The generator matrix

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

generates a code of length 35 over Z5[X]/(X^2) who´s minimum homogenous weight is 137.

Homogenous weight enumerator: w(x)=1x^0+120x^137+80x^139+212x^140+180x^142+20x^144+8x^150+4x^155

The gray image is a linear code over GF(5) with n=175, k=4 and d=137.
This code was found by Heurico 1.16 in 0.18 seconds.