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

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+680x^311+1040x^312+700x^313+800x^314+416x^315+2460x^316+2980x^317+1600x^318+2040x^319+628x^320+4780x^321+4800x^322+2600x^323+2780x^324+512x^325+5440x^326+5140x^327+2640x^328+2460x^329+576x^330+4720x^331+5160x^332+2480x^333+2440x^334+428x^335+4120x^336+4140x^337+2020x^338+1620x^339+428x^340+2420x^341+1520x^342+460x^343+360x^344+76x^345+380x^346+220x^347+32x^350+8x^355+12x^360+4x^375+4x^385

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 15.6 seconds.