The generator matrix

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

generates a code of length 29 over F4[X]/(X^2) who´s minimum homogenous weight is 77.

Homogenous weight enumerator: w(x)=1x^0+72x^77+600x^78+396x^79+438x^80+528x^81+1440x^82+768x^83+618x^84+864x^85+2160x^86+696x^87+813x^88+1008x^89+2400x^90+864x^91+561x^92+600x^93+1080x^94+348x^95+99x^96+12x^100+9x^104+9x^108

The gray image is a linear code over GF(4) with n=116, k=7 and d=77.
This code was found by Heurico 1.16 in 93.6 seconds.