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Register a new userNew families of Hadamard matrices with maximum excess
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In this paper, we find regular or biregular Hadamard matrices with maximum excess by negating some rows and columns of known Hadamard matrices obtained from quadratic residues of finite fields. In particular, we show that if either $4m^2+4m+3$ or $2m^2+2m+1$ is a prime power, then there exists a biregular Hadamard matrix of order $n=4(m^2+m+1)$ with maximum excess. Furthermore, we give a sufficient condition for Hadamard matrices obtained from quadratic residues being transformed to be regular in terms of four-class translation association schemes on finite fields.
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In this paper, we generalize classical constructions of skew Hadamard difference families with two or four blocks in the additive groups of finite fields given by Szekeres (1969, 1971), Whiteman (1971) and Wallis-Whiteman (1972). In particular, we show that there exists a skew Hadamard difference family with $2^{u-1}$ blocks in the additive group of the finite field of order $q^e$ for any prime power $qequiv 2^u+1,({mathrm{mod, , }2^{u+1}})$ with $uge 2$ and any positive integer $e$. In the aforementioned work of Szekeres, Whiteman, and Wallis-Whiteman, the constructions of skew Hadamard difference families with $2^{u-1}$ ($u=2$ or $3$) blocks in $({mathbb F}_{q^e},+)$ depend on the exponent $e$, with $eequiv 1,2,$ or $3,({mathrm{mod, , }4})$ when $u=2$, and $eequiv 1,({mathrm{mod, , }2})$ when $u=3$, respectively. Our more general construction, in particular, removes the dependence on $e$. As a consequence, we obtain new infinite families of skew Hadamard matrices.
In this paper, we obtain a number of new infinite families of Hadamard matrices. Our constructions are based on four new constructions of difference families with four or eight blocks. By applying the Wallis-Whiteman array or the Kharaghani array to the difference families constructed, we obtain new Hadamard matrices of order $4(uv+1)$ for $u=2$ and $vin Phi_1cup Phi_2 cup Phi_3 cup Phi_4$; and for $uin {3,5}$ and $vin Phi_1cup Phi_2 cup Phi_3$. Here, $Phi_1={q^2:qequiv 1pmod{4}mbox{ is a prime power}}$, $Phi_2={n^4in mathbb{N}:nequiv 1pmod{2}} cup {9n^4in mathbb{N}:nequiv 1pmod{2}}$, $Phi_3={5}$ and $Phi_4={13,37}$. Moreover, our construction also yields new Hadamard matrices of order $8(uv+1)$ for any $uin Phi_1cup Phi_2$ and $vin Phi_1cup Phi_2 cup Phi_3$.
Let $q$ be a prime power of the form $q=12c^2+4c+3$ with $c$ an arbitrary integer. In this paper we construct a difference family with parameters $(2q^2;q^2,q^2,q^2,q^2-1;2q^2-2)$ in ${mathbb Z}_2times ({mathbb F}_{q^2},+)$. As a consequence, by applying the Wallis-Whiteman array, we obtain Hadamard matrices of order $4(2q^2+1)$ for the aforementioned $q$s.
If $q = p^n$ is a prime power, then a $d$-dimensional emph{$q$-Butson Hadamard matrix} $H$ is a $dtimes d$ matrix with all entries $q$th roots of unity such that $HH^* = dI_d$. We use algebraic number theory to prove a strong constraint on the dimension of a circulant $q$-Butson Hadamard matrix when $d = p^m$ and then explicitly construct a family of examples in all possible dimensions. These results relate to the long-standing circulant Hadamard matrix conjecture in combinatorics.
We investigate polynomials, called m-polynomials, whose generator polynomial has coefficients that can be arranged as a matrix, where q is a positive integer greater than one. Orthogonality relations are established and coefficients are obtained for the expansion of a polynomial in terms of m-polynomials. We conclude this article by an implementation in MATHEMATICA of m-polynomials and the results obtained for them.
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