The generator matrix

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

generates a code of length 84 over Z5[X]/(X^2) who´s minimum homogenous weight is 310.

Homogenous weight enumerator: w(x)=1x^0+64x^310+20x^314+316x^315+200x^316+320x^317+200x^318+640x^319+420x^320+1620x^321+900x^322+920x^323+2240x^324+392x^325+2700x^326+1440x^327+1480x^328+3680x^329+424x^330+4380x^331+2220x^332+2780x^333+5480x^334+256x^335+5780x^336+3120x^337+3580x^338+7060x^339+240x^340+5880x^341+2700x^342+2780x^343+4540x^344+244x^345+3760x^346+1520x^347+760x^348+1340x^349+160x^350+680x^351+280x^352+180x^355+148x^360+92x^365+72x^370+72x^375+16x^380+12x^385+12x^390+4x^395

The gray image is a linear code over GF(5) with n=420, k=7 and d=310.
This code was found by Heurico 1.16 in 15.5 seconds.