The generator matrix

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

generates a code of length 82 over Z5[X]/(X^2) who´s minimum homogenous weight is 311.

Homogenous weight enumerator: w(x)=1x^0+940x^311+660x^312+160x^313+1560x^314+108x^315+3720x^316+1640x^317+840x^318+3500x^319+164x^320+7240x^321+2800x^322+940x^323+3700x^324+96x^325+7740x^326+3360x^327+1160x^328+4020x^329+76x^330+7460x^331+3060x^332+880x^333+3640x^334+48x^335+6740x^336+2500x^337+840x^338+3000x^339+36x^340+3080x^341+920x^342+180x^343+580x^344+44x^345+580x^346+60x^347+12x^350+8x^355+20x^360+4x^365+4x^375+4x^380

The gray image is a linear code over GF(5) with n=410, k=7 and d=311.
This code was found by Heurico 1.16 in 12.7 seconds.