The generator matrix

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

generates a code of length 78 over Z2[X]/(X^3) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+194x^68+4x^69+402x^70+68x^71+565x^72+152x^73+684x^74+348x^75+867x^76+440x^77+926x^78+524x^79+828x^80+248x^81+562x^82+196x^83+494x^84+52x^85+264x^86+16x^87+197x^88+90x^90+45x^92+16x^94+8x^96+1x^112

The gray image is a linear code over GF(2) with n=312, k=13 and d=136.
This code was found by Heurico 1.16 in 7.74 seconds.