The generator matrix

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

generates a code of length 83 over Z5[X]/(X^2) who´s minimum homogenous weight is 310.

Homogenous weight enumerator: w(x)=1x^0+528x^310+940x^311+1260x^312+800x^313+2320x^314+5276x^315+3760x^316+4660x^317+2420x^318+6100x^319+11916x^320+7960x^321+8420x^322+4720x^323+10600x^324+18900x^325+11500x^326+13020x^327+6660x^328+14340x^329+27140x^330+17240x^331+16860x^332+9660x^333+18100x^334+30060x^335+15980x^336+16600x^337+8420x^338+15240x^339+22776x^340+11940x^341+10500x^342+3860x^343+7020x^344+9844x^345+5120x^346+3440x^347+920x^348+1280x^349+1640x^350+560x^351+240x^352+40x^353+16x^355+12x^360+8x^365+4x^375+4x^380

The gray image is a linear code over GF(5) with n=415, k=8 and d=310.
This code was found by Heurico 1.16 in 337 seconds.