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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]

A family of two-unitary complex Hadamard matrices (CHMs) of size 36 stemming from a particular matrix is constructed. Every matrix in this orbit remains unitary after operations of partial transpose and reshuffling which makes it a distinguished subset of CHM. It provides a novel solution to the quantum version of the Euler problem, in which each field of the Graeco-Latin square of size six contains a symmetric superposition of all 36 officers with phases being multiples of sixth root of unity. This simplifies previously known solutions as all amplitudes of the superposition are equal and the set of phases consists of six elements only. Multidimensional parameterization allows for more flexibility in a potential experimental realization.

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]

By assuming a type of balance for length ℓ=87\ell =87 and nontrivial subgroups of multiplier groups of Legendre pairs (LPs) for length ℓ=85\ell =85, we find LPs of these lengths. We then study the power spectral density (PSD) values of mm compressions of LPs of length 5m5m. We also formulate a conjecture for LPs of lengths ℓ≡0\ell \equiv 0 (mod 5) and demonstrate how it can be used to decrease the search space and storage requirements for finding such LPs. The newly found LPs decrease the number of integers in the range ≤200\le 200 for which the existence question of LPs remains unsolved from 12 to 10.

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]