The generator matrix

 1  0  1  1  1  1  1  1  1  1  0  1  1  1  X  1  1  1  1  X  1  1  1  1  1  1  1  1  X a*X a*X  1  1  1  1  1  1  0
 0  1  1  a a^2*X+a^2  0 a^2*X+1 a^2*X+a^2  X  a  1 a^2*X+1 X+a a*X+a^2  1  X a*X+1 X+a a*X+a^2  1 a*X+1  1  0  X  a X+a a^2*X+a^2 a*X+a^2  1  1  1 a*X a*X+a X+a^2 a*X a*X+a X+a^2  1
 0  0 a^2*X a*X  X  X  0 a^2*X a*X a^2*X a^2*X  X  0 a*X a*X a^2*X a^2*X  X  0  X  0  X a*X  X  0 a*X a*X a^2*X a^2*X  0 a*X  0 a^2*X  X a^2*X  X  0  X

generates a code of length 38 over F4[X]/(X^2) who´s minimum homogenous weight is 110.

Homogenous weight enumerator: w(x)=1x^0+288x^110+180x^112+288x^114+96x^118+72x^120+96x^122+3x^128

The gray image is a linear code over GF(4) with n=152, k=5 and d=110.
This code was found by Heurico 1.16 in 3.23 seconds.