Comment on revised version of "The Hadamard circulant conjecture"

Circulant q-Butson Hadamard matrices

If $q = p^n$ is a prime power, then a $d$-dimensional \emph{$q$-Butson Hadamard matrix} $H$ is a $d\times d$ 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.

A generalization of circulant Hadamard and conference matrices

We study the existence and construction of circulant matrices $C$ of order $n\geq2$ with diagonal entries $d\geq0$, off-diagonal entries $\pm1$ and mutually orthogonal rows. These matrices generalize circulant conference ($d=0$) and circulant Hadamard ($d=1$) matrices. We demonstrate that matrices $C$ exist for every order $n$ and for $d$ chosen such that $n=2d+2$, and we find all solutions $C$ with this property. Furthermore, we prove that if $C$ is symmetric, or $n-1$ is prime, or $d$ is not an odd integer, then necessarily $n=2d+2$. Finally, we conjecture that the relation $n=2d+2$ holds fo…

Analytic aspects of the circulant Hadamard conjecture

We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for $|q_0|=...=|q_{N-1}|=1$ the quantity $Φ=\sum_{i+k=j+l}\frac{q_iq_k}{q_jq_l}$ satisfies $Φ\geq N^2$, with equality if and only if $q=(q_i)$ is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems, namely: (1) the brute-force minimization of $Φ$, (2) the study of the critical points of $Φ$, and (3) the computation of the moments of $Φ$. We explore here these questions, with so…

An Approximation to Proof of the Circulant Hadamard Conjecture

Turyn prove that if a circulant Hadamard matrix of order $n$ exists then $n$ must be of the form $n=4m^{2}$ for some odd integer $m$. In this paper we use the structure constant of Schur ring of $\Z_{2}^{4m^{2}}$ to prove that there is no circulant Hadamard matrix in $\Z_{2}^{4m^{2}}$ except possibly for sequences with Hamming weight $a+b$, with $\frac{m^{2}-m}{2}\leq a\leq\frac{3m^{2}-m}{2}$ and $b=2m^{2}-m-a$ and with $\frac{m^{2}+m}{2}\leq 2m^{2}-a\leq\frac{3m^{2}+m}{2}$ and $b=m+a$.

Small circulant complex Hadamard matrices of Butson type

We study the circulant complex Hadamard matrices of order $n$ whose entries are $l$-th roots of unity. For $n=l$ prime we prove that the only such matrix, up to equivalence, is the Fourier matrix, while for $n=p+q,l=pq$ with $p,q$ distinct primes there is no such matrix. We then provide a list of equivalence classes of such matrices, for small values of $n,l$.

Hadamard full propelinear codes with associated group $C_{2t}\times C_2$; rank and kernel

We introduce the Hadamard full propelinear codes that factorize as direct product of groups such that their associated group is $C_{2t}\times C_2$. We study the rank, the dimension of the kernel, and the structure of these codes. For several specific parameters we establish some links from circulant Hadamard matrices and the nonexistence of the codes we study. We prove that the dimension of the kernel of these codes is bounded by 3 if the code is nonlinear. We also get an equivalence between circulant complex Hadamard matrix and a type of Hadamard full propelinear code, and we find a new examp…

阶数为94的复Hadamard矩阵

A complex Hadamard matrix of order 94

本文首次利用计算机辅助方法构造了阶数为94的复Hadamard矩阵，通过修改基本块构造方法并使用特定的循环矩阵实现。

Schur Ring, Run Structure and Periodic Compatible Binary Sequences

In this paper three Schur ring are discussed, namenly: Hamming, circulant orbists and decimated circulant orbits Schur ring. By using autocorrelation function and the run structure of binary sequences we proof the relation between this Schur ring and combinatorial structures such as Hadamard matrices, periodic compatible binary sequences and perfect binary sequences. Cai proved for binary sequences that the autocorrelation function is in fact completely determined by its run structure. Also, he characterised the structure of the circulant Hadamard matrices. We characterise a more general struc…

寻找Williamson类型的Hadamard矩阵

A search for Hadamard matrices of Williamson type

本文研究了一种特殊的Williamson类型矩阵，即近Williamson矩阵，通过计算机搜索发现所有奇数阶至35的非等价矩阵，并证明所有奇数阶至63的此类矩阵存在，从而找到首个118阶四元Hadamard矩阵。

Ryser 猜想与随机矩阵

Ryser's Conjecture and Stochastic matrices

本文通过构造依赖于循环 Hadamard 矩阵 H 的随机矩阵 S，证明当 n≥4 时必有 n=4，从而得出循环 Hadamard 矩阵仅有 10 个，支持 Ryser 猜想。

关于近似 Hadamard 矩阵的一个注记

A Note on Approximate Hadamard Matrices

本文指出，基于平坦 Littlewood 多项式的存在性，对于所有 $n \geq 1$ 均存在循环近似 Hadamard 矩阵。