The generator matrix

 1  0  0  1  1  1  0 X^3  1  1 X^3 X^2  1  1  1  1 X^3+X X^3+X  1 X^3+X  1  1  1 X^3+X^2+X X^2+X  1  1 X^3+X^2  1  1 X^3+X^2+X  1 X^3+X X^2+X  1  1 X^3+X^2+X  1  1 X^3+X^2  1  X  1  1 X^2  1  1  1 X^3+X^2  1 X^2  X X^2+X  1 X^2 X^3  1  1  0  0 X^2+X  1  1  1  1 X^2  1  1  1  1  1  1  1  X  1  1 X^3+X^2  1  1  1  1  1  1
 0  1  0  0 X^2+1 X^2+1  1 X^3+X^2+X X^3 X^3+X^2+1  1  1 X^3+X^2  1 X^2+X X+1  1 X^2+X  X  1 X^3+X+1 X^3+X^2+X+1 X^2+X+1  0  1 X^3+X^2 X^3  1 X^3+X^2 X^3+X+1  1 X^3+X^2+X X^3+X  1 X^3+X  X  1 X^3+1  1  0 X^2+X  1  X  1  1 X^3+X+1 X^3+X^2+X X^3+X^2+X  1 X^2 X^3+X^2+X  1 X^3+X X^3+X^2+X+1  1  1 X^3+1  X X^3 X^3+X^2  1 X^3 X^3+X^2+X+1 X^2+1 X^3+X^2 X^3+X X+1 X^3+X^2+1  0 X^2  X  X X^3+X^2  0 X+1 X^3+1  1 X^3  X X^3 X^2 X^2  0
 0  0  1 X+1 X^3+X+1 X^3 X^3+X^2+X+1  1 X^3+X^2+X X^2+1  1 X^3+X X^3+X^2+1  X X^3+X+1 X^2 X^3+X^2+1  1  X X^3+X^2+X X^2+X+1 X^3+X^2+X X^2+1  1  0 X+1 X^3+X^2 X^2+1  1 X^3+X X^2+X+1 X^3+X^2  1 X^3+X^2+X X^3+X^2 X+1 X^2+1 X^3 X^3+X^2+X+1  1 X^3+X^2+1 X^3 X^2+X+1  1 X+1  0 X^2+X X^3+X^2+1  0 X^3+1  1 X^3+X^2  1 X^2+X+1 X^3+X X^3+X^2+1 X^3+X+1 X^3+X  1  1 X^3+X X^3+1 X^3 X^3+X^2+X X^3+X  1 X^3+X^2+1  0 X^2 X^2+1 X^3+X^2+1  1 X+1  1 X^3+1 X^3+X+1 X^2+1 X^2+X+1 X^2+1 X^3+X^2+1 X^2+X+1 X^2+X+1 X^2+X
 0  0  0 X^3 X^3  0 X^3 X^3 X^3  0  0 X^3  0 X^3  0 X^3 X^3  0  0  0  0  0 X^3  0 X^3  0 X^3 X^3 X^3 X^3  0  0 X^3  0 X^3 X^3  0 X^3  0 X^3  0  0 X^3 X^3  0  0  0 X^3 X^3  0  0  0 X^3 X^3  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0  0 X^3  0 X^3  0 X^3 X^3  0 X^3  0 X^3  0  0  0  0 X^3  0 X^3 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+107x^78+720x^79+1075x^80+1200x^81+999x^82+880x^83+879x^84+748x^85+481x^86+324x^87+309x^88+220x^89+69x^90+84x^91+36x^92+40x^93+8x^94+8x^95+1x^100+3x^104

The gray image is a linear code over GF(2) with n=664, k=13 and d=312.
This code was found by Heurico 1.16 in 5.42 seconds.