The generator matrix

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

generates a code of length 28 over F4[X]/(X^2) who´s minimum homogenous weight is 64.

Homogenous weight enumerator: w(x)=1x^0+72x^64+267x^68+417x^72+993x^76+3159x^80+5772x^84+4521x^88+546x^92+384x^96+189x^100+54x^104+9x^108

The gray image is a linear code over GF(4) with n=112, k=7 and d=64.
This code was found by Heurico 1.16 in 0.941 seconds.