A class of N-parameter Gaussian rocesses are introduced, which are more general than the -parameter Wiener process. The definition of the set generated by xceptional oscillations of a class of these processes is given, and hen the Hausdorff dimension of this set is defined. The Hausdorff imensions of these processes are studied and an exact epresentative for them is given, which is similar to that for the wo-parameter Wiener process by Zacharie (2001). Moreover, the time et considered is a hyperrectangle which is more general than a yper-square used by Zacharie (2001). For this more general case, a ernique-type inequality is established and then using this nequality and the Slepian lemma, a Lévy's continuity modulus theorem is shown. Independence of increments is required for showing he representative of the Hausdorff dimension by Zacharie (2001). his property is absent for the processes introduced here, so we ave to find a different way.