In the present paper the so-called \( (\mathrm{Vil}_{B_\nu}; \alpha; \gamma) \)-diaphony as a quantitative measure for the distribution of sequences and nets is considered. A class of two-dimensional nets \( Z^{\kappa,\mu}_{B_2,\nu} \) of type of Zaremba–Halton constructed in a generalized \( B_2 \)-adic system or Cantor system is introduced and the \( (\mathrm{Vil}_{B_2}; \alpha; \gamma) \)-diaphony of these nets is studied. The influence of the vector \( \alpha = (\alpha_1, \alpha_2) \) of exponential parameters to the exact order of the \( (\mathrm{Vil}_{B_2}; \alpha; \gamma) \)-diaphony of the nets \( Z^{\kappa,\mu}_{B_2,\nu} \) is shown. If \( \alpha_1 = \alpha_2 \), then the following holds: if \( 1 < \alpha_2 0 \), if \( \alpha_2 = 2 \) the exact order is \( O\!\left(\dfrac{\sqrt{\log N}}{N}\right) \) and if \( \alpha_2 > 2 \) the exact order is \( O\!\left(\dfrac{\sqrt{\log N}}{N^{1+\frac{1}{\alpha}}}\right) \) for some \( \varepsilon > 0 \). If \( \alpha_1 > \alpha_2 \), then the following holds: if \( 1 < \alpha_2 0 \), if \( \alpha_2 = 2 \) the exact order is \( O\!\left(\dfrac{1}{N}\right) \) and if \( \alpha_2 > 2 \) the exact order is \( O\!\left(\dfrac{1}{N^{1+\frac{1}{\alpha}}}\right) \) for some \( \varepsilon > 0 \). Here \( N = B_\nu \), where \( B_\nu \) denotes the number of the points of the nets \( Z^{\kappa,\mu}_{B_2,\nu} \).