The generator matrix

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

generates a code of length 83 over Z5[X]/(X^2) who´s minimum homogenous weight is 310.

Homogenous weight enumerator: w(x)=1x^0+304x^310+60x^311+320x^313+1572x^315+1020x^316+900x^318+3328x^320+2680x^321+1280x^323+6128x^325+4120x^326+2540x^328+8912x^330+5420x^331+3040x^333+10380x^335+6660x^336+2620x^338+7208x^340+4000x^341+1440x^343+2156x^345+1040x^346+360x^348+188x^350+136x^355+104x^360+80x^365+56x^370+28x^375+32x^380+12x^385

The gray image is a linear code over GF(5) with n=415, k=7 and d=310.
This code was found by Heurico 1.16 in 54.6 seconds.