The generator matrix

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

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+268x^290+300x^291+120x^292+180x^293+580x^294+388x^295+1320x^296+720x^297+520x^298+1320x^299+436x^300+2080x^301+1400x^302+1400x^303+3440x^304+408x^305+4500x^306+2360x^307+2400x^308+5440x^309+332x^310+5380x^311+2660x^312+4000x^313+7140x^314+288x^315+5780x^316+3040x^317+3020x^318+5560x^319+224x^320+4060x^321+1840x^322+980x^323+1520x^324+164x^325+1400x^326+360x^327+184x^330+180x^331+136x^335+100x^340+80x^345+64x^350+28x^355+12x^360+4x^365+8x^370

The gray image is a linear code over GF(5) with n=390, k=7 and d=290.
This code was found by Heurico 1.16 in 27.4 seconds.