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

generates a code of length 78 over Z5[X]/(X^2) who´s minimum homogenous weight is 275.

Homogenous weight enumerator: w(x)=1x^0+112x^275+512x^280+804x^285+1176x^290+1820x^295+4164x^300+10504x^305+19844x^310+22500x^315+11976x^320+1424x^325+1200x^330+884x^335+596x^340+344x^345+164x^350+64x^355+28x^360+8x^365

The gray image is a linear code over GF(5) with n=390, k=7 and d=275.
This code was found by Heurico 1.16 in 32.4 seconds.