The generator matrix

 1  0  0  1  1  1  1  1  1  1  1  1  1  1  1 2X  1  1 3X  1 3X  1 2X  1  1  1  1  1  1  1  1  1  1 4X  1  X  0  0  1  1  1  1  1  1  1  1  1  1  1  1  X  1  1  0  1  1  1  1  1  1  1  1  1  1  1  1 4X  1  1  1  1  1  1  X 2X  1  X  1  1  1  1  1  1  1  1  1  1  1  1
 0  1  0  0  X 4X  X 3X+1 4X+1 3X+3  1 3X+2 X+4  2  3  1 3X+2 3X+4  1 2X+3  1  2  1 4X+3 3X+3 3X+4 4X+4 4X+1  2 X+4 X+1 2X+2 3X+2  1 X+1  1  X  1 4X+2  0  3 2X+2  1 4X+3  4  X 4X+1 3X  3 X+3  1 3X+3  1  1 3X+1 3X X+3 3X 3X 4X+1  4 4X+2 2X+1 3X+1 X+4 4X  1 4X+2  0  3 X+3 X+2 2X+1  1  X 2X+3  1 X+1  2 X+4  X 3X+2  X 2X+4  2 X+3 3X+4  3  0
 0  0  1  1 3X+2  4 3X+3 4X+3  X 2X+4  4 X+4 X+4 4X  2 4X+2 3X+1  3 4X+1 X+1 X+3 3X+3 4X+4 X+3  0  X 2X+2 4X+2 2X+2 2X+1 3X+1 2X+1 2X  1 2X+4 4X+4  1 2X+2 X+4 X+3 2X+3 3X+2 3X+1 2X  4  4 3X  2 4X+2 4X X+3 4X+4 2X 3X+2 X+1 3X+1 3X+2  2  0 2X+2 X+1  0 X+4 2X 3X+2  0 3X+3 2X  3  2 4X+3  1 3X+3  4  1 4X X+1  4 4X+2 X+2  2 4X+3 X+3  X X+2 4X 3X+3  3 X+4
 0  0  0 3X 3X 3X  0  0  0  0  0 2X  X  0 3X  0 3X  X 4X 3X  X 3X 3X  X  X 3X 2X 3X 4X 4X  0 2X  X  X 2X  X 3X 4X 4X 2X 4X  0 4X 2X  0  X 2X 2X  0 4X 4X  X 3X 3X 2X  X 2X  0 3X  X 2X 3X  X  X  X  X 3X 2X 4X  X 3X  X 2X  0  X 3X 3X 4X  X 3X 4X  X 3X  0 2X 2X 2X 4X 4X

generates a code of length 89 over Z5[X]/(X^2) who´s minimum homogenous weight is 339.

Homogenous weight enumerator: w(x)=1x^0+640x^339+1196x^340+620x^341+500x^342+520x^343+3160x^344+3972x^345+1380x^346+1260x^347+780x^348+5320x^349+5488x^350+2120x^351+1640x^352+1040x^353+5380x^354+5104x^355+1860x^356+1620x^357+1140x^358+5200x^359+5544x^360+2160x^361+1200x^362+940x^363+4580x^364+4036x^365+1140x^366+880x^367+520x^368+2680x^369+2164x^370+580x^371+400x^372+60x^373+540x^374+572x^375+140x^376+16x^380+16x^385+4x^390+4x^395+4x^400+4x^415

The gray image is a linear code over GF(5) with n=445, k=7 and d=339.
This code was found by Heurico 1.16 in 14.7 seconds.