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.