The generator matrix

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

generates a code of length 76 over Z2[X]/(X^4) who´s minimum homogenous weight is 71.

Homogenous weight enumerator: w(x)=1x^0+136x^71+621x^72+1046x^73+1108x^74+1132x^75+998x^76+916x^77+613x^78+496x^79+386x^80+290x^81+248x^82+100x^83+40x^84+44x^85+13x^86+1x^88+1x^90+1x^96+1x^98

The gray image is a linear code over GF(2) with n=608, k=13 and d=284.
This code was found by Heurico 1.16 in 4.75 seconds.