**Abstarct **
In [7] Rains has shown that for any linear code $C$ over $Z_4, d_H,$ the minimum
Hamming distance of $C$ and $d_L$, the minimum Lee distance of $C$ satisfy
$d_H > = \lceil \frac{d_L}{2} \rceil$. $C$ is said to be of type $\alpha$
($\beta$) if $d_H = \lceil \frac{d_L}{2} \rceil$ ( $d_H > \lceil \frac{d_L}{2} \rceil$).
In this paper we define Simplex codes of type $\alpha$ and $\beta$, namely,
$S_k^{\alpha}$ and $S_k^{\beta}$ respectively over $Z_4$. Some fundamental
properties like $2$-dimension, Hamming and Lee weight distributions, weight hierarchy
etc. are determined for these codes. It is shown that binary images of $S_k^{\alpha}$
and $S_k^{\beta}$ by the Gray map give rise to some interesting binary codes.