The generator matrix

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

generates a code of length 78 over Z2[X]/(X^3) who´s minimum homogenous weight is 69.

Homogenous weight enumerator: w(x)=1x^0+62x^69+138x^70+340x^71+241x^72+576x^73+449x^74+778x^75+495x^76+890x^77+555x^78+834x^79+454x^80+686x^81+372x^82+516x^83+162x^84+306x^85+113x^86+72x^87+37x^88+28x^89+26x^90+18x^91+15x^92+6x^93+9x^94+2x^95+3x^96+6x^97+1x^102+1x^106

The gray image is a linear code over GF(2) with n=312, k=13 and d=138.
This code was found by Heurico 1.16 in 5.34 seconds.