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

generates a code of length 84 over Z2[X]/(X^4) who´s minimum homogenous weight is 86.

Homogenous weight enumerator: w(x)=1x^0+24x^86+5x^88+2x^92

The gray image is a linear code over GF(2) with n=672, k=5 and d=344.
This code was found by Heurico 1.16 in 0.391 seconds.