The generator matrix

 1  0  0  1  1  1  X  1  1 X^2+X  1  1  X X^2  0  0  1  1 X^2+X  1 X^2+X  1  1  1  1 X^2+X  1 X^2  1  1  1 X^2  X  1  1  1  1 X^2+X  X X^2+X  1  0  1 X^2 X^2 X^2+X  1  0  1  X  1  1  1  1 X^2  1  0 X^2  X  1 X^2  X  X X^2 X^2  1 X^2+X  1  1  0 X^2+X  1  1  1  1  1  1  1
 0  1  0  X  1 X^2+X+1  1 X^2+X  0 X^2  1 X+1 X^2+X  1  1  1 X^2+X X^2+1  1 X^2+X+1  1 X^2 X^2+1  X  0  1 X+1  X X+1  X X^2+X+1  1  1 X^2 X+1 X^2 X+1  1  1  1 X^2+X  1  1 X^2  1  X  X X^2 X^2+X+1  1  1  1  X X^2+X  1 X^2+X+1 X^2  1  1 X^2+1  1  1 X^2+X X^2+X  1 X^2+1  1 X^2+1  X  0  1 X^2 X^2+1 X+1  X  X X^2+X+1  0
 0  0  1  1 X^2+X+1 X^2+X  1 X+1 X^2+X  1 X^2+1  0  1  X X+1 X^2+1 X^2 X^2+X  0  1 X+1  1  0  X X+1 X^2 X+1  1  1  1 X^2  1  X X^2  1 X^2+1 X^2+X  X X^2+1  0  X X+1 X^2+X  1 X^2+X+1  1 X+1  1 X^2+X X^2+X+1 X^2+X+1  1  0  X X^2+X  X  1  X X^2+X X+1  0  X  1  1  1  0  1 X^2+1 X^2+X+1  1 X^2 X^2+1  1 X^2+X+1 X^2+1  1  X X+1
 0  0  0 X^2  0  0  0  0 X^2 X^2  0  0  0 X^2  0 X^2  0  0  0 X^2 X^2 X^2 X^2  0  0 X^2 X^2 X^2  0 X^2 X^2 X^2 X^2  0 X^2 X^2  0 X^2 X^2  0 X^2 X^2 X^2 X^2  0  0  0 X^2 X^2  0 X^2  0  0 X^2 X^2  0  0 X^2  0  0 X^2  0  0 X^2 X^2 X^2  0  0 X^2 X^2  0  0  0  0 X^2  0  0 X^2
 0  0  0  0 X^2  0  0 X^2 X^2 X^2  0 X^2 X^2  0 X^2 X^2  0  0  0  0  0  0 X^2 X^2 X^2  0 X^2  0  0  0  0  0 X^2 X^2 X^2 X^2  0 X^2 X^2 X^2 X^2 X^2 X^2 X^2  0 X^2  0 X^2  0 X^2 X^2  0  0  0 X^2 X^2 X^2 X^2 X^2 X^2  0 X^2  0  0 X^2 X^2 X^2 X^2 X^2  0 X^2 X^2 X^2 X^2 X^2 X^2 X^2 X^2
 0  0  0  0  0 X^2  0  0  0  0  0 X^2  0  0  0  0  0 X^2  0  0  0 X^2  0 X^2 X^2 X^2 X^2 X^2 X^2 X^2  0 X^2  0  0  0 X^2  0 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 X^2 X^2  0 X^2  0  0  0 X^2  0  0  0 X^2  0 X^2 X^2  0 X^2 X^2  0  0 X^2  0 X^2  0 X^2 X^2  0
 0  0  0  0  0  0 X^2  0  0  0 X^2 X^2 X^2 X^2 X^2  0 X^2 X^2 X^2 X^2  0 X^2  0 X^2  0 X^2  0 X^2 X^2  0  0  0 X^2 X^2  0  0  0  0 X^2  0 X^2  0 X^2 X^2  0  0 X^2 X^2 X^2 X^2 X^2  0  0 X^2 X^2 X^2  0  0  0 X^2  0  0  0  0 X^2 X^2  0 X^2 X^2 X^2 X^2  0 X^2  0  0  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+86x^70+260x^71+451x^72+496x^73+746x^74+544x^75+744x^76+572x^77+801x^78+550x^79+680x^80+486x^81+473x^82+322x^83+343x^84+214x^85+176x^86+106x^87+73x^88+18x^89+12x^90+6x^91+8x^92+6x^93+9x^94+4x^95+3x^96+1x^98+1x^100

The gray image is a linear code over GF(2) with n=312, k=13 and d=140.
This code was found by Heurico 1.16 in 4.51 seconds.