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

generates a code of length 69 over Z5[X]/(X^2) who´s minimum homogenous weight is 270.

Homogenous weight enumerator: w(x)=1x^0+12x^270+84x^275+500x^276+20x^280+4x^295+4x^345

The gray image is a linear code over GF(5) with n=345, k=4 and d=270.
This code was found by Heurico 1.16 in 0.0417 seconds.