The generator matrix

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

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+680x^290+820x^291+580x^292+600x^293+2800x^294+4696x^295+3640x^296+3300x^297+3760x^298+6280x^299+9552x^300+8500x^301+6600x^302+7060x^303+10500x^304+15940x^305+13440x^306+10080x^307+11440x^308+17060x^309+23880x^310+18620x^311+13380x^312+14520x^313+20560x^314+26492x^315+19140x^316+13680x^317+13300x^318+16760x^319+19908x^320+13660x^321+8040x^322+6020x^323+7280x^324+8184x^325+4040x^326+1620x^327+800x^328+1260x^329+1228x^330+640x^331+220x^332+28x^335+20x^340+8x^345+4x^355+4x^360

The gray image is a linear code over GF(5) with n=390, k=8 and d=290.
This code was found by Heurico 1.16 in 314 seconds.