The generator matrix

 1  0  1  1  1  1  1  1  1  0  1  1  1  0  1  1  X  1  1  1  1  1  1  0  1  1  1  1  1  1  1  1  1 (a+1)X  1  1  X  1  1  1  1 aX aX  1  1  1  0  1  1  X  1 (a+1)X  1  X  1  1  1  1  1  1  1  1  1 (a+1)X  1  1  1  X  1  1  1 aX  1 (a+1)X  1  1  1  1  1  1 (a+1)X  1  1  X  1  1  X  1
 0  1  1  a a+1  0 (a+1)X+1 (a+1)X+a+1  a  1  0 (a+1)X+1 (a+1)X+a+1  1  a  X  1 (a+1)X+a+1 (a+1)X+1  a aX+a+1  0 aX+1  1  a (a+1)X+a+1 aX+1 (a+1)X+a+1  0 X+a aX+a+1 (a+1)X  1  1 (a+1)X+1  a  1 aX+a  X X+1 aX  1  1  1 X+a+1 aX  1 (a+1)X+a  X  1 aX+1  1 (a+1)X+a+1  1 X+a aX+a+1 a+1 aX+a+1 (a+1)X+1 aX+a+1 a+1 (a+1)X+a aX  1 (a+1)X+a X+a+1  1  1 aX+1 aX+a (a+1)X+a+1  1 aX+1  1  1 (a+1)X+1 X+a (a+1)X+a (a+1)X+a aX  1  X (a+1)X  1 aX+1  1  1  0
 0  0 (a+1)X  0  0  0  X aX  X  X  X (a+1)X (a+1)X aX aX aX  0  0  X aX (a+1)X aX (a+1)X (a+1)X (a+1)X aX  X aX  X aX (a+1)X  X  X aX aX (a+1)X aX (a+1)X aX  0 aX (a+1)X  0  X (a+1)X  0  0  X  0  0 (a+1)X (a+1)X  X  0  0  0  X aX  X  0  0 (a+1)X  0 (a+1)X  X (a+1)X  0 aX aX aX (a+1)X  0 (a+1)X aX  0 (a+1)X  X (a+1)X  X  X aX  X aX (a+1)X aX  X (a+1)X  0
 0  0  0  X  0  X (a+1)X (a+1)X  X (a+1)X  0 (a+1)X  X  X  0  X  0 (a+1)X (a+1)X  0  0  X (a+1)X (a+1)X  X aX (a+1)X (a+1)X  X (a+1)X (a+1)X  0  0  X aX  0 (a+1)X  X  X aX  0  X  X  X aX (a+1)X (a+1)X (a+1)X  0  0  X (a+1)X  X aX  X aX  0  X aX  0 aX  0 (a+1)X  0  X aX (a+1)X  0 aX (a+1)X  0  X  X aX  0  X aX aX aX (a+1)X  X aX  0  X  0  X (a+1)X  X
 0  0  0  0 (a+1)X (a+1)X (a+1)X (a+1)X  0 aX  X aX  0 aX (a+1)X  X (a+1)X  X  X aX  X (a+1)X  0 (a+1)X aX  X  0  0  X aX (a+1)X (a+1)X aX (a+1)X (a+1)X  0 aX  X aX aX  X  X aX aX aX (a+1)X aX  X aX aX  0  0  X (a+1)X  X  0 (a+1)X (a+1)X  0 aX aX aX aX (a+1)X (a+1)X  0 (a+1)X aX  0  0 (a+1)X (a+1)X (a+1)X (a+1)X  X  X  X aX aX  0  0  X  0  0 aX (a+1)X (a+1)X  X

generates a code of length 88 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 248.

Homogenous weight enumerator: w(x)=1x^0+216x^248+12x^249+216x^251+921x^252+444x^253+420x^255+1422x^256+300x^257+540x^259+1545x^260+744x^261+504x^263+2088x^264+384x^265+540x^267+1713x^268+588x^269+420x^271+1368x^272+444x^273+324x^275+684x^276+144x^277+96x^279+144x^280+12x^281+12x^283+27x^284+27x^288+36x^292+18x^296+9x^300+9x^304+9x^308+3x^320

The gray image is a linear code over GF(4) with n=352, k=7 and d=248.
This code was found by Heurico 1.16 in 2.09 seconds.