The generator matrix

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

generates a code of length 34 over F4[X]/(X^2) who´s minimum homogenous weight is 88.

Homogenous weight enumerator: w(x)=1x^0+123x^88+432x^91+267x^92+1344x^95+516x^96+2928x^99+987x^100+4176x^103+1368x^104+2976x^107+606x^108+432x^111+93x^112+75x^116+27x^120+33x^124

The gray image is a linear code over GF(4) with n=136, k=7 and d=88.
This code was found by Heurico 1.16 in 0.659 seconds.