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  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  1  1  1  1  1  1  1  1 X^3+X^2  1  1  1  1  1  1  X  1  0  1  1  X X^2 X^3  1  1  1 X^3  X  1 X^3+X^2  X  1  0  1  1  1  0  X  1 X^3+X^2
 0  X  0  X X^3  0 X^2+X X^3+X^2+X  0 X^3  X X^3+X X^3 X^2+X X^3 X^2+X X^3  X X^3+X^2 X^3+X^2+X X^2  X X^3+X^2 X^3+X X^2+X X^3+X^2  X X^3 X^2+X  0 X^2+X X^3+X^2 X^3+X^2 X^3+X X^3+X^2+X X^3+X^2  X  X X^2  0 X^3+X X^2 X^3 X^3+X^2 X^3+X^2+X X^3 X^3+X^2+X  0 X^2 X^3+X X^2+X  X X^2 X^3+X  0 X^3+X X^2+X X^2+X X^3+X X^3+X^2+X  0 X^2 X^2  X  0  X X^3+X^2+X X^3+X X^3+X X^2  X  X  X X^2 X^2+X  X  0 X^3 X^3 X^3+X^2 X^2 X^3+X  X
 0  0  X  X  0 X^3+X^2+X X^2+X X^3 X^2 X^3+X^2+X X^3+X^2+X X^3+X^2 X^3+X^2  X X^3+X X^3+X^2 X^3 X^3+X  X X^3+X^2  0 X^3+X  X X^2 X^3+X^2+X X^2 X^2 X^3+X X^3 X^2 X^3+X^2+X  X X^3+X^2+X  0 X^2+X  0 X^3 X^3+X X^3+X^2+X  0 X^2+X X^3 X^3+X^2+X X^3+X^2+X  X X^2 X^3+X^2 X^2+X X^3+X^2 X^3  0 X^3+X^2 X^3+X^2 X^3+X^2+X X^2+X X^2+X X^3+X^2 X^3+X X^3  X  X X^2 X^3 X^3+X^2  X  0 X^3 X^3+X  X  X X^2 X^3+X^2 X^3+X^2 X^3+X X^3+X  0 X^2 X^2 X^3+X^2  X X^3+X X^3+X^2  X
 0  0  0 X^2 X^2 X^3+X^2  0 X^3+X^2 X^2 X^3 X^2 X^3  0  0 X^2 X^2 X^3 X^3+X^2 X^2 X^3 X^2  0  0 X^2 X^3+X^2 X^3 X^3+X^2 X^3  0 X^3+X^2 X^3 X^3+X^2 X^2 X^2 X^2 X^3 X^3 X^3 X^3 X^3+X^2 X^3  0  0  0 X^3 X^3  0 X^2 X^3+X^2 X^3+X^2 X^2 X^3+X^2 X^2 X^3+X^2  0  0 X^3+X^2 X^3+X^2 X^3 X^2  0  0 X^3+X^2  0 X^2 X^2  0 X^3 X^3  0 X^3+X^2  0 X^3 X^3 X^2 X^3 X^3 X^3+X^2 X^3+X^2 X^3+X^2  0 X^2 X^3+X^2

generates a code of length 83 over Z2[X]/(X^4) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+384x^78+40x^79+587x^80+240x^81+790x^82+224x^83+724x^84+208x^85+448x^86+56x^87+254x^88+86x^90+24x^92+20x^94+8x^96+1x^104+1x^136

The gray image is a linear code over GF(2) with n=664, k=12 and d=312.
This code was found by Heurico 1.16 in 91.7 seconds.