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

generates a code of length 79 over Z2[X]/(X^3) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+50x^68+84x^69+161x^70+260x^71+242x^72+400x^73+395x^74+458x^75+536x^76+584x^77+688x^78+626x^79+729x^80+580x^81+525x^82+420x^83+334x^84+308x^85+195x^86+212x^87+104x^88+72x^89+65x^90+62x^91+44x^92+16x^93+11x^94+6x^95+4x^96+4x^97+7x^98+4x^99+4x^100+1x^110

The gray image is a linear code over GF(2) with n=316, k=13 and d=136.
This code was found by Heurico 1.16 in 7.47 seconds.