cocycle, defined over a group of order congruent to 2 modulo 4, is naturally analogous to an orthogonal cocycle (i.e., one defined over a group of order divisible by 4, and whose display matrix is Hadamard). Here we extend the theory of quasi-orthogonal cocycles in new directions, using equivalences with various optimal binary and quaternary sequences. [ABSTRACT FROM AUTHOR]

Let G be a finite group and k ⩾ 2 be an integer. A (G, k, 1)-difference matrix (DM) is a k × | G | matrix D = (d ij) with entries from G, such that for all distinct rows x and y, the multiset of differences { d xi d yi - 1 : 1 ⩽ i ⩽ | G | } contains each element of G exactly once. This paper examines the existence of difference matrices with five rows over a finite abelian group. It is proved that if G is a finite abelian group and the Sylow 2-subgroup of G is trivial or noncyclic, then a (G, 5, 1)-DM exists, except for G ∈ { Z 3 , Z 2 ⊕ Z 2 , Z 4 ⊕ Z 2 , Z 9 } and possibly for some groups whose Sylow 2-subgroup lies in { Z 2 ⊕ Z 2 , Z 4 ⊕ Z 2 , Z 32 ⊕ Z 2 , Z 16 ⊕ Z 4 } , and some cyclic groups of order 9p with p prime. [ABSTRACT FROM AUTHOR]

We show that the maximum cardinality of an equiangular line system in 14 and 16 dimensions is 28 and 40, respectively, thereby solving a longstanding open problem. We also improve the upper bounds on the cardinality of equiangular line systems in 19 and 20 dimensions to 74 and 94, respectively. [ABSTRACT FROM AUTHOR]

Designs over edge-regular, co-edge-regular and amply regular graphs are investigated. Using linear algebra, we obtain lower bounds in certain inequalities involving the parameters of the designs. Some results on designs meeting the bounds are obtained. These designs are over connected regular graphs with least eigenvalue - 2 , have the minimal number of blocks and do not appear in an earlier work. Partial classification such designs over strongly regular graphs with least eigenvalue - 2 is given. [ABSTRACT FROM AUTHOR]

A simple (2 , 3 , v) trade is a pair (T 0 , T 1) of disjoint sets of 3 -subsets (blocks) of a v -set such that any two elements meet the same number of times in the blocks of T 0 and the blocks of T 1 . The size of T 0 equals the size of T 1 and is called the volume of the trade. In this paper, for v = 4 n + 2 , we introduce a new partitioning of the set of all 3-subsets of a v-set into the trades of volume 4 , 6 , and 8 . [ABSTRACT FROM AUTHOR]

Let G be a finite group and let h be a positive integer. A BH(G, h) matrix is a G-invariant ∣G∣ × ∣G∣ matrix H whose entries are complex hth roots of unity such that H H* = ∣G∣I∣G∣, where H* denotes the complex conjugate transpose of H, and I∣G∣ denotes the identity matrix of order ∣G∣. In this paper, we give three new constructions of BH(G, h) matrices. The first construction is the first known family of BH(G, h) matrices in which G does not need to be abelian. The second and the third constructions are two families of BH(G, h) matrices in which G is a finite local ring. [ABSTRACT FROM AUTHOR]

The concept of balancedly splittable orthogonal designs is introduced along with a recursive construction. As an application, equiangular tight frames over the real, complex, and quaternions meeting the Delsarte–Goethals–Seidel upper bound are obtained. [ABSTRACT FROM AUTHOR]

We consider the class A s y m 0 (n , k) of symmetric (0 , 1) -matrices with zero trace and constant row sums k which can be identified with the class of the adjacency matrices of k-regular undirected graphs. In a previous paper, two partial orders, the Bruhat and the Bruhat-graph order, have been introduced in this class. In fact, when k = 1 or k = 2, it was shown that the two orders coincide, while for k ≥ 3 the two orders are distinct. In this paper we give general properties of minimal and maximal matrices for these orders on A s y m 0 (n , k) and study the minimal and maximal matrices when k = 1, 2 or 3. [ABSTRACT FROM AUTHOR]

For every prime power q ≡ 7 m o d 16 , there are (q; a, b, c, d)-partitions of GF(q), with odd integers a, b, c, and d, where a ≡ ± 1 m o d 8 such that q = a 2 + 2 (b 2 + c 2 + d 2) and d 2 = b 2 + 2 a c + 2 b d . Many results for the existence of 4 - { q 2 ; q (q - 1) 2 ; q (q - 2) } SDS which are simple homogeneous polynomials of parameters a, b, c and d of degree at most 2 have been found. Hence, for each value of q, the construction of SDS becomes equivalent to building a (q ; a , b , c , d) -partition. Once this is done, the verification of the construction only involves verifying simple conditions on a, b, c and d which can be done manually. [ABSTRACT FROM AUTHOR]

We provide a search method for Legendre pairs of composite length based on generating binary matrices with fixed row and column sums from compressed, complementary integer vectors. This approach yielded the first construction of a Legendre pair of length 77, as well as the first exhaustive generation of Legendre pairs of length 55. [ABSTRACT FROM AUTHOR]

In this paper, we present necessary and sufficient conditions for an additive derivation of an incidence algebra of a connected finite partially ordered set X to be inner. These conditions are related to the structure of X as a directed graph and can be applied for finite partially ordered sets that are not connected or even for some that are not finite. [ABSTRACT FROM AUTHOR]

Codes from generalized Hadamard matrices have already been introduced. Here we deal with these codes when the generalized Hadamard matrices are cocyclic. As a consequence, a new class of codes that we call generalized Hadamard full propelinear codes turns out. We prove that their existence is equivalent to the existence of central relative (v, w, v, v/w)-difference sets. Moreover, some structural properties of these codes are studied and examples are provided. Some of the propelinear codes constructed for the examples perform better than any comparable linear code. [ABSTRACT FROM AUTHOR]

Linear codes with complementary duals are linear codes whose intersection with their duals are trivial, shortly named LCD codes. In this paper we outline a construction for LCD codes over finite fields of order q using weighing matrices and their orbit matrices. The LCD codes constructed can be of any length dimension according to the choice of matrices used in their construction. As a special case, LCD codes of length 2n and dimension n are constructed which also have the property of being formally self-dual. Alternatively, under a condition depending on q that the codes are not LCD, this method constructs self-dual codes. To illustrate the method we construct LCD codes from weighing matrices, including the Paley conference matrices and Hadamard matrices. We also extend the construction to Hermitian LCD codes over the finite field of order 4. In addition, we propose a decoding algorithm that can be feasible for the LCD codes obtained from some of the given methods. [ABSTRACT FROM AUTHOR]

In the context of Riordan arrays, the problem of determining the square root of a Bell matrix R = R (f (t) / t , f (t)) defined by a formal power series f (t) = ∑ k ≥ 0 f k t k with f (0) = f 0 = 0 is presented. It is proved that if f ′ (0) = 1 and f ″ (0) ≠ 0 then there exists another Bell matrix H = R (h (t) / t , h (t)) such that H ∗ H = R ; in particular, function h(t) is univocally determined by a symbolic computational method which in many situations allows to find the function in closed form. Moreover, it is shown that function h(t) is related to the solution of Schröder’s equation. We also compute a Riordan involution related to this kind of matrices. [ABSTRACT FROM AUTHOR]

The group testing problem is that we are asked to identify all the defects with the minimum number of tests when given a set of n items with at most d defects. In this paper, as a generalization of Liu et al.'s construction in the paper (Liu and Gao in Discret Math 338:857–862, 2015), new pooling designs are constructed from singular linear spaces over finite fields. Then we make comparisons with Liu et al.'s construction in the aspects of parameters of pooling designs. By choosing appropriate parameters in our pooling designs, the performance of test efficiency in our pooling designs is better than that given by Liu et al. Finally, the analysis of parameters in our pooling designs is provided. [ABSTRACT FROM AUTHOR]

A mixed graph is obtained from an unoriented graph by orienting a subset of its edges. Yu, Liu and Qu in 2017 have established the expression for the determinant of the Hermitian (quasi-) Laplacian matrix of a mixed graph. Here we find general expressions for all minors of Hermitian (quasi-) Laplacian matrix of a mixed graph. [ABSTRACT FROM AUTHOR]

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