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

generates a code of length 98 over Z2[X]/(X^3) who´s minimum homogenous weight is 87.

Homogenous weight enumerator: w(x)=1x^0+96x^87+118x^88+208x^89+237x^90+260x^91+345x^92+476x^93+498x^94+468x^95+573x^96+626x^97+691x^98+526x^99+623x^100+516x^101+389x^102+384x^103+244x^104+214x^105+136x^106+114x^107+100x^108+106x^109+71x^110+58x^111+30x^112+24x^113+20x^114+12x^115+8x^116+6x^117+4x^118+2x^119+5x^120+2x^126+1x^144

The gray image is a linear code over GF(2) with n=392, k=13 and d=174.
This code was found by Heurico 1.16 in 96.7 seconds.