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

generates a code of length 78 over Z5[X]/(X^2) who´s minimum homogenous weight is 290.

Homogenous weight enumerator: w(x)=1x^0+248x^290+396x^295+544x^300+1704x^305+5052x^310+6608x^315+268x^320+192x^325+192x^330+136x^335+80x^340+92x^345+64x^350+24x^355+12x^360+4x^365+4x^370+4x^375

The gray image is a linear code over GF(5) with n=390, k=6 and d=290.
This code was found by Heurico 1.16 in 1.35 seconds.