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

generates a code of length 31 over F4[X]/(X^2) who´s minimum homogenous weight is 76.

Homogenous weight enumerator: w(x)=1x^0+264x^76+378x^80+459x^84+768x^87+387x^88+4608x^91+711x^92+6912x^95+540x^96+573x^100+393x^104+285x^108+93x^112+12x^116

The gray image is a linear code over GF(4) with n=124, k=7 and d=76.
This code was found by Heurico 1.16 in 25.8 seconds.