The generator matrix

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

generates a code of length 39 over F4[X]/(X^2) who´s minimum homogenous weight is 96.

Homogenous weight enumerator: w(x)=1x^0+84x^96+264x^100+366x^104+666x^108+2214x^112+5694x^116+5679x^120+492x^124+456x^128+222x^132+171x^136+54x^140+21x^144

The gray image is a linear code over GF(4) with n=156, k=7 and d=96.
This code was found by Heurico 1.16 in 1.52 seconds.