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

generates a code of length 75 over Z5[X]/(X^2) who´s minimum homogenous weight is 279.

Homogenous weight enumerator: w(x)=1x^0+720x^279+1284x^280+1580x^281+2020x^282+1240x^283+3360x^284+6060x^285+4740x^286+6680x^287+3480x^288+7580x^289+13716x^290+9540x^291+10460x^292+5540x^293+13220x^294+20776x^295+15200x^296+16080x^297+9360x^298+17580x^299+27808x^300+19060x^301+18280x^302+10840x^303+18280x^304+28736x^305+16940x^306+15060x^307+7480x^308+12160x^309+16548x^310+8940x^311+7700x^312+1820x^313+4160x^314+3124x^315+1500x^316+1220x^317+240x^318+440x^319+28x^320+20x^325+4x^330+16x^335+4x^340

The gray image is a linear code over GF(5) with n=375, k=8 and d=279.
This code was found by Heurico 1.16 in 368 seconds.