The generator matrix

 1  0  0  0  1  1  1  1  1  1  1  1 a^2*X  1  1  1 a*X a^2*X  1 a^2*X a*X  1  1  X  1  1  1  0  1  1  1  1  1  1  1  1  1  1  X  1  X  1  1  1  0  1  1  1  1  1  1 a^2*X  1  1  1  1  1  1  1  1  1  1 a^2*X  0  1  1  1  1  X  1  1  0  1  1  1  1  1  1 a*X  1  1  1 a*X  X  1  1  1
 0  1  0  0 a^2*X  0 a^2*X  1 a^2*X+a a^2  X  0  X  X a^2*X+a^2 X+1  1  1 X+a  1  1  a a*X+a^2  1 a^2 a^2*X+1 a^2*X a*X a*X+a^2  1  1  X a*X+1 a^2 a*X+1 X+a a^2 a*X+a  1 X+a^2  1 a*X a^2*X+a X+1  1 a*X+a X+a a*X+1 X+1  1 a^2  1 a^2*X a*X+a^2 a^2*X+a^2  0 a^2*X a*X a*X+a^2 X+a a*X+1 a*X+a a^2*X  1 X+1 a*X+1 a*X+1 X+1  1  1 a*X+a  1 a^2*X+a^2  X a^2*X+1 a*X+1 a^2*X+a a^2*X  1 a^2*X+a^2 X+a  0  1  1 a^2*X+a^2 a^2*X+1 a^2*X+a^2
 0  0  1  0  X a^2*X  0 a*X a*X a*X a^2*X+1 a^2*X+1  1 a^2*X+a X+a^2 X+1  1 X+1  a  1 a^2*X+a a*X+1  a X+a^2  1 a^2 X+a  1  1 a*X+a  1 X+1 X+1  0  a X+a^2 a^2*X a*X+1 a^2*X+a^2 X+a  0 a^2*X+a^2  X X+a^2 a^2*X+a  X a^2*X+a  0 a*X+a^2  a a^2*X+a^2 a*X+a^2 a^2*X+a X+1  X a^2 a^2 a*X a^2*X+a^2 a^2*X+1 a^2*X+a a*X  1 X+1  X a*X+a^2 a*X a*X+1 a*X+a a^2*X a*X+a a*X a^2*X+a a*X+a^2  X a^2*X+1 X+1 a^2 a^2*X+a^2  a  1 a^2*X+1 a*X+a a^2*X  1 a*X+a^2 a^2
 0  0  0  1 a^2*X+1 a^2*X+a a^2 X+a^2 a^2*X+a^2 a*X+a^2 a^2  X a^2  a  a a*X+a a^2*X+a a*X a^2*X+a^2  1 a^2  1 a*X+a^2 X+1  X  0 a^2*X+a^2 X+a X+a  0 a*X a^2*X+a X+a^2 a^2*X X+a  1 a*X+a a^2*X a^2*X+a^2  0 a*X+a  1 a*X+1 a^2 X+1 a*X+a a*X X+1 X+a a*X+1 a^2  a a*X+1 X+1  1 X+a a^2*X a^2*X+a a^2*X a^2*X+a^2 X+a a*X X+1 a*X+1 a*X X+1 a^2*X+a a^2*X+1 a^2*X X+a^2 a^2*X+a  1  a X+a^2  X a*X a*X+a^2  1 a^2*X X+1 a^2*X+a X+1 a*X a^2*X a*X+a^2 a^2*X+1  1

generates a code of length 87 over F4[X]/(X^2) who´s minimum homogenous weight is 244.

Homogenous weight enumerator: w(x)=1x^0+306x^244+516x^245+576x^246+744x^247+1497x^248+1584x^249+1944x^250+1512x^251+2229x^252+2532x^253+2460x^254+1980x^255+2700x^256+2892x^257+2724x^258+2112x^259+2844x^260+3444x^261+3156x^262+1908x^263+3162x^264+3012x^265+2532x^266+1704x^267+2496x^268+2484x^269+1956x^270+1428x^271+1689x^272+1284x^273+1116x^274+720x^275+786x^276+504x^277+396x^278+180x^279+207x^280+156x^281+36x^282+3x^284+24x^285

The gray image is a linear code over GF(4) with n=348, k=8 and d=244.
This code was found by Heurico 1.16 in 28 seconds.