The generator matrix

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

generates a code of length 76 over Z2[X]/(X^4) who´s minimum homogenous weight is 69.

Homogenous weight enumerator: w(x)=1x^0+356x^69+1396x^70+3190x^71+3938x^72+5854x^73+6532x^74+7882x^75+7651x^76+8054x^77+6534x^78+5798x^79+3585x^80+2462x^81+1062x^82+722x^83+299x^84+90x^85+94x^86+24x^87+6x^88+6x^90

The gray image is a linear code over GF(2) with n=608, k=16 and d=276.
This code was found by Heurico 1.16 in 39.5 seconds.