The generator matrix

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

generates a code of length 84 over Z5[X]/(X^2) who´s minimum homogenous weight is 320.

Homogenous weight enumerator: w(x)=1x^0+1000x^320+1780x^321+1360x^322+300x^323+3492x^325+4620x^326+2880x^327+400x^328+4688x^330+6500x^331+3400x^332+560x^333+5608x^335+6740x^336+3720x^337+440x^338+4800x^340+6300x^341+3240x^342+420x^343+3552x^345+4980x^346+2200x^347+300x^348+2060x^350+1500x^351+700x^352+80x^353+384x^355+80x^356+20x^360+12x^365+4x^375+4x^385

The gray image is a linear code over GF(5) with n=420, k=7 and d=320.
This code was found by Heurico 1.16 in 27.8 seconds.