**semi-ring of binary spaces**

This page reports numerical facts related on the product of spaces.

Let K be a finite field. Let n be a positive integer. The Hadamard product of K^n is denoted by *. The set of subspaces of K^n becomes a semiring when it is equipped of the laws :

n: number of idempotents of dimension k 2: 1 1 3: 1 3 1 4: 1 7 6 1 5: 1 15 25 10 1 6: 1 31 90 65 15 1 7: 1 63 301 350 140 21 1 8: 1 127 966 1701 1050 266 28 1Stirling numbers of the second kind appear by column. It easy to prove there is only one projective idempotent.

n : number of irreducibles of dimension k 2 : 1 3 : 1 1 4 : 1 6 5 : 1 25 6 : 1 90 30 7 : 1 301 660 30More infos in the corresponding file.

[irred-6] [irred-7]

#length: 7 #number of spaces: 29212 ( 2^14.83 ) #number of projs : 2960 ( 2^11.53 ) --- 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 #projective code #dim. of the square 7 #dimension of square roots 0 0 0 1605 1225 99 1 #orbit size : 1 --- 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 1 #projective code #dim. of the square 7 #dimension of square roots 0 0 30 0 0 0 0 #orbit size : 1 #class number of proj. square :2 @runtime=0.00

that means there 29212 subspaces, 2960 are projectives whose 2 ( 2 class of size 1) are squares. One of dimension 6 has 30 square roots, the other one has 20930 square roots : 1605 of dimension 4, 1225 of dimension 5, 99 of dimension 6 and 1 of dimension 1.

[square-6] [square-7] [square-8]

[square-9] [square-10]

[dim-8] [dim-9] [dim-10] [dim-11] [dim-12]

[dim-13] [dim-14]

#dimension sequences of binary projective codes of length 8 2 dim : 4.7.8 4 dim : 4.8 15 dim : 5.8 14 dim : 6.8 6 dim : 7.8

that means there are 2 class of code of dimension 4 such that

jeu. janv. 26 09:57:29 CET 2017 #binary projective codes of length 8 --- 0 1 1 1 0 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 dim : 4.7.8 --- 0 1 1 1 0 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 1 0 0 1 1 1 0 1 0 0 0 dim : 4.7.8

See also : http://www.mathe2.uni-bayreuth.de/frib/codes/tables_8.html

#number of [7,4] class : 5 :1:2:4:8:7:11:13 :1:2:4:8:3:5:14 :1:2:4:8:3:5:10 :1:2:4:8:3:5:9 :1:2:4:8:3:5:6 #number of [7,4] class : 5that means there are 5 class of projective [7,4] code. One gets a generator matrix decomposing the integers using basis 2 e.g. the matrix described by the last line :1:2:4:8:3:5:6

1 0 0 0 1 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 0 0 0

Philippe Langevin, Last modification, January 2017, January 2018.