In this paper we consider the problem of the asymptotic expansion of double Laplace-type integrals, in the case when the set γ of points where the phase achieves its absolute minimum is a simple curve. It will be shown that the asymptotic behaviour of such integrals is governed by the order of degeneracy of normal derivatives of the phase with respect to the curve γ. Complete asymptotic expansions will be constructed if that order is constant along γ, and the first two coefficients will be explicitly computed. If not, a uniform asymptotic expansion method, involving special functions, is suggested.

In this paper, we introduce the homogeneous weight and homogeneous Gray map over the ring R_q=F₂[u₁,u₂,…,u_q]/⟨u²_i=0,u_iu_j=u_ju_i⟩ for q≥1q≥1. We also consider the construction of simplex and MacDonald codes of types α and β over this ring and their covering radius.

In this paper, we define the simplex and MacDonald codes of types αα and β over Z₂Z₄. We also examine the covering radii of these codes. Further, we study the binary images of these codes and prove that the binary image of the simplex codes of type αα meets the Gilbert bound.