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

generates a code of length 78 over Z5[X]/(X^2) who´s minimum homogenous weight is 311.

Homogenous weight enumerator: w(x)=1x^0+160x^311+400x^312+8x^315+40x^316+8x^320+8x^340

The gray image is a linear code over GF(5) with n=390, k=4 and d=311.
This code was found by Heurico 1.16 in 0.0617 seconds.