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 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 X X X^2 1 X^3 X^2 0 X X^2 X X^2 X X^2 X^3 X^3 0 X X^2 X^3 X^2 X^2
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 0 X^3 X^3 0 0 X^3 0 X^3 X^3 0 X^3 X^3 0 0 X^3 0 X^3 X^3 X^3 0 X^3 0 0 X^3 X^3 0 0 X^3 0 X^3 0 0 0 X^3 0
generates a code of length 70 over Z2[X]/(X^4) who´s minimum homogenous weight is 72.
Homogenous weight enumerator: w(x)=1x^0+27x^72+4x^74
The gray image is a linear code over GF(2) with n=560, k=5 and d=288.
This code was found by Heurico 1.16 in 0.203 seconds.