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Hadi Kharaghani

Publications and source records attributed to Hadi Kharaghani.

At least 19 recordsLinked to original sources
arXiv2026

A search for Hadamard matrices of Williamson type

In this article, we consider a special class of Williamson type matrices which we call them near Williamson matrices. They are in fact four $n\times n$ $(-1, 1)$-matrices $A, B, C, D$ so that $A$ is circulant, $B,C,D$ are symmetric circulant, and they satisfy $AA^\top+BB^\top+CC^\top+DD^\top=4nI$. Using a computer search, we find all inequivalent near Williamson matrices for all odd orders at most $35$. We also show that such matrices exist for all odd orders up to $63$. As a consequence, we find the first known example of a quaternary Hadamard matrix of order $118$.

math.CO↗
arXiv2026

A class of skew-regular quaternary Hadamard matrices

This paper introduces and investigates a novel class of skew-regular Quaternary Hadamard matrices. For every odd prime power $p$, we establish the existence of these matrices for all orders $1+p^2$, each characterized by a constant row sum of $1-pi$. Motivated by the growing importance of large-excess matrices in the maximum determinant problem and quantum nonlocality, we demonstrate that this class provides a robust framework for generating Hadamard matrices with exceptionally large excess.

math.CO↗
arXiv2025

Unbiased weighing matrices of weight $9$

We investigate unbiased weighing matrices of weight $9$ and provide a construction method using mutually suitable Latin squares. For $n \le 16$, we determine the maximum size among sets of mutually unbiased weighing matrices of order $n$ and weight $9$. Notably, our findings reveal that $13$ is the smallest order where such pairs exist, and $16$ is the first order for which a maximum class of unbiased weighing matrices is found.

math.CO↗
arXiv2025

A generalisation of bent vectors for Butson Hadamard matrices

An $n\times n$ complex matrix $M$ with entries in the $k^{\textrm{th}}$ roots of unity which satisfies $MM^{\ast} = nI_{n}$ is called a Butson Hadamard matrix. While a matrix with entries in the $k^{\textrm{th}}$ roots typically does not have an eigenvector with entries in the same set, such vectors and their generalisations turn out to have multiple applications. A bent vector for $M$ satisfies $M{\bf x} = λ{\bf y}$ where ${\bf x}$ has entries in the $k^{\textrm{th}}$ roots of unity and all entries of $\textbf{y}$ are complex numbers of norm $1$. Such a bent vector ${\bf x}$ is self-dual if ${\bf y} = μ{\bf x}$ and conjugate self-dual if ${\bf y} = μ\overline{\bf x}$ for some $μ$ of norm $1$. Using techniques from algebraic number theory, we prove some order conditions and non-existence results for self-dual and conjugate self-dual bent vectors; using tensor constructions and Bush-type matrices we give explicit examples. We conclude with an application to the covering radius of certain non-linear codes generalising the Reed Muller codes.

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arXiv2024

Constructions of self-orthogonal and LCD subspace codes

Recently, the notions of self-orthogonal subspace codes and LCD subspace codes were introduced, and LCD subspace codes obtained from mutually unbiased weighing matrices were studied. In this paper, we provide a method of constructing self-orthogonal and LCD subspace codes from a set of matrices under certain conditions. In particular, we give constructions of self-orthogonal and LCD subspace codes from mutually quasi-unbiased weighing matrices, linked systems of symmetric designs, and linked systems of symmetric group divisible designs, Deza graphs and their equitable partitions.

math.CO↗
arXiv2024

On the classification of skew Hadamard matrices of order 36 and related structures

Two skew Hadamard matrices are considered {\sf SH}-equivalent if they are similar by a signed permutation matrix. This paper determines the number of {\sf 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 $36$.

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arXiv2023

Ternary extremal four-negacirculant self-dual codes

In this note, we give basic properties of ternary four-negacirculant self-dual codes. By exhaustive computer search based on the properties, we complete a classification of ternary extremal four-negacirculant self-dual codes of lengths 40, 44, 48, 52 and 60.

math.CO↗
arXiv2022

Hadamard matrices related to the projective planes

Let $n$ be the order of a (quaternary) Hadamard matrix. It is shown that the existence of a projective plane of order $n$ is equivalent to the existence of a balancedly multi-splittable (quaternary) Hadamard matrix of order $n^2$.

math.CO↗
arXiv2022

Quasi-balanced weighing matrices, signed strongly regular graphs and association schemes

A weighing matrix $W$ is quasi-balanced if $|W||W|^\top=|W|^\top|W|$ has at most two off-diagonal entries, where $|W|_{ij}=|W_{ij}|$. A quasi-balanced weighing matrix $W$ signs a strongly regular graph if $|W|$ coincides with its adjacency matrix. Among other things, signed strongly regular graphs and their equivalent association schemes are presented.

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arXiv2021

On a class of optimal constant weight ternary codes

A weighing matrix $W$ of order $n=\frac{p^{m+1}-1}{p-1}$ and weight $p^m$ is constructed and shown that the rows of $W$ and $-W$ form optimal constant weight ternary codes of length $n$, weight $p^m$ and minimum distance $p^{m-1}(\frac{p+3}{2})$ for each odd prime power $p$ and integer $m\ge 1$ and thus $$A_3\left(\frac{p^{m+1}-1}{p-1},p^{m-1}\big(\frac{p+3}{2}\big),p^{m}\right)=2\big(\frac{p^{m+1}-1}{p-1}\big).$$

math.CO↗
arXiv2021

A family of balanced generalized weighing matrices

Balanced weighing matrices with parameters $$ \left(1+18\cdot\frac{9^{m+1}-1}{8},9^{m+1},4\cdot 9^m\right), $$ for each nonzero integer $m$ is 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.

math.CO↗
arXiv2021

Balanced Weighing Matrices

A unified approach to the construction of weighing matrices and certain symmetric designs is presented. Assuming the weight $p$ in a weighing matrix $W(n,p)$ is a prime power, it is shown that there is a $$W\left(\frac{p^{m+1}-1}{p-1}(n-1)+1,p^{m+1}\right)$$ for each positive integer $m$. The case of $n=p+1$ reduces to the balanced weighing matrices with classical parameters $$W\left(\frac{p^{m+2}-1}{p-1},p^{m+1}\right).$$ The equivalence with certain classes of association schemes is discussed in details.

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arXiv2020

Disjoint weighing matrices

The notion of disjoint weighing matrices is introduced as a generalization of orthogonal designs. A recursive construction along with a computer search lead to some infinite classes of disjoint weighing matrices, which in turn are shown to form commutative association schemes with 3 or 4 classes.

math.CO↗
arXiv2020

Divisible design digraphs and association schemes

Divisible design digraphs are constructed from skew balanced generalized weighing matrices and generalized Hadamard matrices. Commutative and non-commutative association schemes are shown to be attached to the constructed divisible design digraphs.

math.CO↗
arXiv2020

New constructions of Deza digraphs

Deza digraphs were introduced in 2003 by Zhang and Wang as directed graph version of Deza graphs, that also generalize the notion of directed strongly regular graphs. In this paper we give several new constructions of Deza digraphs. Further, we introduce twin and Siamese twin (directed) Deza graphs and construct several examples. Moreover, we classify directed Deza graphs with parameters $(n,k,b,a,t)$ having the property that $b=t$. Finally, we introduce a variation of directed Deza graphs and provide a construction from finite fields.

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