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 X 1 1 X 1 1 X 1 1 1 X 1 X X 1 X 1 X^2 X X 1 X 1 X^2 X X X X X^2 X^3 X^2 0 1 X X^2 X^2 X X X^2 X^3 X^3 0 1 X^2 X^3 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 X^3 0 X^3 X^3 0 X^3 X^3 0 X^3 0 X^3 X^3 0 X^3 0 0 X^3 X^3 0 X^3 0 0 X^3 X^3 X^3 0 X^3 0 X^3 X^3 0 X^3 0 0 0 X^3 X^3 0 0 X^3 0 X^3 X^3 0 0 0
generates a code of length 68 over Z2[X]/(X^4) who´s minimum homogenous weight is 70.
Homogenous weight enumerator: w(x)=1x^0+28x^70+3x^72
The gray image is a linear code over GF(2) with n=544, k=5 and d=280.
This code was found by Heurico 1.16 in 0.171 seconds.