24分10-D正割10。由WDS和NJAS在1990年4月发现。这里的WDS,RHH,是一个相当于一个华伦发现的美丽代码,有24个向量在10-D和割线10。这里有两个版本。版本1:每个向量用14个实数来描述,它的坐标在14个空间中,每行一个数。Thus there are 336 lines To verify that this has the correct min angle , feed it into secsum AS IF IT WERE A 14-D CODE. (Give it a file name temp.14.24 in other words) To verify that it really is a 10-D code, note that the first 12 coords come in 4 sets of 3, and each set of 3 sums to zero. So the first 12 coords are really only in an 8-D space. This can be seen more easily in the second version. Of course if you really want 10 coords, just do this to each triple: x -x 0 -> (x, 0 )*sqrt(2) 0 x -x -> (-x/2, +x*sqrt(3)/2 )*sqrt(2) -x 0 x -> (-x/2, -x*sqrt(3)/2 )*sqrt(2) Version 2 is a symbolic description of the EXACTLY the same vectors This is at the end of the file. 这说明代码是多么漂亮,解释了它的构造。Neil Version 1 2.121320343560 -2.121320343560 0 0 0 0 0 0 0 0 0 0 1 0 0 2.121320343560 -2.121320343560 0 0 0 0 0 0 0 0 0 1 0 -2.121320343560 0 2.121320343560 0 0 0 0 0 0 0 0 0 1 0 0 0 0 2.121320343560 -2.121320343560 0 0 0 0 0 0 0 1 0 0 0 0 0 2.121320343560 -2.121320343560 0 0 0 0 0 0 1 0 0 0 0 -2.121320343560 0 2.121320343560 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2.121320343560 -2.121320343560 0 0 0 0 1 0 0 0 0 0 0 0 0 2.121320343560 -2.121320343560 0 0 0 1 0 0 0 0 0 0 0 -2.121320343560 0 2.121320343560 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1.732050807568877 -1.732050807568877 0 1 +1.732050807568877 0 0 0 0 0 0 0 0 0 0 1.732050807568877 -1.732050807568877 1 +1.732050807568877 0 0 0 0 0 0 0 0 0 -1.732050807568877 0 1.732050807568877 1 +1.732050807568877 0 0 0 0 0 0 0 0 0 -1.732050807568877 1.732050807568877 0 1 -1.732050807568877 0 0 0 0 0 0 0 0 0 0 -1.732050807568877 1.732050807568877 1 -1.732050807568877 0 0 0 0 0 0 0 0 0 1.732050807568877 0 -1.732050807568877 1 -1.732050807568877 0 -1.0606601717798212866 1.0606601717798212866 0 -1.0606601717798212866 1.0606601717798212866 0 -1.0606601717798212866 1.0606601717798212866 0 -.86602540378443864676 .86602540378443864676 -1.25 .43301270189221932338 0 -1.0606601717798212866 1.0606601717798212866 1.0606601717798212866 0 -1.0606601717798212866 1.0606601717798212866 0 -1.0606601717798212866 .86602540378443864676 0 -.86602540378443864676 -1.25 .43301270189221932338 0 -1.0606601717798212866 1.0606601717798212866 -1.0606601717798212866 1.0606601717798212866 0 -1.0606601717798212866 1.0606601717798212866 0 -.86602540378443864676 .86602540378443864676 0 -1.25 .43301270189221932338 1.0606601717798212866 0 -1.0606601717798212866 1.0606601717798212866 0 -1.0606601717798212866 -1.0606601717798212866 1.0606601717798212866 0 0 -.86602540378443864676 .86602540378443864676 -1.25 .43301270189221932338 -1.0606601717798212866 1.0606601717798212866 0 -1.0606601717798212866 1.0606601717798212866 0 1.0606601717798212866 0 -1.0606601717798212866 0 -.86602540378443864676 .86602540378443864676 -1.25 .43301270189221932338 1.0606601717798212866 0 -1.0606601717798212866 -1.0606601717798212866 1.0606601717798212866 0 0 -1.0606601717798212866 1.0606601717798212866 .86602540378443864676 0 -.86602540378443864676 -1.25 .43301270189221932338 -1.0606601717798212866 1.0606601717798212866 0 1.0606601717798212866 0 -1.0606601717798212866 0 -1.0606601717798212866 1.0606601717798212866 -.86602540378443864676 .86602540378443864676 0 -1.25 .43301270189221932338 1.0606601717798212866 0 -1.0606601717798212866 0 -1.0606601717798212866 1.0606601717798212866 1.0606601717798212866 0 -1.0606601717798212866 -.86602540378443864676 .86602540378443864676 0 -1.25 .43301270189221932338 -1.0606601717798212866 1.0606601717798212866 0 0 -1.0606601717798212866 1.0606601717798212866 -1.0606601717798212866 1.0606601717798212866 0 .86602540378443864676 0 -.86602540378443864676 -1.25 .43301270189221932338 Version 2: Symbolic version: 10 24 a -a 0 0 0 0 0 0 0 0 0 0 t 0 0 a -a 0 0 0 0 0 0 0 0 0 t 0 -a 0 a 0 0 0 0 0 0 0 0 0 t 0 0 0 0 a -a 0 0 0 0 0 0 0 t 0 0 0 0 0 a -a 0 0 0 0 0 0 t 0 0 0 0 -a 0 a 0 0 0 0 0 0 t 0 0 0 0 0 0 0 a -a 0 0 0 0 t 0 0 0 0 0 0 0 0 a -a 0 0 0 t 0 0 0 0 0 0 0 -a 0 a 0 0 0 t 0 0 0 0 0 0 0 0 0 0 u -u 0 t +u 0 0 0 0 0 0 0 0 0 0 u -u t +u 0 0 0 0 0 0 0 0 0 -u 0 u t +u 0 0 0 0 0 0 0 0 0 -u u 0 t -u 0 0 0 0 0 0 0 0 0 0 -u u t -u 0 0 0 0 0 0 0 0 0 u 0 -u t -u 0 -c c 0 -c c 0 -c c 0 -d d e f 0 -c c c 0 -c c 0 -c d 0 -d e f 0 -c c -c c 0 -c c 0 -d d 0 e f c 0 -c c 0 -c -c c 0 0 -d d e f -c c 0 -c c 0 c 0 -c 0 -d d e f c 0 -c -c c 0 0 -c c d 0 -d e f -c c 0 c 0 -c 0 -c c -d d 0 e f c 0 -c 0 -c c c 0 -c -d d 0 e f -c c 0 0 -c c -c c 0 d 0 -d e f where a=3/sqrt(2), t=1, u=sqrt(3), c=3/sqrt(8), d=sqrt(3)/2, e=-5/4, f=sqrt(3)/4. The last 9 rows are the tetracode which is the 9-word linear code over GF(3) generated by 0111 and 1120, with 0 replaced by 0 c -c (or 0 d -d), 1 replaced by -c 0 c, and 2 by c -c 0. Description: The first 9 points are the join of 3 equilateral triangles in 3 perpendicular 2-spaces. The next 6 points are a triangle and its negative in another perpendicular 2-space. The last 9 points are equilateral triangles at a different scale in these 4 2-spaces, the particular triangles used being determined by the tetracode. The last 2 coords in each vector specify 2 stacking coords to keep the layers apart. The group has order 108. min norm = 1.80000000 between 1 16 min angle = 8.426082952e+01 between 1 16 secant = 10.00000000