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  1  1  1  X  1  1  1  1  X  1
 0  X  0  0  0  0  X  X  X a*X  X a^2*X  0 a*X a^2*X a*X  0 a*X  X  X  0  X a*X a^2*X a*X  X  0  X  X a*X a^2*X  X  X  0
 0  0  X  0  0  X a^2*X a*X a*X a*X  X  0  X a^2*X a*X  0 a*X a^2*X  X  X  X  X  0  0  0  0  X  X a*X  X  0 a^2*X  0  X
 0  0  0  X  0 a^2*X  0  X a*X a*X a*X a^2*X a*X  0 a*X  X  0  X  X a*X a*X  0 a^2*X  0  X  0 a^2*X a^2*X  X  X  X  X  0  X
 0  0  0  0  X  X  X a^2*X  X  0 a^2*X a*X  0 a^2*X  0 a*X  X a^2*X  X  X a*X a^2*X a^2*X  X  0 a^2*X a*X  0  0 a*X a^2*X  0 a*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+72x^88+177x^92+351x^96+1296x^100+1869x^104+114x^108+84x^112+63x^116+33x^120+27x^124+6x^128+3x^132

The gray image is a linear code over GF(4) with n=136, k=6 and d=88.
This code was found by Heurico 1.16 in 0.0892 seconds.