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

generates a code of length 82 over Z2[X]/(X^3) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+214x^72+16x^73+402x^74+52x^75+581x^76+128x^77+676x^78+372x^79+886x^80+456x^81+866x^82+452x^83+831x^84+368x^85+582x^86+140x^87+439x^88+56x^89+288x^90+8x^91+185x^92+106x^94+51x^96+20x^98+11x^100+4x^102+1x^120

The gray image is a linear code over GF(2) with n=328, k=13 and d=144.
This code was found by Heurico 1.16 in 50.6 seconds.