Multiuser systems suffer from multiple access interference (MAI) due to the correlations between users signature codes. The decorrelator detector uses the inverse correlation matrix of the signature codes to eliminate the MAI. The major disadvantage of the decorrelator detector is the enhancement of the noise power at its outputs as mentioned in many previous works. This statement is not always true for non-orthogonal correlation matrix. The decorrelator detector can reduce the noise power at its outputs at certain conditions for non-orthogonal correlation matrix. If the sum of the eigenvalues of the non-orthogonal correlation matrix is greater than the sum of the eigenvalues of its inverse, the decorrelator detector will not enhance the noise power at its output. Moreover, the exponential of the determinant of the correlation matrix should be greater than the exponential of the determinant of its inverse. These two conditions are presented and proved in this work. A new family of signatures codes is also proposed to match the proposed conditions for the correlation matrix. The new signature codes are based on pulse position modulation (PPM). The paper shows that the new codes have three degrees of freedom to change the correlation coefficients between them.