On the equitable partitions of H ( 12 , 2 ) [ 3 9 ] with quotient - PowerPoint PPT Presentation

On the equitable partitions of H ( 12 , 2 ) [︃ 3 9 ]︃ with quotient matrix 7 5 Den´ is Krotov Sobolev Institute of Mathematics, Novosibirsk, Russia (joint work with K. Vorob’ev) Shanghai Jiao Tong University June 17, 2018, Shanghai

Perfect colorings -> [[3,9],[7,5]] topic: perfect colorings = equitable partitions subtopic: perfect 2-colorings (2 colors) subsubtopic: perfect 2-colorings of H ( n , 2 ) subsubsubtopic: perfect 2-colorings of H ( n , 2 ) that attain the correlation-immunity bound subsubsubsubtopic: perfect 2-colorings of H ( 12 , 2 ) with parameters [︃ 3 ]︃ 9 7 5

Perfect coloring (equitable partition) Γ = ( V (Γ) , E (Γ)) — graph. Deҥnition A function f : V (Γ) → { C 0 , . . . , C m − 1 } is called a perfect coloring with parameter (quotient) matrix S = ( S ij ) m − 1 i , j = 0 if every vertex of color C i has exactly S ij neighbors of color C j . perfect coloring ∼ еquitable partition ∼ regular partition ∼ partition design ∼ . . .

0 BBBBB@ 1 CCCCCA Example: perfect 3-coloring 1 2 0 S = 1 0 2 0 2 1

Adjacency matrix of a graph 0 1 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 A : 0 0 0 0 1 0 0 1 1 0

Incidence matrix of a coloring 1 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 C : 0 0 1

Matrix deҥnition of perfect colorings A ҫ the adjacency matrix of a graph; C ҫ the incidence matrix of the perfect coloring with parameter matrix S . AC = CS 0 1 1 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 1 2 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 1 0 · = · 1 0 2 0 0 0 1 1 0 1 0 0 0 0 0 1 0 0 1 0 2 1 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1

Perfect coloring of distance-regular graphs Perfect colorings of distance-regular graphs are of great interest because of their algebraico-combinatorial properties. Many famous classes can be deҥned as perfect colorings: Completely regular codes, including perfect codes, nearly perfect codes, distance-2 and distance-3 MDS (in Hamming and Doob graphs) and MRD (in bilinear-form graphs) codes and many others; latin squares and latin hypercubes (in Hamming graph), transversals of latin squares (in latin-square graphs), Steiner systems S ( t , t + 1 , v ) , S ( t , t + 2 , v ) , t - ( v , t + 1 , λ ) -designs (in Johnson graphs); their subspace analogs S q ( t , t + 1 , v ) , S q ( t , t + 2 , v ) , t - ( v , t + 1 , λ ) -designs (in Gra ß mann graphs); spreads, Cameron-Liebler line classes (in Gra ß mann graphs); . . .

Example: perfect coloring of the inҥnite hexagonal grid 3 2 3 3 2 1 1 2 3 3 2 3 3 3 3 2 3 3 2 3 2 3 3 3 3 2 3 3 2 1 1 2 3 3 2 3 3 3 2 1 1 2 3 3 2 3 3 3 3 2 3 3 2 1 1 2 3 3 3 2 3 3 2 1 1 2 3 3 2 3 3 3 3 2 3 3 3 3 2 3 3 3 3 2 3 3 2 1 1 2 3 3 2 3 3 3 2 1 1 2 3 3 2 3 3 3 3 2 3 3 2 1 1 2 3 3 3 2 3 3 2 1 1 2 3 3 2 3 3 3 3 2 3 2 3 3 2 3 3 3 3 2 3 3 2 1 1 2 3 3 2 3 3 3 2 1 1 2 3 3 2 3 3 3 3 2 3 3 2 1 1

Perfect 2-colorings and eigenfunction Let Γ be a regular graph. Let f : V (Γ) → { color 0 , color 1 } be a perfect 2-coloring with parameter matrix [︃ a ]︃ b S = c d Put color 0 = b , color 1 = − c . Then, f is an eigenfunction of Γ (equivalently, an eigenvector of the adjacency matrix) with eigenvalue a − c = d − b (the second eigenvalue of S ). Inversely any two-valued eigenfunction of Γ is a perfect 2-coloring.

n -cube, Hamming graph H ( n , 2 ) n -cube (hypercube) H ( n , 2 ) is a graph over the set of all words of length n over the binary alphabet { 0 , 1 } . Two words are adjacent if they difger in exactly one position. H ( 3 , 2 ) : 011 111 001 101 010 110 000 100 [︃ 0 3 ]︃ At the picture: perfect coloring of H ( 3 , 2 ) with matrix 1 2

Graph coverings If the parameter matrix of the graph Γ is the adjacency matrix of some graph γ (in general, multiplied by some coeffjcient t ), then such coloring is called a covering (a t -fold covering) of γ by Γ . In other words, covering is a map f from V (Γ) to V ( γ ) such that the neighborhood of every vertex x is bijectively mapped onto the neighborhood of f ( x ) . Theorem If f is a perfect coloring of a graph γ and h : V (Γ) → V ( γ ) is a covering of γ by Γ , then f ( h ( · )) is a perfect coloring of Γ with the same parameter matrix (in the case of a t -fold covering, the parameter matrix is multiplied by t ).

Covering H ( 6 , 2 ) → Sh The Hamming graph H ( 6 , 2 ) is (isomorphic to) a Cayley graph on Z 4 × Z 4 × Z 4 with the connecting set {± ( 1 , 0 , 0 ) , ± ( 0 , 1 , 0 ) , ± ( 0 , 0 , 1 ) The Shrikhande graph Sh is a Cayley graph Z 4 × Z 4 with the connecting set ± ( 1 , 0 ) , ± ( 0 , 1 ) , ± ( 1 , 1 ) . 03 13 23 33 02 12 22 32 01 11 21 31 00 10 20 30 Homomorphism: ( x , y , z ) → ( x + z , y + z ) .

A perfect coloring of H ( 6 , 2 ) with parameters [[1,5],[3,3]] [︃ 1 ]︃ 5 A perfect 2-coloring of Sh with parameter matrix : 3 3 Covering H ( 6 , 2 ) → Sh

Perfect 2-colorings of H ( n , 2 ) , publications Parameters of perfect 2-colorings of H ( n , 2 ) Fon-Der-Flaass. Perfect 2-Colorings of a Hypercube. Sib. Math. J. 2007 Fon-Der-Flaass. A bound on correlation immunity. Sib. El. Math. Rep. 2007. Fon-Der-Flaass. Perfect Colorings of the 12-Cube That Attain the Bound on Correlation Immunity. Sib. El. Math. Rep. 2007 [in Russian]. http://arxiv.org/abs/1403.8091 [English transl.] The minimal dimension for which there are open parameters: 24 [︃ 1 ]︃ [︃ 2 ]︃ [︃ 3 ]︃ [︃ 5 ]︃ [︃ 7 ]︃ 23 22 21 19 17 , , , , 9 15 10 14 11 13 13 11 15 9

Perfect 2-colorings of H ( n , 2 ) , bounds a + b [FDF,2007] gcd ( a , b ) is a power of 2 (the number 2 n c b + c of vertices of ҥrst color must be integer). [FDF,2007] Correlation-immunity bound: if b ̸ = c , then a − c ≥ − n 3 [︃ 1 ]︃ 11 [FDF,2007] do not exist (sporadic proof) 5 7 NEW! if b ̸ = c and a − c = − n bc 3 , then gcd ( b , c ) 2 ≡ 0 mod 3. Theorem (D.G. Fon-Der-Flaass, 2007) b + c Assume that gcd ( b , c ) is a power of 2 . Then there exists a 0 such that [︂ ]︂ a b perfect 2 -colorings of H ( n , 2 ) with parameter matrix , where c d d = a + b − c , exist for all a ≥ a 0 .

Perfect 2-colorings of H ( n , 2 ) , constructions S → S + Id (trivial extension) S → t · S (from t -fold covering H ( tn , 2 ) → H ( n , 2 ) ) [︃ 0 ]︃ [︃ k − 1 ]︃ n n − k + 1 1-Perfect codes: , . 1 n − 1 n − k k [︃ a ]︃ [︃ a − 1 ]︃ b 2 b + c Splitting: → 2 b + a − 1 c d c [︃ 3 ]︃ 9 Special: 7 5

Parameters table for perfect 2-colorings of H ( n , 2 )

Attaining the Bound on Correlation Immunity Perfect Colorings of the 12-Cube That Attain the Bound on Correlation Immunity: Known parameters: [︃ 0 ]︃ [︃ 1 ]︃ [︃ 3 ]︃ 3 5 9 t · , t · , t · 1 2 3 3 7 5 First questions: [︃ 1 ]︃ [︃ 2 ]︃ [︃ 3 ]︃ [︃ 5 ]︃ [︃ 7 ]︃ 23 22 21 19 17 , , , , 9 15 10 14 11 13 13 11 15 9

New results Perfect Colorings of the 12-Cube That Attain the Bound on Correlation Immunity: Proposition. There are exactly 2 equivalence classes of perfect [︃ 3 ]︃ 9 colorings with parameter matrix (Fourier analysis, exact- 7 5 covering software, solving systems of linear equations over GF(2) ) Theorem. For every perfect 2-coloring attaining the correlation- immunity bound, the number of two-color edges of ҥxed direction does not depend on the direction Corollary. For every perfect 2-coloring attaining the correlation- b c immunity bound, either gcd ( b , c ) or gcd ( b , c ) is divisible by 3. [︃ 1 [︃ 2 [︃ 5 ]︃ ]︃ ]︃ 11 22 19 Do not exist: , , 5 7 10 14 13 11

Fourier transform ( − ) ( + ) ( + ) ( − ) ( − ) ( + ) ( + ) ( − ) Basis from eigenfunctions: χ y ( x ) = ( − 1 ) ⟨ x , y ⟩ , y ∈ V ( H ( n , 2 )) χ y ( x ) corresponds to the eigenvalue n − 2 wt ( y ) ∑︂ ̂︁ f ( x ) = f ( y ) χ y ( x ) y ∈ V ( H ( n , 2 )) ∑︂ f ( y ) = 2 − n ̂︁ χ y ( x ) f ( x ) x ∈ V ( H ( n , 2 ))

correlation-immunity bound Any { c , − b } -valued ( b ̸ = c ) function satisҥes: f 2 = ( c − b ) f + cb f * ̂︁ ̂︁ f = ( c − b ) ̂︁ = ⇒ f + cb · δ ¯ 0 where the convolution * is deҥned by ∑︂ g * h ( x ) = g ( y ) h ( z ) y , z : y + z = x It follows that supp ( ̂︁ 0 } ⊆ supp ( ̂︁ f ) + supp ( ̂︁ f ) ∪ { ¯ f ) Hence, ̂︁ f must have a nonzero of weight at most 2 n / 3 (correlation-immunity bound). Moreover, if it has no nonzeros of weight less than 2 n / 3, then it is a perfect coloring.