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

generates a code of length 80 over Z5[X]/(X^2) who´s minimum homogenous weight is 310.

Homogenous weight enumerator: w(x)=1x^0+188x^310+152x^315+500x^316+104x^320+2000x^321+52x^325+36x^330+28x^335+24x^340+8x^345+8x^350+20x^355+4x^395

The gray image is a linear code over GF(5) with n=400, k=5 and d=310.
This code was found by Heurico 1.16 in 0.135 seconds.