The generator matrix

 1  1  1  1  1  1  1  X  1  1  X  1  X X^2  X  X X^2 X^3  0
 0 X^3  0 X^3  0 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3  0 X^3  0 X^3 X^3

generates a code of length 19 over Z2[X]/(X^4) who´s minimum homogenous weight is 19.

Homogenous weight enumerator: w(x)=1x^0+20x^19+5x^20+4x^21+2x^22

The gray image is a linear code over GF(2) with n=152, k=5 and d=76.
As d=78 is an upper bound for linear (152,5,2)-codes, this code is optimal over Z2[X]/(X^4) for dimension 5.
This code was found by Heurico 1.16 in -3.24e-008 seconds.