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

generates a code of length 83 over Z5[X]/(X^2) who´s minimum homogenous weight is 310.

Homogenous weight enumerator: w(x)=1x^0+268x^310+388x^315+524x^320+1648x^325+5156x^330+6664x^335+212x^340+168x^345+144x^350+164x^355+92x^360+56x^365+48x^370+28x^375+40x^380+16x^385+4x^390+4x^400

The gray image is a linear code over GF(5) with n=415, k=6 and d=310.
This code was found by Heurico 1.16 in 4.3 seconds.