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

generates a code of length 73 over Z2[X]/(X^4) who´s minimum homogenous weight is 70.

Homogenous weight enumerator: w(x)=1x^0+46x^70+24x^71+281x^72+336x^73+260x^74+24x^75+45x^76+6x^78+1x^140

The gray image is a linear code over GF(2) with n=584, k=10 and d=280.
This code was found by Heurico 1.16 in 22.9 seconds.