The generator matrix

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

generates a code of length 78 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+208x^68+56x^69+550x^70+308x^71+802x^72+564x^73+1272x^74+972x^75+1581x^76+1152x^77+1600x^78+1232x^79+1570x^80+956x^81+1222x^82+540x^83+675x^84+272x^85+384x^86+84x^87+178x^88+8x^89+100x^90+56x^92+22x^94+16x^96+2x^98+1x^104

The gray image is a code over GF(2) with n=312, k=14 and d=136.
This code was found by Heurico 1.16 in 19.3 seconds.