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

generates a code of length 80 over Z2[X]/(X^3) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+116x^68+4x^70+215x^72+82x^74+308x^76+680x^78+1369x^80+692x^82+243x^84+68x^86+154x^88+10x^90+90x^92+47x^96+11x^100+5x^104+1x^128

The gray image is a linear code over GF(2) with n=320, k=12 and d=136.
This code was found by Heurico 1.16 in 2.61 seconds.