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

generates a code of length 78 over Z5[X]/(X^2) who´s minimum homogenous weight is 300.

Homogenous weight enumerator: w(x)=1x^0+92x^300+100x^304+188x^305+800x^309+112x^310+1600x^314+108x^315+32x^320+20x^325+24x^330+8x^335+12x^340+8x^345+12x^350+4x^355+4x^380

The gray image is a linear code over GF(5) with n=390, k=5 and d=300.
This code was found by Heurico 1.16 in 0.118 seconds.