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

generates a code of length 31 over F4[X]/(X^2) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+93x^80+186x^84+192x^87+183x^88+1152x^91+117x^92+1728x^95+147x^96+105x^100+78x^104+63x^108+39x^112+9x^116+3x^120

The gray image is a linear code over GF(4) with n=124, k=6 and d=80.
This code was found by Heurico 1.16 in 0.0806 seconds.