The generator matrix

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

generates a code of length 70 over Z5[X]/(X^2) who´s minimum homogenous weight is 279.

Homogenous weight enumerator: w(x)=1x^0+400x^279+112x^280+100x^284+4x^300+8x^305

The gray image is a linear code over GF(5) with n=350, k=4 and d=279.
This code was found by Heurico 1.16 in 2.05 seconds.