Incidence structures, codes, and Galois geometry Vladimir D. Tonchev Michigan Technological University 1 / 35 2 / 35 3 / 35 4 / 35 Incidence Structures A finite incidence structure D=(X ; B) is a finite set X of points and a collection B of subsets called blocks. An incidence structure is simple if all blocks are distinct as sets. The complementary structure D∗ of an incidence structure D has as blocks the complements of the blocks of D. Design A t-(v; k; λ) design is an incidence structure D=(X ; B) such that: jX j = v, jBj = k for each B 2 B, and Every t-subset of X s contained in exactly λ blocks. 5 / 35 Incidence Structures A finite incidence structure D=(X ; B) is a finite set X of points and a collection B of subsets called blocks. An incidence structure is simple if all blocks are distinct as sets. The complementary structure D∗ of an incidence structure D has as blocks the complements of the blocks of D. Design A t-(v; k; λ) design is an incidence structure D=(X ; B) such that: jX j = v, jBj = k for each B 2 B, and Every t-subset of X s contained in exactly λ blocks. 5 / 35 Incidence Structures A finite incidence structure D=(X ; B) is a finite set X of points and a collection B of subsets called blocks. An incidence structure is simple if all blocks are distinct as sets. The complementary structure D∗ of an incidence structure D has as blocks the complements of the blocks of D. Design A t-(v; k; λ) design is an incidence structure D=(X ; B) such that: jX j = v, jBj = k for each B 2 B, and Every t-subset of X s contained in exactly λ blocks. 5 / 35 Incidence Structures A finite incidence structure D=(X ; B) is a finite set X of points and a collection B of subsets called blocks. An incidence structure is simple if all blocks are distinct as sets. The complementary structure D∗ of an incidence structure D has as blocks the complements of the blocks of D. Design A t-(v; k; λ) design is an incidence structure D=(X ; B) such that: jX j = v, jBj = k for each B 2 B, and Every t-subset of X s contained in exactly λ blocks. 5 / 35 Incidence Structures A finite incidence structure D=(X ; B) is a finite set X of points and a collection B of subsets called blocks. An incidence structure is simple if all blocks are distinct as sets. The complementary structure D∗ of an incidence structure D has as blocks the complements of the blocks of D. Design A t-(v; k; λ) design is an incidence structure D=(X ; B) such that: jX j = v, jBj = k for each B 2 B, and Every t-subset of X s contained in exactly λ blocks. 5 / 35 Incidence Structures A finite incidence structure D=(X ; B) is a finite set X of points and a collection B of subsets called blocks. An incidence structure is simple if all blocks are distinct as sets. The complementary structure D∗ of an incidence structure D has as blocks the complements of the blocks of D. Design A t-(v; k; λ) design is an incidence structure D=(X ; B) such that: jX j = v, jBj = k for each B 2 B, and Every t-subset of X s contained in exactly λ blocks. 5 / 35 Incidence Structures A finite incidence structure D=(X ; B) is a finite set X of points and a collection B of subsets called blocks. An incidence structure is simple if all blocks are distinct as sets. The complementary structure D∗ of an incidence structure D has as blocks the complements of the blocks of D. Design A t-(v; k; λ) design is an incidence structure D=(X ; B) such that: jX j = v, jBj = k for each B 2 B, and Every t-subset of X s contained in exactly λ blocks. 5 / 35 Incidence Structures A finite incidence structure D=(X ; B) is a finite set X of points and a collection B of subsets called blocks. An incidence structure is simple if all blocks are distinct as sets. The complementary structure D∗ of an incidence structure D has as blocks the complements of the blocks of D. Design A t-(v; k; λ) design is an incidence structure D=(X ; B) such that: jX j = v, jBj = k for each B 2 B, and Every t-subset of X s contained in exactly λ blocks. 5 / 35 A small example A B G D C E F A 2-(7; 3; 1) design 6 / 35 Incidence Matrix The incidence matrix of a t-(v; k; λ) design is a b × v (0; 1) matrix whose (i; j) entry is 1 if block i contains point j, and 0 otherwise. The 2-(7,3,1) Design: ABCDEFG B1 1 1 1 0 0 0 0 B2 1 0 0 1 1 0 0 B3 1 0 0 0 0 1 1 B4 0 1 0 1 0 1 0 B5 0 1 0 0 1 0 1 B6 0 0 1 1 0 0 1 B7 0 0 1 0 1 1 0 7 / 35 Incidence Matrix The incidence matrix of a t-(v; k; λ) design is a b × v (0; 1) matrix whose (i; j) entry is 1 if block i contains point j, and 0 otherwise. The 2-(7,3,1) Design: ABCDEFG B1 1 1 1 0 0 0 0 B2 1 0 0 1 1 0 0 B3 1 0 0 0 0 1 1 B4 0 1 0 1 0 1 0 B5 0 1 0 0 1 0 1 B6 0 0 1 1 0 0 1 B7 0 0 1 0 1 1 0 7 / 35 Finite Geometries Projective Geometry PG(n; q) n+1 points of PG(n; q) are the 1-dimensional subspaces of Fq . n+1 lines of PG(n; q) are the 2-dimensional subspaces of Fq . d-dimensional projective subspaces are the (d + 1)-dimensional n+1 subspaces of Fq . Affine Geometry AG(n; q) n points of AG(n; q) are the vectors of Fq. n lines of AG(n; q) are the 1-dimensional subspaces of Fq and their cosets. d-dimensional affine subspaces are the d-dimensional n subspaces of Fq and their cosets. 8 / 35 Finite Geometries Projective Geometry PG(n; q) n+1 points of PG(n; q) are the 1-dimensional subspaces of Fq . n+1 lines of PG(n; q) are the 2-dimensional subspaces of Fq . d-dimensional projective subspaces are the (d + 1)-dimensional n+1 subspaces of Fq . Affine Geometry AG(n; q) n points of AG(n; q) are the vectors of Fq. n lines of AG(n; q) are the 1-dimensional subspaces of Fq and their cosets. d-dimensional affine subspaces are the d-dimensional n subspaces of Fq and their cosets. 8 / 35 Finite Geometries Projective Geometry PG(n; q) n+1 points of PG(n; q) are the 1-dimensional subspaces of Fq . n+1 lines of PG(n; q) are the 2-dimensional subspaces of Fq . d-dimensional projective subspaces are the (d + 1)-dimensional n+1 subspaces of Fq . Affine Geometry AG(n; q) n points of AG(n; q) are the vectors of Fq. n lines of AG(n; q) are the 1-dimensional subspaces of Fq and their cosets. d-dimensional affine subspaces are the d-dimensional n subspaces of Fq and their cosets. 8 / 35 Finite Geometries Projective Geometry PG(n; q) n+1 points of PG(n; q) are the 1-dimensional subspaces of Fq . n+1 lines of PG(n; q) are the 2-dimensional subspaces of Fq . d-dimensional projective subspaces are the (d + 1)-dimensional n+1 subspaces of Fq . Affine Geometry AG(n; q) n points of AG(n; q) are the vectors of Fq. n lines of AG(n; q) are the 1-dimensional subspaces of Fq and their cosets. d-dimensional affine subspaces are the d-dimensional n subspaces of Fq and their cosets. 8 / 35 Finite Geometries Projective Geometry PG(n; q) n+1 points of PG(n; q) are the 1-dimensional subspaces of Fq . n+1 lines of PG(n; q) are the 2-dimensional subspaces of Fq . d-dimensional projective subspaces are the (d + 1)-dimensional n+1 subspaces of Fq . Affine Geometry AG(n; q) n points of AG(n; q) are the vectors of Fq. n lines of AG(n; q) are the 1-dimensional subspaces of Fq and their cosets. d-dimensional affine subspaces are the d-dimensional n subspaces of Fq and their cosets. 8 / 35 Finite Geometries Projective Geometry PG(n; q) n+1 points of PG(n; q) are the 1-dimensional subspaces of Fq . n+1 lines of PG(n; q) are the 2-dimensional subspaces of Fq . d-dimensional projective subspaces are the (d + 1)-dimensional n+1 subspaces of Fq . Affine Geometry AG(n; q) n points of AG(n; q) are the vectors of Fq. n lines of AG(n; q) are the 1-dimensional subspaces of Fq and their cosets. d-dimensional affine subspaces are the d-dimensional n subspaces of Fq and their cosets. 8 / 35 Finite Geometries Projective Geometry PG(n; q) n+1 points of PG(n; q) are the 1-dimensional subspaces of Fq .