Introduction
Results
On Field Size and Success Probability
in Network Coding
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen
Workshop on Groebner bases and geometric codes,
held at Dept. of Mathematics, University of Trento
June 10, 2009
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
The Usual Example
u
• Information network;
DAG.
v1
v2
• Unit capacity edges
• Two unit source
w1
v3
processes over F2m
located at the source
vertex u.
v4
• Both receivers w1 and
w2
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
w2 wants information
from both source
processes.
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
The Usual Example
X (1) u X (2)
• Information network;
DAG.
v1
v2
• Unit capacity edges
• Two unit source
w1
v3
processes over F2m
located at the source
vertex u.
v4
• Both receivers w1 and
w2
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
w2 wants information
from both source
processes.
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
The Usual Example
X (1) u X (2)
• Information network;
DAG.
v1
v2
• Unit capacity edges
• Two unit source
w1
v3
processes over F2m
located at the source
vertex u.
v4
• Both receivers w1 and
w2
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
w2 wants information
from both source
processes.
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Using Switching
a X (1) u X (2) b
1
2
v1
v2
3
4
5
v3
v4
w1
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
w2
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Using Switching
1
X (1) u X (2)
2
a
b
v1
v2
3
4
5
v3
v4
w1
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
w2
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Using Switching
1
X (1) u X (2)
2
v1
a
b
v2
4
5
v3
a
3
b
v4
w1
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
w2
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Using Switching
1
X (1) u X (2)
2
v1
v2
b
v3
3
4
5
a
v4
w1
a
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
w2
b
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Using Switching
1
X (1) u X (2)
2
v1
v2
3
4
5
v3
b
v4
a
w1
a
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
w2
b
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Using Switching
1
X (1) u X (2)
2
v1
v2
3
4
5
v3
v4
b
w1
a
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
w2
b
a
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Using Switching
1
X (1) u X (2)
2
v1
v2
3
4
5
v3
v4
w1
a
b
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
w2
b
a
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Using Network Coding
a X (1) u X (2) b
1
2
v1
v2
3
4
v3
v4
w1
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
w2
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Using Network Coding
1
X (1) u X (2)
2
a
b
v1
v2
3
4
v3
v4
w1
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
w2
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Using Network Coding
1
X (1) u X (2)
2
v1
a
b
v2
3
4
v3
a
b
v4
w1
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
w2
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Using Network Coding
1
X (1) u X (2)
2
v1
v2
3
4
v3
a+b
v4
w1
a
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
w2
b
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Using Network Coding
1
X (1) u X (2)
2
v1
v2
3
4
v3
v4
a+b
w1
a
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
w2
b
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Using Network Coding
1
X (1) u X (2)
2
v1
v2
3
4
v3
v4
w1
a
a+b
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
w2
b
a+b
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
What About “Decoding”?
wi must know how a and b is linearly combined.
a
1
=a+b
b
1
• Local coding vector
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
What About “Decoding”?
wi must know how a and b is linearly combined.
a
1
=a+b
b
1
• Local coding vector
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Formalizing . . .
u
v1
v2
v3
v4
w1
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
w2
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Formalizing . . .
u
i
fi,j
v1
v2
j
v3
v4
w1
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
w2
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Formalizing . . .
u
i
fi,j
1
2
3
4
v1
j
v2
v3
5
6
7
v4
8
w1
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
9
w2
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Formalizing . . .
u
1
i
fi,j
j
2
f1,5 v1 f1,3
3
f2,4 v2 f2,7
4
f3,6 v3 f4,6
5
6
7
f6,8 v4 f6,9
8
w1
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
9
w2
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Formalizing . . .
u
i
fi,j
1
2
3
4
v1
j
v2
v3
5
6
X (i) u
j
7
v4
ai,j
8
w1
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
9
w2
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Formalizing . . .
u
a1,2
a1,1
2 a2,2
a2,1 1
i
fi,j
v1
v2
3
j
4
v3
5
6
X (i) u
j
7
v4
ai,j
8
w1
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
9
w2
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Formalizing . . .
X (i) u
u
1
2
3
4
v1
v2
(w )
bi,j 1
v3
5
6
j
v4
w1
8
w1
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
7
9
w2
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Formalizing . . .
X (i) u
u
1
2
3
4
v1
v2
(w )
bi,j 1
v3
5
j
7
v4
w1
(w )
b1,51
(w )
b2,51
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
6
w1
8 9
(w )
b1,81
(w )
b2,81
w2
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Formalizing . . .
X (i) u
u
1
2
3
4
v1
v2
(w )
bi,j 1
v3
5
6
j
v4
w1
8
w1
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
7
9
(w )
b1,91
(w )
b2,91
(w )
w2 b1,71
(w )
b2,71
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Formalizing . . . (contd)
u
• Y (1) = a1,1 X (1) + a2,1 X (2)
• Y (3) = f1,3 Y (1)
• Y (6) = f3,6 Y (3) + f4,6 Y (4)
In general:
X
X
Y (j) =
ai,j X (u) +
fi,j Y (i)
u
(w )
2
3
4
v1
v2
v3
5
6
i
=
8
X
(w )
bi,j Y (j)
7
v4
“Decoding”:
bi
1
w1
9
w2
j
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Formalizing . . . (contd)
u
• Y (1) = a1,1 X (1) + a2,1 X (2)
• Y (3) = f1,3 Y (1)
• Y (6) = f3,6 Y (3) + f4,6 Y (4)
In general:
X
X
Y (j) =
ai,j X (u) +
fi,j Y (i)
u
(w )
2
3
4
v1
v2
v3
5
6
i
=
8
X
(w )
bi,j Y (j)
7
v4
“Decoding”:
bi
1
w1
9
w2
j
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Formalizing . . . (contd)
u
• Y (1) = a1,1 X (1) + a2,1 X (2)
• Y (3) = f1,3 Y (1)
• Y (6) = f3,6 Y (3) + f4,6 Y (4)
In general:
X
X
ai,j X (u) +
fi,j Y (i)
Y (j) =
u
(w )
2
3
4
v1
v2
v3
5
6
i
=
8
X
(w )
bi,j Y (j)
7
v4
“Decoding”:
bi
1
w1
9
w2
j
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Formalizing . . . (contd)
u
• Y (1) = a1,1 X (1) + a2,1 X (2)
• Y (3) = f1,3 Y (1)
• Y (6) = f3,6 Y (3) + f4,6 Y (4)
In general:
X
X
Y (j) =
ai,j X (u) +
fi,j Y (i)
u
(w )
2
3
4
v1
v2
v3
5
6
i
=
8
X
(w )
bi,j Y (j)
7
v4
“Decoding”:
bi
1
w1
9
w2
j
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Formalizing . . . (contd)
u
• Y (1) = a1,1 X (1) + a2,1 X (2)
• Y (3) = f1,3 Y (1)
• Y (6) = f3,6 Y (3) + f4,6 Y (4)
In general:
X
X
Y (j) =
ai,j X (u) +
fi,j Y (i)
u
(w )
2
3
4
v1
v2
v3
5
6
i
=
8
X
(w )
bi,j Y (j)
7
v4
“Decoding”:
bi
1
w1
9
w2
j
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Success Criterion
Theorem
#
(ai,j )
0
Mw :=
;
(w )
I − (fi,j ) (bi,j )T
|Mw | =
6 0 in
"
F2
⇐⇒
m
Receiver w can decode.
(Koetter et al. (2003) and Ho et al. (2006))
P :=
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
Q
w
|Mw | =
6 0 in
F2
m
?
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Success Criterion
Theorem
#
(ai,j )
0
Mw :=
;
(w )
I − (fi,j ) (bi,j )T
|Mw | =
6 0 in
"
F2
⇐⇒
m
Receiver w can decode.
(Koetter et al. (2003) and Ho et al. (2006))
P :=
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
Q
w
|Mw | =
6 0 in
F2
m
?
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
Results
1. Minimum field size of given characteristic. (Imply
that finding the polynomial is NP-complete.)
2. Improving a bound for random network coding.
3. Topological interpretation of |Mw |.
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
Minimum field size of given
characteristic
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
What Is known
(w )
Choose ai,j ’s, fi,j ’s and bi,j ’s from a finite field.
Field size ≥ no. of receivers
(Koetter et al. (2003))
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
• Utilizing the fact that for
F (X1 , . . . , Xn ) ∈ Fq [X1 , . . . , Xn ]
then there exists
(x1 , . . . , xn ) ∈ Fnq
such that
F (x1 , . . . , xn ) 6= 0
if and only if
F (X1 , . . . , Xn ) rem (X1q − X1 , . . . , Xnq − Xn ) 6= 0.
• Reduce P modulo
(w )
(w ) q
ai,j
− ai,j , . . . , fi,jq − fi,j , . . . , (bi,j )q − bi,j
.
(Fast!)
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
• Utilizing the fact that for
F (X1 , . . . , Xn ) ∈ Fq [X1 , . . . , Xn ]
then there exists
(x1 , . . . , xn ) ∈ Fnq
such that
F (x1 , . . . , xn ) 6= 0
if and only if
F (X1 , . . . , Xn ) rem (X1q − X1 , . . . , Xnq − Xn ) 6= 0.
• Reduce P modulo
(w )
(w ) q
ai,j
− ai,j , . . . , fi,jq − fi,j , . . . , (bi,j )q − bi,j
.
(Fast!)
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
• Utilizing the fact that for
F (X1 , . . . , Xn ) ∈ Fq [X1 , . . . , Xn ]
then there exists
(x1 , . . . , xn ) ∈ Fnq
such that
F (x1 , . . . , xn ) 6= 0
if and only if
F (X1 , . . . , Xn ) rem (X1q − X1 , . . . , Xnq − Xn ) 6= 0.
• Reduce P modulo
(w )
) (w
q
q
ai,j
− ai,j , . . . , fi,jq − fi,j , . . . , (b
i,j) − bi,j
.
(Fast!)
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
The Method
• For characteristic p: If P rem p reduces to 0 modulo
the ai,j ’s and fi,j ’s in Fp , then try Fp2 . Etc.
• Max logp (no. receivers) trials.
• Easy to find the ai,j ’s and fi,j ’s afterwards:
F
• q large enough.
• F ∈ q (X1 , . . . , Xn−1 )[Xn ].
• Choose Xn = zn s.t. F is nonzero; possible because
F
degXn (F ) ≤ q − 1.
• Find F (X1 , . . . , Xn−1 , zn ) ∈
• Continue.
Fq [X1 , . . . , Xn−1 ].
• Quite fast. (Finding P is NP-complete.)
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
The Method
• For characteristic p: If P rem p reduces to 0 modulo
the ai,j ’s and fi,j ’s in Fp , then try Fp2 . Etc.
• Max logp (no. receivers) trials.
• Easy to find the ai,j ’s and fi,j ’s afterwards:
F
• q large enough.
• F ∈ q (X1 , . . . , Xn−1 )[Xn ].
• Choose Xn = zn s.t. F is nonzero; possible because
F
degXn (F ) ≤ q − 1.
• Find F (X1 , . . . , Xn−1 , zn ) ∈
• Continue.
Fq [X1 , . . . , Xn−1 ].
• Quite fast. (Finding P is NP-complete.)
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
The Method
• For characteristic p: If P rem p reduces to 0 modulo
the ai,j ’s and fi,j ’s in Fp , then try Fp2 . Etc.
• Max logp (no. receivers) trials.
• Easy to find the ai,j ’s and fi,j ’s afterwards:
F
• q large enough.
• F ∈ q (X1 , . . . , Xn−1 )[Xn ].
• Choose Xn = zn s.t. F is nonzero; possible because
F
degXn (F ) ≤ q − 1.
• Find F (X1 , . . . , Xn−1 , zn ) ∈
• Continue.
Fq [X1 , . . . , Xn−1 ].
• Quite fast. (Finding P is NP-complete.)
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
Improving a bound for random
network coding
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
Introducing Randomized Network Coding
Choose the ai,j ’s and fi,j ’s uniformly i.i.d.
(w )
• . . . then hope that you can choose the bi,j ’s such
that P is nonzero.
• Success probability.
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
Introducing Randomized Network Coding
Choose the ai,j ’s and fi,j ’s uniformly i.i.d.
(w )
• . . . then hope that you can choose the bi,j ’s such
that P is nonzero.
• Success probability.
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
Introducing Randomized Network Coding
Choose the ai,j ’s and fi,j ’s uniformly i.i.d.
(w )
• . . . then hope that you can choose the bi,j ’s such
that P is nonzero.
• Success probability.
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
What Is Known
q The field size.
d The number of receivers.
µ The number of randomly chosen variables.
η The number of edges j where Y (j) depends
on randomly chosen ai,j ’s and fi,j ’s.
Success probability ≥
(q − d)η
qη
(Ho et al. (2006))
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
What Is Known
q The field size.
d The number of receivers.
µ The number of randomly chosen variables.
η The number of edges j where Y (j) depends
on randomly chosen ai,j ’s and fi,j ’s.
Success probability ≥
(q − d)η
qη
(Ho et al. (2006))
Success probability ≥
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
(q − 1)η
qη
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
What Is Known
q The field size.
d The number of receivers.
µ The number of randomly chosen variables.
η The number of edges j where Y (j) depends
on randomly chosen ai,j ’s and fi,j ’s.
Success probability ≥
(q − d)η
qη
(Ho et al. (2006))
(
(
η
(q(−(1)
((
(
Success
(((( ≥
(probability
(((
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
qη
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
Our bound
Fq [X1 , . . . , Xµ ] the remainder of P modulo
X1q − X1 , . . . , Xµq − Xµ .
b = X j1 · · · Xµjµ . (Any monomial
• Assume lm(P)
1
ordering.)
b∈
• P
Success probability ≥
since
(q−j1 )···(q−jµ )
qµ
≥
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
(q − j1 ) · · · (q − jµ )
qµ
q−(j1 +···+jµ )
.
q
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
Our bound
Fq [X1 , . . . , Xµ ] the remainder of P modulo
X1q − X1 , . . . , Xµq − Xµ .
b = X j1 · · · Xµjµ . (Any monomial
• Assume lm(P)
1
ordering.)
b∈
• P
Success probability ≥
since
(q−j1 )···(q−jµ )
qµ
≥
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
(q − j1 ) · · · (q − jµ )
qµ
q−(j1 +···+jµ )
.
q
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
Our bound
Fq [X1 , . . . , Xµ ] the remainder of P modulo
X1q − X1 , . . . , Xµq − Xµ .
b = X j1 · · · Xµjµ . (Any monomial
• Assume lm(P)
1
ordering.)
b∈
• P
Success probability ≥
since
(q−j1 )···(q−jµ )
qµ
≥
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
(q − j1 ) · · · (q − jµ )
(q − d)η
≥
qµ
qη
q−(j1 +···+jµ )
.
q
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
Improvement?
u
v1
w1
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
v2
v3
v4
v5
v6
v6
v7
v8
w2
w3
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
Improvement?
X (2)
X (1) u X (3)
v1
w1
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
v2
v3
v4
v5
v6
v6
v7
v8
w2
w3
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
Improvement?
X (2)
X (1) u X (3)
v1
w1
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
v2
v3
v4
v5
v6
v6
v7
v8
w2
w3
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
Improvement? (contd)
q
Ptrue (q)
Pnew (q)
PHo (q)
2
0.039
0.610 · 10−4
0
4
0.252
0.133
0.156 · 10−1
8
0.521
0.392
0.244
16
0.768
0.636
0.536
32
?
0.800
0.744
0.8
0.7
Field size
0.6
0.5
0.4
0.3
0.2
PTrue
Pnew
PHo
0.1
0
0
5
10
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
15
20
Success probability
25
30
35
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
Topological Interpretation of
|Mw |
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
u
v1
w1
v2
v3
v4
v5
v6
v6
v7
v8
w2
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
w3
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
u
1
v1
5
2
7
4
6
16
w1
v4
13
3
v2
v3
10
9
8
11
v5
v6
14
15 12
v6
v7
v8
22
19
17
20
21 w
18
w2
3
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
|Mw1 | = Q1
u
1
v1
5
2
7
4
6
16
w1
v4
13
3
v2
v3
10
9
8
11
v5
v6
14
15 12
v6
v7
v8
22
19
17
20
21 w
18
w2
3
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
|Mw1 | = Q1 f1,6
u
1
v1
5
2
7
4
6
16
w1
v4
13
3
v2
v3
10
9
8
11
v5
v6
14
15 12
v6
v7
v8
22
19
17
20
21 w
18
w2
3
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
|Mw1 | = Q1 f1,6
·f2,7 f7,13 f13,16
u
1
v1
5
2
7
4
6
16
w1
v4
13
3
v2
v3
10
9
8
11
v5
v6
14
15 12
v6
v7
v8
22
19
17
20
21 w
18
w2
3
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
u
1
v1
5
2
7
4
6
16
w1
v4
13
3
v2
v3
10
9
8
11
v5
v6
14
15 12
|Mw1 | = Q1 f1,6
·f2,7 f7,13 f13,16
·f3,10 f10,14 f14,18
v6
v7
v8
22
19
17
20
21 w
18
w2
3
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
u
1
v1
5
2
7
4
6
16
w1
v4
13
3
v2
v3
10
9
8
11
v5
v6
14
15 12
|Mw1 | = Q1 f1,6
·f2,7 f7,13 f13,16
·f3,10 f10,14 f14,18
|Mw3 | = Q2
f1,5 f5,14 f14,21
·f2,9 f9,15 f15,22 ·f3,12
v6
v7
v8
22
19
17
20
21 w
18
w2
3
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
u
1
v1
5
2
7
4
6
16
w1
v4
13
3
v2
v3
10
9
8
11
v5
v6
14
15 12
|Mw1 | = Q1 f1,6
·f2,7 f7,13 f13,16
·f3,10 f10,14 f14,18
|Mw3 | = Q2
f1,5 f5,14 f14,21
·f2,9 f9,15 f15,22 ·f3,12
|Mw2 | = Q3
v6
v7
v8
22
19
17
20
21 w
18
w2
3
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
u
1
v1
5
2
7
4
6
16
w1
v4
13
3
v2
v3
10
9
8
11
v5
v6
14
15 12
v6
v7
v8
22
19
17
20
21 w
18
w2
3
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
|Mw1 | = Q1 f1,6
·f2,7 f7,13 f13,16
·f3,10 f10,14 f14,18
|Mw3 | = Q2
f1,5 f5,14 f14,21
·f2,9 f9,15 f15,22 ·f3,12
|Mw2 | = Q3
f1,4 f4,13 f13,17
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
u
1
v1
5
2
7
4
6
16
w1
v4
13
3
v2
v3
10
9
8
11
v5
v6
14
15 12
v6
v7
v8
22
19
17
20
21 w
18
w2
3
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
|Mw1 | = Q1 f1,6
·f2,7 f7,13 f13,16
·f3,10 f10,14 f14,18
|Mw3 | = Q2
f1,5 f5,14 f14,21
·f2,9 f9,15 f15,22 ·f3,12
|Mw2 | = Q3
f1,4 f4,13 f13,17
·f2,8 f8,14 f14,19
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
u
1
v1
5
2
7
4
6
16
w1
v4
13
3
v2
v3
10
9
8
11
v5
v6
14
15 12
v6
v7
v8
22
19
17
20
21 w
18
w2
3
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
|Mw1 | = Q1 f1,6
·f2,7 f7,13 f13,16
·f3,10 f10,14 f14,18
|Mw3 | = Q2
f1,5 f5,14 f14,21
·f2,9 f9,15 f15,22 ·f3,12
|Mw2 | = Q3
f1,4 f4,13 f13,17
·f2,8 f8,14 f14,19
·f3,11 f11,15 f15,20
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
u
1
v1
5
2
7
4
6
16
w1
v4
13
3
v2
v3
10
9
8
11
v5
v6
14
15 12
v6
v7
v8
22
19
17
20
21 w
18
w2
3
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
|Mw1 | = Q1 f1,6
·f2,7 f7,13 f13,16
·f3,10 f10,14 f14,18
|Mw3 | = Q2
f1,5 f5,14 f14,21
·f2,9 f9,15 f15,22 ·f3,12
|Mw2 | = Q3
f1,4 f4,13 f13,17
·f2,8 f8,14 f14,19
·f3,11 f11,15 f15,20
+f1,5 f5,14 f14,19
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
u
1
v1
5
2
7
4
6
16
w1
v4
13
3
v2
v3
10
9
8
11
v5
v6
14
15 12
v6
v7
v8
22
19
17
20
21 w
18
w2
3
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
|Mw1 | = Q1 f1,6
·f2,7 f7,13 f13,16
·f3,10 f10,14 f14,18
|Mw3 | = Q2
f1,5 f5,14 f14,21
·f2,9 f9,15 f15,22 ·f3,12
|Mw2 | = Q3
f1,4 f4,13 f13,17
·f2,8 f8,14 f14,19
·f3,11 f11,15 f15,20
+f1,5 f5,14 f14,19
·f2,7 f7,13 f13,17
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
u
1
v1
5
2
7
4
6
16
w1
v4
13
3
v2
v3
10
9
8
11
v5
v6
14
15 12
v6
v7
v8
22
19
17
20
21 w
18
w2
3
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
|Mw1 | = Q1 f1,6
·f2,7 f7,13 f13,16
·f3,10 f10,14 f14,18
|Mw3 | = Q2
f1,5 f5,14 f14,21
·f2,9 f9,15 f15,22 ·f3,12
|Mw2 | = Q3
f1,4 f4,13 f13,17
·f2,8 f8,14 f14,19
·f3,11 f11,15 f15,20
+f1,5 f5,14 f14,19
·f2,7 f7,13 f13,17
·f3,11 f11,15 f15,20
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
u
1
v1
5
2
7
4
6
16
w1
v4
13
3
v2
v3
10
9
8
11
v5
v6
14
15 12
v6
v7
v8
22
19
17
20
21 w
18
w2
3
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
|Mw1 | = Q1 f1,6
·f2,7 f7,13 f13,16
·f3,10 f10,14 f14,18
|Mw3 | = Q2
f1,5 f5,14 f14,21
·f2,9 f9,15 f15,22 ·f3,12
|Mw2 | = Q3
f1,4 f4,13 f13,17
·f2,8 f8,14 f14,19
·f3,11 f11,15 f15,20
+f1,5 f5,14 f14,19
·f2,7 f7,13 f13,17
·f3,11 f11,15 f15,20
+f1,4 f4,13 f13,17
·f2,9 f9,15 f15,20
·f3,10 f10,14 f14,19
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
u
1
v1
5
2
7
4
6
16
w1
v4
13
3
v2
v3
10
9
8
11
v5
v6
14
15 12
v6
v7
v8
22
19
17
20
21 w
18
w2
3
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
|Mw1 | = Q1 f1,6
·f2,7 f7,13 f13,16
·f3,10 f10,14 f14,18
|Mw3 | = Q2
f1,5 f5,14 f14,21
·f2,9 f9,15 f15,22 ·f3,12
|Mw2 | = Q3
f1,4 f4,13 f13,17
·f2,8 f8,14 f14,19
·f3,11 f11,15 f15,20
+f1,5 f5,14 f14,19
·f2,7 f7,13 f13,17
·f3,11 f11,15 f15,20
+f1,4 f4,13 f13,17
·f2,9 f9,15 f15,20
·f3,10 f10,14 f14,19
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
u
1
v1
5
2
7
4
6
16
w1
v4
13
3
v2
v3
10
9
8
11
v5
v6
14
15 12
v6
v7
v8
22
19
17
20
21 w
18
w2
3
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
f1,6
|Mw1 | = Q1 f7,13 f13,16
·
f2,7
f10,14 f14,18
·
f3,10
|Mw3 | = Q2
f5,14 f14,21
f
1,5
f f9,15 f15,22
f2,9
·
3,12
·
|Mw2 | = Q3
f4,13 f13,17
f
1,4
f8,14 f14,19
·
f2,8
f11,15 f15,20
·
f3,11
f5,14 f14,19
+
f1,5
f7,13 f13,17
·
f2,7
f11,15 f15,20
·
f3,11
f4,13 f13,17
+
f1,4
f9,15 f15,20
·
f2,9
·
f3,10 f10,14
f14,19
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
|Mw1 | = Q1 f7,13 ·f10,14
u
1
v1
5
2
7
4
v4
13
6
16
w1
3
|Mw3 | = Q2 f5,14 ·f9,15
v2
v3
10
9
8
11
v5
v6
14
15 12
|Mw2 | = Q3
f4,13 ·f8,14 ·f11,15
+f5,14 ·f7,13 ·f11,15
+f4,13 ·f9,15 ·f10,14
v6
v7
v8
22
19
17
20
21 w
18
w2
3
P = Q1 Q2 Q3 · f7,13 f10,14 · f5,14 f9,15 ·
(f4,13 f8,14 f11,15 + f5,14 f7,13 f11,15 + f4,13 f9,15 f10,14 )
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Introduction
Results
Min. field size
Improving bound
Interpret |Mw |
End
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Definitions and Theorems
Theorem
Given a network connection problem as outlined above
where the min-cut from the source vertex u to each
receiver wi is at least h, the number of sources processes.
Then there exists a large enough finite field Fq in which
inner vertices linearly combine the input symbols and
send the linear combination to the out-edges such that
each receiver wi receive information at a rate equal to h.
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Definitions and Theorems
u
Definition (min-cut)
The min-cut from a vertex u
to a vertex w in a directed
graph is the least number of
edges that can be removed
from the graph such that
there is no path from u to
w.
v1
v3
v4
w1
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
v2
w2
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g
Definitions and Theorems
Theorem
Given a network connection problem as outlined above
where the min-cut from the source vertex u to each
receiver wi is at least h, the number of sources processes.
Then there exists a large enough finite field Fq in which
inner vertices linearly combine the input symbols and
send the linear combination to the out-edges such that
each receiver wi receive information at a rate equal to h.
Olav Geil, Ryutaroh Matsumoto, Casper Thomsen[1em]
On Field Size and Success
[1em]Probability
Workshop in
onNetwork
GroebnerCoding
bases and g