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

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+40x^310+324x^315+440x^320+100x^324+432x^325+1200x^329+364x^330+4800x^334+300x^335+6400x^339+272x^340+196x^345+156x^350+176x^355+108x^360+112x^365+84x^370+36x^375+56x^380+12x^385+12x^390+4x^405

The gray image is a linear code over GF(5) with n=420, k=6 and d=310.
This code was found by Heurico 1.16 in 1.52 seconds.