Certain nonlinear binary codes can be constructed as binary images of Z4-linear codes under the Gray map. Examples include the second-order Reed-Muller code and the Kerdock and Preparata codes. In this paper, we consider a new quaternary code which is an additive subcode of the Z 4-linear Kerdock code. The Kerdock code is the direct sum of a one-dimensional quaternary code and the quaternary subcode examined in this paper. This paper calculates the weight distribution of the projective Kerdock code from which the weight distribution of the dual code can be computed. The dual code is a supercode of the quaternary Preparata code. The projective Kerdock code is used to construct a deterministic measurement matrix for compressed sensing. Numerical experiments are presented for sparse reconstruction using the LASSO that show improvement over random Gaussian matrices of the same size.