The generator matrix

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

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+175x^78+314x^79+435x^80+478x^81+515x^82+404x^83+462x^84+386x^85+402x^86+302x^87+163x^88+30x^89+8x^90+4x^91+8x^92+2x^93+1x^96+2x^104+2x^106+1x^114+1x^118

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 0.719 seconds.