The generator matrix

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

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+54x^69+636x^70+1488x^71+2223x^72+2842x^73+3446x^74+3994x^75+4242x^76+3736x^77+3274x^78+2644x^79+1840x^80+1022x^81+607x^82+394x^83+176x^84+54x^85+53x^86+24x^87+6x^88+4x^89+7x^90+1x^94

The gray image is a linear code over GF(2) with n=608, k=15 and d=276.
This code was found by Heurico 1.16 in 16.8 seconds.