The
$$\sum_{k=1}^N H_{i,k}\bar{H}_{j,k}\left(R_{i,k}-R_{j,k}\right) = 0\quad :\quad 1\leqslant i < j \leqslant N.$$
The dephased defect $d(H)$ of $H$ reads
$$d(H) = D(H) - (2N-1).$$
The above values safisty $D(H) \geqslant 2N-1$ and $d(H) \geqslant 0$. See sec. 2 in [88], subsec. 3.1 in [38] and sec. 1, 2 in [140].
$D(H)$ is an upper bound on the dimension of a smooth family $\mathcal{F}$ of complex Hadamard matrices stemming from a complex Hadamard matrix $H$. If $H$ and all the other members of $\mathcal{F}$ are dephased, the dimension of $\mathcal{F}$ is bounded by the smaller value $d(H)$.
A dephased complex Hadamard matrix $H$ satisfying $d(H)=0$ is isolated. See subsec. 4.1 in [88], sec. 3 in [38] and sec. 2 in [140].
If $H_1$ and $H_2$ are equivalent, then $D(H_1)=D(H_2)$ and $d(H_1)=d(H_2)$. Thus $H$ and the dephased version of it have equal defects. See also sec. 3 in [88] and subsec. 3.2 in [140].
Examples:
| $H$ | $D(H)$ | $d(H)$ | remarks |
|---|---|---|---|
| $D_6^{(1)}(c)$ | $15$ | $4$ | for typical $c$, although this family has dimension $1$ |
| $C_6^{(0)}$ | $15$ | $4$ | - |
| $S_6^{(0)}$ | $11$ | $0$ | so this is isolated |
| $B_6^{(1)}$$(y = 0)$ | $15$ | $4$ | - |
| $P_7^{(1)}$$(a)$ | $15$ | $2$ | for generic values of $a$ (compare below with specific values) |
| $P_7^{(1)}$$(a)$ | $16$ | $3$ | for $a \in \{ 0, \pi/3, 2\pi/3, \pi\}$ |
| $B_9^{(0)}$ | $19$ | $2$ | it might be a part of some family |
| $D_{10}^{(3)}$$(a,b,c)$ | $35$ | $16$ | for generic values of $a$, $b$ and $c$ |
Fourier examples - a list of the dephased defects of all inequivalent Fourier matrices of size $N=2, ..., 51$ can be found here. See also [28].
A table (pdf file) of the defects of all inequivalent Fourier matrices of size up to $10000$, obtained using the formulas in sec. 5 of [140].
Further examples with Fourier matrices:
| $H$ | $D(H)$ | $d(H)$ | remarks |
|---|---|---|---|
| $F$$_p$ | $2p-1$ | $0$ | for $p$ prime |
| $F$$_p\otimes$$F$$_p$ | $p(p^2+p-1)$ | $(p-1)^2(p+1)$ | for $p$ prime |
| $F$$_{p{^2}}\otimes$ $F$$_p$ | $p^2(2p^2-1)$ | $(p-1)^2(2p^2+2p+1)$ | for $p$ prime |
| $F$$_{p{^2}}\otimes$ $F$$_{p{^2}}$ | $p^3(p^3+p^2-1)$ | $(p-1)^2(p+1)(p^3+2p^2+p+1)$ | for $p$ prime, for other examples of this type see Table 1 in sec. 5 in [140] |
| $F_4$ | $8$ | $1$ | - |
| $F_2$ $\otimes$ $F_2$ | $10$ | $3$ | - |
| $F_6$ | $15$ | $4$ | although $F_6^{(2)}$ and $\left(F_6^{(2)}\right)^{\rm T}$ are $2$-dim. |
| $F_8$ | $20$ | $5$ | - |
| $F_4$ $\otimes$ $F_2$ | $28$ | $13$ | - |
| $F_2$ $\otimes$ $F_2$ $\otimes$ $F_2$ | $36$ | $21$ | - |
| $F_9$ | $21$ | $4$ | - |
| $F_3$ $\otimes$ $F_3$ | $33$ | $16$ | - |
| $F_{10}$ $\simeq$ $F_2$ $\otimes$ $F_5$ | $27$ | $8$ | - |
| $F_{12}$ $\simeq$ $F_3$ $\otimes$ $F_4$ | $40$ | $17$ | - |
| $F_2$ $\otimes$ $F_6$ $\simeq$ $F_2$ $\otimes$ $F_2$ $\otimes$ $F_3$ | $50$ | $27$ | - |
| $F_{14}$ $\simeq$ $F_2$ $\otimes$ $F_7$ | $39$ | $12$ | - |
| $F_{15}$ $\simeq$ $F_3$ $\otimes$ $F_5$ | $45$ | $16$ | - |
| $F_{16}$ | $48$ | $17$ | - |
| $F_2$ $\otimes$ $F_8$ | $72$ | $41$ | - |
| $F_4$ $\otimes$ $F_4$ | $88$ | $57$ | - |
| $F_2$ $\otimes$ $F_2$ $\otimes$ $F_4$ | $104$ | $73$ | - |
| $F_2$ $\otimes$ $F_2$ $\otimes$ $F_2$ $\otimes$ $F_2$ | $136$ | $105$ | - |