The generator matrix

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

generates a code of length 78 over Z2[X]/(X^3) who´s minimum homogenous weight is 70.

Homogenous weight enumerator: w(x)=1x^0+70x^70+278x^71+336x^72+754x^73+594x^74+648x^75+640x^76+780x^77+551x^78+678x^79+519x^80+622x^81+365x^82+358x^83+294x^84+242x^85+113x^86+188x^87+56x^88+32x^89+29x^90+26x^91+10x^92+2x^93+6x^94

The gray image is a linear code over GF(2) with n=312, k=13 and d=140.
This code was found by Heurico 1.11 in 1.39 seconds.