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

generates a code of length 88 over Z2[X]/(X^3) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+39x^78+90x^79+114x^80+128x^81+163x^82+198x^83+276x^84+302x^85+300x^86+378x^87+371x^88+310x^89+290x^90+242x^91+190x^92+214x^93+109x^94+62x^95+83x^96+46x^97+37x^98+34x^99+44x^100+20x^101+15x^102+14x^103+7x^104+4x^105+6x^106+6x^107+2x^108+1x^134

The gray image is a linear code over GF(2) with n=352, k=12 and d=156.
This code was found by Heurico 1.16 in 2.17 seconds.