The generator matrix

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

generates a code of length 84 over Z5[X]/(X^2) who´s minimum homogenous weight is 300.

Homogenous weight enumerator: w(x)=1x^0+144x^300+604x^305+860x^310+1292x^315+1268x^320+500x^324+1348x^325+6000x^329+1428x^330+24000x^334+1328x^335+32000x^339+1448x^340+1492x^345+1248x^350+1060x^355+904x^360+580x^365+320x^370+184x^375+84x^380+28x^385+4x^405

The gray image is a linear code over GF(5) with n=420, k=7 and d=300.
This code was found by Heurico 1.16 in 34.4 seconds.