A complex $n\times n$ matrix $A=[a_{ij}]$ is called a Hadamard matrix if $A^{+}A=nI$ and $|a_{ij}|=1$ holds for all $i,j$, where $A^{+}$ denotes the conjugate transposed matrix of $A$, and a vector $x=(x_1,x_2.....x_n)\in C^n$ is called unit if $|x_i|=1$ holds for all $i$.

For $n>5$, are there $d\times n$ matrix $F=[f_{ij}]$ which satisfies the following properties:

1) $FF^{+}=dI$,

2) $|f_{ij}|=1$ , holds for all $i,j$,

3) there is no unit vector $x\in C^n$ satisfies $F x^{+}=0$.

Which can be regarded as the difficulty of generate complex Hadamard Matrix, actually, it is wanted to show the following method to find a complex Hadamard Matrix does not always work:

Let $S$ be an empty set, choose a unit vector $x$ which is orthogonal to all the elements in $S$, and put $x$ in $S$, until there is no unit vector which orthogonal to all elements in $S$. Now what we ask can be regarded as is it possible that the algorithm stops but $|S|< n$.

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