The generator matrix

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

generates a code of length 77 over Z2[X]/(X^4) who´s minimum homogenous weight is 69.

Homogenous weight enumerator: w(x)=1x^0+302x^69+1409x^70+3046x^71+4972x^72+8058x^73+10687x^74+13520x^75+14925x^76+16940x^77+15519x^78+14198x^79+10732x^80+7450x^81+4432x^82+2520x^83+1226x^84+632x^85+294x^86+120x^87+45x^88+22x^89+11x^90+4x^91+3x^92+2x^93+2x^97

The gray image is a linear code over GF(2) with n=616, k=17 and d=276.
This code was found by Heurico 1.16 in 174 seconds.