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

generates a code of length 84 over Z2[X]/(X^4) who´s minimum homogenous weight is 79.

Homogenous weight enumerator: w(x)=1x^0+152x^79+772x^80+926x^81+1207x^82+1000x^83+1028x^84+752x^85+701x^86+508x^87+433x^88+254x^89+185x^90+48x^91+112x^92+68x^93+33x^94+4x^95+4x^96+1x^98+2x^100+1x^106

The gray image is a linear code over GF(2) with n=672, k=13 and d=316.
This code was found by Heurico 1.16 in 2.92 seconds.