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

generates a code of length 70 over Z5[X]/(X^2) who´s minimum homogenous weight is 270.

Homogenous weight enumerator: w(x)=1x^0+176x^270+100x^272+132x^275+800x^277+148x^280+1600x^282+68x^285+28x^290+20x^295+4x^300+12x^305+12x^310+8x^315+4x^320+8x^325+4x^340

The gray image is a linear code over GF(5) with n=350, k=5 and d=270.
This code was found by Heurico 1.16 in 3.17 seconds.