[PPT] - New upper bounds for nonbinary codes based on semidefinite PowerPoint Presentation

SLIDE 1

New upper bounds for nonbinary codes based on semidefinite programming and parity

Sven Polak

Partly based on joint work with Bart Litjens and Lex Schrijver

Korteweg-de Vries Institute for Mathematics Faculty of Science University of Amsterdam

Plzeˇ n, October 6th, 2016

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 2

Definitions and notation

Fix q, n, d ∈ N with q ≥ 2. Define [q] := {0, . . . , q − 1}.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 3

Definitions and notation

Fix q, n, d ∈ N with q ≥ 2. Define [q] := {0, . . . , q − 1}. A word is an element of [q]n and a code is a subset of [q]n.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 4

Definitions and notation

Fix q, n, d ∈ N with q ≥ 2. Define [q] := {0, . . . , q − 1}. A word is an element of [q]n and a code is a subset of [q]n. The Hamming distance between two words u, v ∈ [q]n is dH(u, v) := |{i : ui = vi}|.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 5

Definitions and notation

Fix q, n, d ∈ N with q ≥ 2. Define [q] := {0, . . . , q − 1}. A word is an element of [q]n and a code is a subset of [q]n. The Hamming distance between two words u, v ∈ [q]n is dH(u, v) := |{i : ui = vi}|. The minimum distance dmin(C) of a code C ⊆ [q]n is the minimum

f dH(u, v) over all distinct u, v ∈ C.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 6

Definitions and notation

Fix q, n, d ∈ N with q ≥ 2. Define [q] := {0, . . . , q − 1}. A word is an element of [q]n and a code is a subset of [q]n. The Hamming distance between two words u, v ∈ [q]n is dH(u, v) := |{i : ui = vi}|. The minimum distance dmin(C) of a code C ⊆ [q]n is the minimum

f dH(u, v) over all distinct u, v ∈ C.

Example

(i) dmin({1112, 2111, 3134}) = 2,

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 7

Definitions and notation

Fix q, n, d ∈ N with q ≥ 2. Define [q] := {0, . . . , q − 1}. A word is an element of [q]n and a code is a subset of [q]n. The Hamming distance between two words u, v ∈ [q]n is dH(u, v) := |{i : ui = vi}|. The minimum distance dmin(C) of a code C ⊆ [q]n is the minimum

f dH(u, v) over all distinct u, v ∈ C.

Example

(i) dmin({1112, 2111, 3134}) = 2,

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 8

The parameter Aq(n, d)

Definition

Aq(n, d) := max{|C| | C ⊆ [q]n, dmin(C) ≥ d}.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 9

The parameter Aq(n, d)

Definition

Aq(n, d) := max{|C| | C ⊆ [q]n, dmin(C) ≥ d}.

Examples

Aq(n, 1) = qn.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 10

The parameter Aq(n, d)

Definition

Aq(n, d) := max{|C| | C ⊆ [q]n, dmin(C) ≥ d}.

Examples

Aq(n, 1) = qn. A5(7, 6) = 15.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 11

The parameter Aq(n, d)

Definition

Aq(n, d) := max{|C| | C ⊆ [q]n, dmin(C) ≥ d}.

Examples

Aq(n, 1) = qn. A5(7, 6) = 15. (i) Tables with bounds on Aq(n, d) on the website of Andries Brouwer.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 12

The parameter Aq(n, d)

Definition

Aq(n, d) := max{|C| | C ⊆ [q]n, dmin(C) ≥ d}.

Examples

Aq(n, 1) = qn. A5(7, 6) = 15. (i) Tables with bounds on Aq(n, d) on the website of Andries Brouwer. (ii) Interesting parameter in cryptography: a code C ⊆ [q]n with dmin(C) = 2e + 1 is e-error correcting.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 13

The parameter Aq(n, d) – II

Definition

Aq(n, d) := max{|C| | C ⊆ [q]n, dmin(C) ≥ d}.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 14

The parameter Aq(n, d) – II

Definition

Aq(n, d) := max{|C| | C ⊆ [q]n, dmin(C) ≥ d}.

Remark

Let G = (V , E) be the graph with V = [q]n and E := {{u, v} | 0 < dH(u, v) < d}.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 15

The parameter Aq(n, d) – II

Definition

Aq(n, d) := max{|C| | C ⊆ [q]n, dmin(C) ≥ d}.

Remark

Let G = (V , E) be the graph with V = [q]n and E := {{u, v} | 0 < dH(u, v) < d}. Then Aq(n, d) = α(G), the stable set number of G.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 16

The parameter Aq(n, d) – II

Definition

Aq(n, d) := max{|C| | C ⊆ [q]n, dmin(C) ≥ d}.

Remark

Let G = (V , E) be the graph with V = [q]n and E := {{u, v} | 0 < dH(u, v) < d}. Then Aq(n, d) = α(G), the stable set number of G.

111 100 110 011 010 101 001 000

n = 3, d = 2 Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 17

The parameter Aq(n, d) – II

Definition

Aq(n, d) := max{|C| | C ⊆ [q]n, dmin(C) ≥ d}.

Remark

Let G = (V , E) be the graph with V = [q]n and E := {{u, v} | 0 < dH(u, v) < d}. Then Aq(n, d) = α(G), the stable set number of G.

111 100 110 011 010 101 001 000

n = 3, d = 2 Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 18

The parameter Aq(n, d) – II

Definition

Aq(n, d) := max{|C| | C ⊆ [q]n, dmin(C) ≥ d}.

Remark

Let G = (V , E) be the graph with V = [q]n and E := {{u, v} | 0 < dH(u, v) < d}. Then Aq(n, d) = α(G), the stable set number of G.

111 100 110 011 010 101 001 000

n = 3, d = 2 Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 19

The parameter Aq(n, d) – II

Definition

Aq(n, d) := max{|C| | C ⊆ [q]n, dmin(C) ≥ d}.

Remark

Let G = (V , E) be the graph with V = [q]n and E := {{u, v} | 0 < dH(u, v) < d}. Then Aq(n, d) = α(G), the stable set number of G.

111 100 110 011 010 101 001

n = 3, d = 2

000

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 20

Delsarte bound

Notation

Let Ck be the collection of codes C ⊆ [q]n with |C| ≤ k. Given x : C2 → R≥0, define the C1 × C1-matrix Mx by (Mx)C,C ′ = x(C ∪ C ′).

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 21

Delsarte bound

Notation

Let Ck be the collection of codes C ⊆ [q]n with |C| ≤ k. Given x : C2 → R≥0, define the C1 × C1-matrix Mx by (Mx)C,C ′ = x(C ∪ C ′). It can be proven that the Delsarte bound equals θq(n, d) := max

v∈[q]n

x({v}) | x : C2 → R≥0 with: (i) x(∅) = 1, (ii) x(C) = 0 if dmin(C) < d, (iii) Mx is positive semidefinite

.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 22

The quadruple bound

Notation

Let Ck be the collection of codes C ⊆ [q]n with |C| ≤ k. Given x : C4 → R≥0, define the C2 × C2-matrix Mx by (Mx)C,C ′ = x(C ∪ C ′).

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 23

The quadruple bound

Notation

Let Ck be the collection of codes C ⊆ [q]n with |C| ≤ k. Given x : C4 → R≥0, define the C2 × C2-matrix Mx by (Mx)C,C ′ = x(C ∪ C ′). Now we define

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 24

The quadruple bound

Notation

Let Ck be the collection of codes C ⊆ [q]n with |C| ≤ k. Given x : C4 → R≥0, define the C2 × C2-matrix Mx by (Mx)C,C ′ = x(C ∪ C ′). Now we define Bq(n, d) := max

v∈[q]n

x({v}) | x : C4 → R≥0 with: (i) x(∅) = 1, (ii) x(C) = 0 if dmin(C) < d, (iii) Mx is positive semidefinite

.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 25

The quadruple bound

Bq(n, d) := max

v∈[q]n

x({v}) | x : C4 → R≥0 with: (i) x(∅) = 1, (ii) x(C) = 0 if dmin(C) < d, (iii) Mx is positive semidefinite

.
Proposition. Aq(n, d) ≤ Bq(n, d)
Proof. Let C ⊆ [q]n be a code of minimum distance at least d

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 26

The quadruple bound

Bq(n, d) := max

v∈[q]n

x({v}) | x : C4 → R≥0 with: (i) x(∅) = 1, (ii) x(C) = 0 if dmin(C) < d, (iii) Mx is positive semidefinite

.
Proposition. Aq(n, d) ≤ Bq(n, d)
Proof. Let C ⊆ [q]n be a code of minimum distance at least d and

maximum size.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 27

The quadruple bound

Bq(n, d) := max

v∈[q]n

x({v}) | x : C4 → R≥0 with: (i) x(∅) = 1, (ii) x(C) = 0 if dmin(C) < d, (iii) Mx is positive semidefinite

.
Proposition. Aq(n, d) ≤ Bq(n, d)
Proof. Let C ⊆ [q]n be a code of minimum distance at least d and

maximum size. Define x by x(S) = 1 if S ⊆ C and x(S) = 0 else.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 28

The quadruple bound

Bq(n, d) := max

v∈[q]n

x({v}) | x : C4 → R≥0 with: (i) x(∅) = 1, (ii) x(C) = 0 if dmin(C) < d, (iii) Mx is positive semidefinite

.
Proposition. Aq(n, d) ≤ Bq(n, d)
Proof. Let C ⊆ [q]n be a code of minimum distance at least d and

maximum size. Define x by x(S) = 1 if S ⊆ C and x(S) = 0 else. Then x is feasible.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 29

The quadruple bound

Bq(n, d) := max

v∈[q]n

x({v}) | x : C4 → R≥0 with: (i) x(∅) = 1, (ii) x(C) = 0 if dmin(C) < d, (iii) Mx is positive semidefinite

.
Proposition. Aq(n, d) ≤ Bq(n, d)
Proof. Let C ⊆ [q]n be a code of minimum distance at least d and

maximum size. Define x by x(S) = 1 if S ⊆ C and x(S) = 0 else. Then x is feasible. Moreover,

u∈[q]n

x({u}) = |C| = Aq(n, d), which yields the proposition.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 30

The quadruple bound

Bq(n, d) := max

v∈[q]n

x({v}) | x : C4 → R≥0 with: (i) x(∅) = 1, (ii) x(C) = 0 if dmin(C) < d, (iii) Mx is positive semidefinite

.
Proposition. Aq(n, d) ≤ Bq(n, d)
Proof. Let C ⊆ [q]n be a code of minimum distance at least d and

maximum size. Define x by x(S) = 1 if S ⊆ C and x(S) = 0 else. Then x is feasible. Moreover,

u∈[q]n

x({u}) = |C| = Aq(n, d), which yields the proposition.

Sven Polak

Code bounds Plzeˇ n, October 6th, 2016

SLIDE 31

Results [L., P. and Schrijver, 2016]

Table: New upper bounds on Aq(n, d)

q n d Best lower bound known New upper bound Best upper bound previously known 4 6 3 164 176 179 4 7 3 512 596 614 4 7 4 128 155 169 5 7 4 250 489 545 5 7 5 53 87 108

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 32

A5(8, 6) = 75?

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 33

A5(7, 6) = 15.

Consider a (q, n, d) = (5, 7, 6)-code C of size 15.

                        1 1 1 1 1 1 2 2 2 2 2 2 1 2 1 3 3 3 1 1 2 4 4 4 1 3 3 3 2 1 2 4 3 2 1 4 2 1 3 4 3 2 2 4 1 4 2 3 3 2 1 4 3 4 3 3 4 2 1 3 3 4 3 1 2 4 4 2 4 3 4 1 4 3 2 4 4 1 4 4 3 1 3 2                         Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 34

A5(7, 6) = 15.

Consider a (q, n, d) = (5, 7, 6)-code C of size 15.

                        1 1 1 1 1 1 2 2 2 2 2 2 1 2 1 3 3 3 1 1 2 4 4 4 1 3 3 3 2 1 2 4 3 2 1 4 2 1 3 4 3 2 2 4 1 4 2 3 3 2 1 4 3 4 3 3 4 2 1 3 3 4 3 1 2 4 4 2 4 3 4 1 4 3 2 4 4 1 4 4 3 1 3 2                        

For u, v ∈ C, write g(u, v) = 7 − dH(u, v).

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 35

A5(7, 6) = 15.

Consider a (q, n, d) = (5, 7, 6)-code C of size 15.

                        1 1 1 1 1 1 2 2 2 2 2 2 1 2 1 3 3 3 1 1 2 4 4 4 1 3 3 3 2 1 2 4 3 2 1 4 2 1 3 4 3 2 2 4 1 4 2 3 3 2 1 4 3 4 3 3 4 2 1 3 3 4 3 1 2 4 4 2 4 3 4 1 4 3 2 4 4 1 4 4 3 1 3 2                        

For u, v ∈ C, write g(u, v) = 7 − dH(u, v). Let cα,j denote the number of times symbol α

ccurs in column j.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 36

A5(7, 6) = 15.

Consider a (q, n, d) = (5, 7, 6)-code C of size 15.

                        1 1 1 1 1 1 2 2 2 2 2 2 1 2 1 3 3 3 1 1 2 4 4 4 1 3 3 3 2 1 2 4 3 2 1 4 2 1 3 4 3 2 2 4 1 4 2 3 3 2 1 4 3 4 3 3 4 2 1 3 3 4 3 1 2 4 4 2 4 3 4 1 4 3 2 4 4 1 4 4 3 1 3 2                        

For u, v ∈ C, write g(u, v) = 7 − dH(u, v). Let cα,j denote the number of times symbol α

ccurs in column j.

Then 15 2

≥
{u,v}⊆C

u=v

g(u, v) =

7

j=1
α∈[5]

cα,j 2

≥ 7·5·

3 2

.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 37

A5(7, 6) = 15.

Consider a (q, n, d) = (5, 7, 6)-code C of size 15.

                        1 1 1 1 1 1 2 2 2 2 2 2 1 2 1 3 3 3 1 1 2 4 4 4 1 3 3 3 2 1 2 4 3 2 1 4 2 1 3 4 3 2 2 4 1 4 2 3 3 2 1 4 3 4 3 3 4 2 1 3 3 4 3 1 2 4 4 2 4 3 4 1 4 3 2 4 4 1 4 4 3 1 3 2                        

For u, v ∈ C, write g(u, v) = 7 − dH(u, v). Let cα,j denote the number of times symbol α

ccurs in column j.

Then 15 2

≥
{u,v}⊆C

u=v

g(u, v) =

7

j=1
α∈[5]

cα,j 2

≥ 7·5·

3 2

.

Important observations

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 38

A5(7, 6) = 15.

Consider a (q, n, d) = (5, 7, 6)-code C of size 15.

                        1 1 1 1 1 1 2 2 2 2 2 2 1 2 1 3 3 3 1 1 2 4 4 4 1 3 3 3 2 1 2 4 3 2 1 4 2 1 3 4 3 2 2 4 1 4 2 3 3 2 1 4 3 4 3 3 4 2 1 3 3 4 3 1 2 4 4 2 4 3 4 1 4 3 2 4 4 1 4 4 3 1 3 2                        

For u, v ∈ C, write g(u, v) = 7 − dH(u, v). Let cα,j denote the number of times symbol α

ccurs in column j.

Then 15 2

≥
{u,v}⊆C

u=v

g(u, v) =

7

j=1
α∈[5]

cα,j 2

≥ 7·5·

3 2

.

Important observations

g(u, v) = 1 for all u = v ∈ C. cα,j = 3 for all α ∈ [5], j = 1, . . . , 7.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 39

A5(7, 6) = 15. Kirkman’s School Girl Problem (1850)

Kirkman’s School Girl Problem (1850)

“Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk twice abreast.”

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 40

A5(7, 6) = 15. Kirkman’s School Girl Problem (1850)

Kirkman’s School Girl Problem (1850)

“Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk twice abreast.” Correspondence between solutions and (5, 7, 6)-codes C of size 15. girls i1 and i2 walk in the same triple on day j ⇐ ⇒ Ci1,j = Ci2,j.

                        1 1 1 1 1 1 2 2 2 2 2 2 1 2 1 3 3 3 1 1 2 4 4 4 1 3 3 3 2 1 2 4 3 2 1 4 2 1 3 4 3 2 2 4 1 4 2 3 3 2 1 4 3 4 3 3 4 2 1 3 3 4 3 1 2 4 4 2 4 3 4 1 4 3 2 4 4 1 4 4 3 1 3 2                         Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 41

A5(8, 6) = 75?

Can we combine 5 Kirkman systems to get a (5, 8, 6)-code of size 75?

1 2 2 2 2 2 1 2 1 3 3 3 . . . 1 3 3 3 2 1 1 1 1 1 1 1 1 1 1 . . . 1 2 . . . 2 3 . . . 4 .

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 42

A5(8, 6) = 75?

Can we combine 5 Kirkman systems to get a (5, 8, 6)-code of size 75?

1 2 2 2 2 2 1 2 1 3 3 3 . . . 1 3 3 3 2 1 1 1 1 1 1 1 1 1 1 . . . 1 2 . . . 2 3 . . . 4 .

Observation

As g(u, v) = 1 for all u = v in a Kirkman system, all distances in a (5, 8, 6)-code C of size 75 are even.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 43

A5(8, 6) = 75?

Can we combine 5 Kirkman systems to get a (5, 8, 6)-code of size 75?

1 2 2 2 2 2 1 2 1 3 3 3 . . . 1 3 3 3 2 1 1 1 1 1 1 1 1 1 1 . . . 1 2 . . . 2 3 . . . 4 .

Observation

As g(u, v) = 1 for all u = v in a Kirkman system, all distances in a (5, 8, 6)-code C of size 75 are even.

Parity argument

Write [0|K] for the first 15 rows of C (starting with 0). Assume 1 ∈ C.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 44

A5(8, 6) = 75?

Can we combine 5 Kirkman systems to get a (5, 8, 6)-code of size 75?

1 2 2 2 2 2 1 2 1 3 3 3 . . . 1 3 3 3 2 1 1 1 1 1 1 1 1 1 1 . . . 1 2 . . . 2 3 . . . 4 .

Observation

As g(u, v) = 1 for all u = v in a Kirkman system, all distances in a (5, 8, 6)-code C of size 75 are even.

Parity argument

Write [0|K] for the first 15 rows of C (starting with 0). Assume 1 ∈ C. Then K contains each symbol 7 · 3 times (odd).

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 45

A5(8, 6) = 75?

Can we combine 5 Kirkman systems to get a (5, 8, 6)-code of size 75?

1 2 2 2 2 2 1 2 1 3 3 3 . . . 1 3 3 3 2 1 1 1 1 1 1 1 1 1 1 . . . 1 2 . . . 2 3 . . . 4 .

Observation

As g(u, v) = 1 for all u = v in a Kirkman system, all distances in a (5, 8, 6)-code C of size 75 are even.

Parity argument

Write [0|K] for the first 15 rows of C (starting with 0). Assume 1 ∈ C. Then K contains each symbol 7 · 3 times (odd). Hence there is a word u ∈ [0|K] with dH(u, 1) odd.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 46

A5(8, 6) = 75?

Can we combine 5 Kirkman systems to get a (5, 8, 6)-code of size 75?

1 2 2 2 2 2 1 2 1 3 3 3 . . . 1 3 3 3 2 1 1 1 1 1 1 1 1 1 1 . . . 1 2 . . . 2 3 . . . 4 .

Observation

As g(u, v) = 1 for all u = v in a Kirkman system, all distances in a (5, 8, 6)-code C of size 75 are even.

Parity argument

Write [0|K] for the first 15 rows of C (starting with 0). Assume 1 ∈ C. Then K contains each symbol 7 · 3 times (odd). Hence there is a word u ∈ [0|K] with dH(u, 1) odd. Therefore A5(8, 6) < 75.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 47

A5(8, 6) = 75?

Can we combine 5 Kirkman systems to get a (5, 8, 6)-code of size 75?

1 2 2 2 2 2 1 2 1 3 3 3 . . . 1 3 3 3 2 1 1 1 1 1 1 1 1 1 1 . . . 1 2 . . . 2 3 . . . 4 .

Observation

As g(u, v) = 1 for all u = v in a Kirkman system, all distances in a (5, 8, 6)-code C of size 75 are even.

Parity argument

Write [0|K] for the first 15 rows of C (starting with 0). Assume 1 ∈ C. Then K contains each symbol 7 · 3 times (odd). Hence there is a word u ∈ [0|K] with dH(u, 1) odd. Therefore A5(8, 6) < 75.

Sven Polak

Code bounds Plzeˇ n, October 6th, 2016

SLIDE 48

A5(8, 6) = 75?

Result [P., 2016]

A5(8, 6) ≤ 65, by exploiting the parity argument further.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 49

A5(8, 6) = 75?

Result [P., 2016]

A5(8, 6) ≤ 65, by exploiting the parity argument further. Also new upper bounds on Aq(n, d) for other (q, n, d) were obtained.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 50

Further research

(i) Constant weight codes: with a bound in between k = 3 and k = 4 nice results can be obtained.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 51

Further research

(i) Constant weight codes: with a bound in between k = 3 and k = 4 nice results can be obtained. (ii) Use the output of the dual program (which variables are forbidden?) to prove uniqueness of codes.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 52

Further research

(i) Constant weight codes: with a bound in between k = 3 and k = 4 nice results can be obtained. (ii) Use the output of the dual program (which variables are forbidden?) to prove uniqueness of codes. (iii) Binary codes: add small constraints (Kim, Toan 2013), or consider larger semidefinite programs.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016

SLIDE 53

Further research

(i) Constant weight codes: with a bound in between k = 3 and k = 4 nice results can be obtained. (ii) Use the output of the dual program (which variables are forbidden?) to prove uniqueness of codes. (iii) Binary codes: add small constraints (Kim, Toan 2013), or consider larger semidefinite programs.

Sven Polak Code bounds Plzeˇ n, October 6th, 2016