The generator matrix

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

generates a code of length 34 over F4[X]/(X^2) who´s minimum homogenous weight is 84.

Homogenous weight enumerator: w(x)=1x^0+225x^84+297x^88+474x^92+192x^93+543x^96+1728x^97+522x^100+5184x^101+504x^104+5184x^105+528x^108+480x^112+309x^116+159x^120+54x^124

The gray image is a linear code over GF(4) with n=136, k=7 and d=84.
This code was found by Heurico 1.16 in 1.8 seconds.