The generator matrix

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

generates a code of length 78 over Z5[X]/(X^2) who´s minimum homogenous weight is 305.

Homogenous weight enumerator: w(x)=1x^0+196x^305+400x^306+200x^307+552x^310+760x^311+200x^312+164x^315+120x^316+40x^317+96x^320+80x^321+20x^322+112x^325+140x^326+40x^327+4x^330

The gray image is a linear code over GF(5) with n=390, k=5 and d=305.
This code was found by Heurico 1.16 in 0.0865 seconds.