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

generates a code of length 75 over Z2[X]/(X^4) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+352x^68+1524x^69+2724x^70+4116x^71+5679x^72+6892x^73+7899x^74+7960x^75+7560x^76+6926x^77+5547x^78+3720x^79+2266x^80+1270x^81+599x^82+296x^83+102x^84+54x^85+30x^86+4x^87+8x^88+6x^89+1x^94

The gray image is a linear code over GF(2) with n=600, k=16 and d=272.
This code was found by Heurico 1.16 in 39.3 seconds.