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

On the equitable partitions of H(12, 2) with quotient matrix [︃ 3 9 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) → {C0, . . . , Cm−1} is called a perfect coloring with parameter (quotient) matrix S = (Sij)m−1

i,j=0 if every vertex of

color Ci has exactly Sij neighbors of color Cj. perfect coloring ∼ еquitable partition ∼ regular partition ∼ partition design ∼ . . .

Example: perfect 3-coloring

S =

BBBBB@

1 2 0 1 0 2 0 2 1

1 CCCCCA

Adjacency matrix of a graph

A:

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

Incidence matrix of a coloring

C :

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

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

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

·

1 1 1 1 1 1 1 1 1 1

=

1 1 1 1 1 1 1 1 1 1

·

1 2 1 2 2 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

• thers;

latin squares and latin hypercubes (in Hamming graph), transversals

• f 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 Sq(t, t + 1, v),Sq(t, t + 2, v), t-(v, t + 1, λ)-designs (in Graßmann graphs); spreads, Cameron-Liebler line classes (in Graßmann 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) → {color0, color1} be a perfect 2-coloring with parameter matrix S = [︃ a b c d ]︃ Put color0 = b, color1 = −c. Then, f is an eigenfunction of Γ (equivalently, an eigenvector

• f 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) :

000 010 100 110 001 011 101 111

At the picture: perfect coloring of H(3, 2) with matrix [︃ 0 3 1 2 ]︃

Graph coverings

If the parameter matrix of the graph Γ is the adjacency matrix

• f 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

• nto 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

• n Z4×Z4×Z4 with the connecting set {±(1, 0, 0), ±(0, 1, 0), ±(0, 0, 1)

The Shrikhande graph Sh is a Cayley graph Z4 × Z4 with the connecting set ±(1, 0), ±(0, 1), ±(1, 1).

00 10 20 30 01 11 21 31 02 12 22 32 03 13 23 33

Homomorphism: (x, y, z) → (x + z, y + z).

A perfect coloring of H(6, 2) with parameters [[1,5],[3,3]]

A perfect 2-coloring of Sh with parameter matrix [︃1 3 5 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 23 9 15 ]︃ , [︃ 2 22 10 14 ]︃ , [︃ 3 21 11 13 ]︃ , [︃ 5 19 13 11 ]︃ , [︃ 7 17 15 9 ]︃

Perfect 2-colorings of H(n, 2), bounds

[FDF,2007]

a+b gcd(a,b) is a power of 2

(the number 2nc

b+c of vertices of ҥrst color must be integer).

[FDF,2007] Correlation-immunity bound: if b ̸= c, then a − c ≥ − n

3

[FDF,2007] [︃ 1 11 5 7 ]︃ do not exist (sporadic proof) NEW! if b ̸= c and a − c = − n

3, then bc gcd(b,c)2 ≡ 0 mod 3.

Theorem (D.G. Fon-Der-Flaass, 2007) Assume that

b+c gcd(b,c) is a power of 2. Then there exists a0 such that

perfect 2-colorings of H(n, 2) with parameter matrix [︂

a c b d

]︂ , where d = a + b − c, exist for all a ≥ a0.

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)) 1-Perfect codes: [︃ 0 n 1 n − 1 ]︃ , [︃ k − 1 n − k + 1 k n − k ]︃ . Splitting: [︃ a b c d ]︃ → [︃ a − 1 2b + c c 2b + a − 1 ]︃ Special: [︃ 3 9 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: t · [︃ 0 3 1 2 ]︃ , t · [︃ 1 5 3 3 ]︃ , t · [︃ 3 9 7 5 ]︃ First questions: [︃ 1 23 9 15 ]︃ , [︃ 2 22 10 14 ]︃ , [︃ 3 21 11 13 ]︃ , [︃ 5 19 13 11 ]︃ , [︃ 7 17 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

colorings with parameter matrix [︃ 3 9 7 5 ]︃ (Fourier analysis, exact- 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-

immunity bound, either

b gcd(b,c) or c gcd(b,c) is divisible by 3.

Do not exist: [︃ 1 11 5 7 ]︃ , [︃ 2 22 10 14 ]︃ , [︃ 5 19 13 11 ]︃

Fourier transform

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

y∈V (H(n,2))

̂︁ f (y)χy(x) ̂︁ f (y) = 2−n ∑︂

x∈V (H(n,2))

χy(x)f (x)

correlation-immunity bound

Any {c, −b}-valued (b ̸= c) function satisҥes: f 2 = (c − b)f + cb = ⇒ ̂︁ f * ̂︁ f = (c − b)̂︁ f + cb · δ¯ where the convolution * is deҥned by g * h(x) = ∑︂

y,z: y+z=x

g(y)h(z) It follows that supp(̂︁ f ) ∪ {¯ 0} ⊆ supp(̂︁ f ) + supp(̂︁ f ) Hence, ̂︁ f must have a nonzero of weight at most 2n/3 (correlation-immunity bound). Moreover, if it has no nonzeros of weight less than 2n/3, then it is a perfect coloring.

On two-color edges, a − c = −n/3

̂︁ f * ̂︁ f = (c − b)̂︁ f + cb · δ¯ For x ̸= ¯ 0 we have (α := 1/(c − b)) ̂︁ f (x) = α · ̂︁ f * ̂︁ f (x) = α · ∑︂

y,z: y+z=x

̂︁ f (y)̂︁ f (z) ̂︁ f (x)2 = α · ∑︂

y,z: y+z+x=0

̂︁ f (x)̂︁ f (y)̂︁ f (z) ∑︂

x: xi=1

̂︁ f (x)2 = α · ∑︂

¯ x,¯ y,¯ z: ¯ x+¯ y+¯ z=¯ xi =1

̂︁ f (x)̂︁ f (y)̂︁ f (z) If f is a perfect coloring with attaining the c.-i. bound, then the right part does not depend on i.

—

does not depend on i: ∑︂

¯ x,¯ y,¯ z: ¯ x+¯ y+¯ z=¯ xi =1

̂︁ f (x)̂︁ f (y)̂︁ f (z) = 1 3 ⎛ ⎜ ⎜ ⎜ ⎝ ∑︂

¯ x,¯ y,¯ z: ¯ x+¯ y+¯ z=¯ xi =1

̂︁ f (x)̂︁ f (y)̂︁ f (z) + ∑︂

¯ x,¯ y,¯ z: ¯ x+¯ y+¯ z=¯ yi =1

̂︁ f (x)̂︁ f (y)̂︁ f (z) + ∑︂

¯ x,¯ y,¯ z: ¯ x+¯ y+¯ z=¯ zi =1

̂︁ f (x)̂︁ f (y)̂︁ f (z) ⎞ ⎟ ⎟ ⎟ ⎠ = 2 3 ∑︂

¯ x,¯ y,¯ z: ¯ x+¯ y+¯ z=¯

̂︁ f (x)̂︁ f (y)̂︁ f (z) because for nonzero ̂︁ f (x)̂︁ f (y)̂︁ f (z) we have wt(x) = wt(y) = wt(z) = 2

3n

—

• So. If f is a perfect coloring attaining the c.-i. bound, then

∑︂

x: xi=1

̂︁ f (x)2 does not depend on i. It follows that the norm of f (x) − f (x + ei) does not depend on i. This norm is connected with the number of two-color edges of direction i. So,we have the theorem.

Fourier transform for [[3,9],[7,5]]

Fourier transform for [︃ 3 9 7 5 ]︃ : Nonzeros of weight 8; integer values; sum of squares = 63 (so, there are at most 63 nonzeros). Every word of weight 9 = 12 − 3 covers at least one nonzero. So, we have (3, 4, 12) covering design. The size 63 is redundant, the minimum is 57 for such covering. Taking into account the Theorem, puncturing with respect to any coordinate leads to a (2, 3, 11) covering with norm 21. It is close to the minimum, and we can conclude that there ̂︁ f has no values other than 0, 1, −1. There are 16 ҡovercoveredә triples; they form 1-(12, 3, 4) design. We can ҥnd all nonisomorphic systems of triples (with some additional conditions) computationally. Then, we can use the exact-covering software to ҥnd the (3, 4, 12) covering of size 63 that overcover given triples.

Fourier transform for [[3,9],[7,5]]

If we know the support of ̂︁ f , we still need to determine the signs (+1 or −1). The signs satisfy the system of linear equations over GF(2), which comes from the convolution equation. This system can be solved computationally.

[[3,9],[7,5]] construction

[︃1.5 3.5 4.5 2.5 ]︃ → [︃3 7 9 5 ]︃ [︃1.5 3.5 4.5 2.5 ]︃ :

Problems: [[0,12],[4,8]], [[2,10],[6,6]]

Problem: can we characterize 4 · [︃ 0 3 1 2 ]︃ = [︃ 0 12 4 8 ]︃

• r 2 ·

[︃ 1 5 3 3 ]︃ = [︃ 2 10 6 6 ]︃ ? The Fourier transform is integer and has norm 48 or 60, respectively. However, there is no 3-covering property and in general the values other than 0, ±1 can occur.

Problems: [[0,12],[4,8]], [[2,10],[6,6]]

Problem: can we ҥnd [︃ 1 23 9 15 ]︃ ? The Fourier transform is integer, has norm 207, and gives a (4,8,24) covering design. The bounds for such design are 189 ≤ ?? ≤ 226, see https://www.ccrwest.org/cover/show_cover.php?v=24k=8t=4 Can we use additional properties (convolution equation) to prove nonexistence? Another parameters: [︃ 3 21 11 13 ]︃ , [︃ 7 17 15 9 ]︃ ?

Problems: t[[0, 3], [1, 2]]

Problem: can we characterize t · [︃ 0 3 1 2 ]︃ for any t? The colorings with these parameters in the Hamming H(t, 4) graph and Doob graph correspond to the maximum independent

• sets. We have a characterization theorem in those cases