The generator matrix

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

generates a code of length 84 over Z5[X]/(X^2) who´s minimum homogenous weight is 314.

Homogenous weight enumerator: w(x)=1x^0+460x^314+1204x^315+1400x^316+600x^317+2700x^318+3500x^319+5564x^320+5380x^321+2260x^322+6760x^323+7440x^324+11472x^325+9500x^326+4000x^327+11020x^328+11920x^329+18364x^330+13520x^331+6400x^332+16400x^333+18120x^334+23668x^335+18960x^336+7380x^337+19640x^338+18440x^339+24780x^340+18080x^341+7520x^342+16780x^343+15280x^344+17684x^345+11220x^346+3500x^347+7860x^348+6220x^349+6800x^350+4020x^351+800x^352+1340x^353+1120x^354+1040x^355+420x^356+40x^357+24x^360+8x^365+4x^370+4x^375+8x^380

The gray image is a linear code over GF(5) with n=420, k=8 and d=314.
This code was found by Heurico 1.16 in 343 seconds.