The generator matrix

 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  1  1  1  1  1  1  1  1  1  1  1  2  X  X  X  2  1  0  1  2  1  0  2  0  1  1  1  0  X  1  1  1  X  X  X  2  2  1  2  1  X  1  0  1  1  1  0  0  1  2  X  1
 0  X  0  0  0  0  0  0  2  2  X X+2  X  0  0  2 X+2 X+2  X  X  X  X  0  X X+2  X  0  X X+2  2 X+2  2  2 X+2  X  0  0  X  2  2 X+2  2  X  X  2  X  X  X  2  0  2 X+2  0  X  X  X  X  2  2  X  X X+2  X  X  0  X  X  X  2  2  2 X+2  2  X  X  X  2  2  0
 0  0  X  0  0  0  0  0  0  0  0  0  2 X+2 X+2 X+2  X X+2 X+2  X  2  2 X+2 X+2  0  0  X  X  X X+2 X+2 X+2  2  2 X+2  X  X  2  2  X  X  0  X  2  X  0 X+2  0  0  X  2 X+2  2 X+2  X  0 X+2  0  2  0  0  2  X  2  X  X  2  X  2  X X+2 X+2  0  2 X+2  X  2  0  0
 0  0  0  X  0  0  2 X+2  X  X  X  X  2 X+2  X  2  2  0  2  2  2  2  2  X X+2  X  2  X X+2 X+2  X X+2  0  2  0  0  0 X+2  2  X X+2  0  0  0  X  0  X  2  X  X  X  X  0  0  X  2  2  0  2  0 X+2  0 X+2  2  0 X+2 X+2  0  0  0  X  0  X X+2  X  0  X  X  0
 0  0  0  0  X  0 X+2 X+2  X  2 X+2 X+2  0  X  X  0  2  X  0 X+2 X+2  X X+2  X  2  2  X  2  2  0 X+2  2  0 X+2  X X+2  2  0  X  0  0  2  0  X  0  X  0  2  0  X X+2 X+2 X+2 X+2  X  0  2  0  2  0  X  2  X X+2  0  0 X+2  X  2  2  0  2 X+2  2  0  0  2  0  0
 0  0  0  0  0  X  X  2 X+2  X X+2  2  X  X  0  X  0 X+2 X+2  0  X  2  2 X+2  2  X X+2 X+2  2  X  2  2 X+2  0  X X+2  0  0  X  X  0 X+2  2  2  2 X+2  X X+2  2  X  2  X  2  2  X  X X+2  X  0  2  X  X X+2 X+2  X  0  X  0  2  0  2 X+2  X X+2 X+2  0 X+2  0  0

generates a code of length 79 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+55x^68+102x^69+164x^70+218x^71+257x^72+310x^73+407x^74+542x^75+568x^76+604x^77+669x^78+668x^79+627x^80+602x^81+515x^82+412x^83+361x^84+316x^85+202x^86+136x^87+125x^88+88x^89+75x^90+60x^91+40x^92+26x^93+12x^94+10x^95+14x^96+3x^98+2x^99+1x^110

The gray image is a code over GF(2) with n=316, k=13 and d=136.
This code was found by Heurico 1.16 in 7.44 seconds.