finite spreads in random network coding - uzhuser.math.uzh.ch/trautmann/spreads_in_rnc.pdf · 2012....
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Finite Spreads in Random Network Coding
Finite Spreads in Random Network Coding
Anna-Lena Trautmann
Institute of Mathematics
University of Zurich
Colloquium on Galois GeometryGent, May 4th 2012
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Finite Spreads in Random Network Coding
Random Network Coding
1 Random Network Coding
2 Finite Spreads as Constant Dimension Codes
3 Decoding Spread CodesFqk-representation decoderMGR-decoderPrimitive orbit code decoderExtended Reed-Solomon like decoder
4 Isometry and Automorphisms of Spread Codes
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Finite Spreads in Random Network Coding
Random Network Coding
channel
sources sinks
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Finite Spreads in Random Network Coding
Random Network Coding
sources sinksinner nodes
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Finite Spreads in Random Network Coding
Random Network Coding
When sending information through a network we can optimizethe throughput by doing linear combinations on theintermediate nodes. ( =⇒ linear network coding)
Example (The Butterfly Network):
R2
R1
a
b
b
S1
S2
a
a
a
a
Received: R1 : (a, a) , R2 : (a, b)4 / 29
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Finite Spreads in Random Network Coding
Random Network Coding
When sending information through a network we can optimizethe throughput by doing linear combinations on theintermediate nodes. ( =⇒ linear network coding)
Example (The Butterfly Network):
R2
R1
a
b
b
S1
S2
a
a + b
a + b
a + b
Received: R1 : (a, a + b) , R2 : (a + b, b)4 / 29
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Finite Spreads in Random Network Coding
Random Network Coding
Random (linear) network coding I:
The source randomly chooses the parameters of all innernodes beforehand and sends these as headers of theinformation packets.
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Finite Spreads in Random Network Coding
Random Network Coding
Random (linear) network coding I:
The source randomly chooses the parameters of all innernodes beforehand and sends these as headers of theinformation packets.
The codewords are rank-metric (matrix) codes and can bedecoded with the knowledge of the header.
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Finite Spreads in Random Network Coding
Random Network Coding
Random (linear) network coding I:
The source randomly chooses the parameters of all innernodes beforehand and sends these as headers of theinformation packets.
The codewords are rank-metric (matrix) codes and can bedecoded with the knowledge of the header.
Random (linear) network coding II:
Inner nodes forward a random linear combination of theincoming information.
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Finite Spreads in Random Network Coding
Random Network Coding
Random (linear) network coding I:
The source randomly chooses the parameters of all innernodes beforehand and sends these as headers of theinformation packets.
The codewords are rank-metric (matrix) codes and can bedecoded with the knowledge of the header.
Random (linear) network coding II:
Inner nodes forward a random linear combination of theincoming information.
Choose linear subspaces of Fnq as codewords since thesestay invariant under linear operations on the basis vectors.
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Finite Spreads in Random Network Coding
Random Network Coding
Definition
1 The projective geometry P(Fnq ) is the set of all subspacesof Fnq . A subspace code is a subset of P(F
nq ).
2 The Grassmannian Gq(k, n) is the set of all k-subspaces ofF
nq . A constant dimension code is a subset of Gq(k, n).
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Finite Spreads in Random Network Coding
Random Network Coding
Definition
1 The projective geometry P(Fnq ) is the set of all subspacesof Fnq . A subspace code is a subset of P(F
nq ).
2 The Grassmannian Gq(k, n) is the set of all k-subspaces ofF
nq . A constant dimension code is a subset of Gq(k, n).
Implementation:
Map the information to a codeword U ∈ Gq(k, n).
Insert a basis of U into the network.
Receive R = Ū ⊕ E .
Decode with minimum distance decoding to U .
Map the codeword back to the information.
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Finite Spreads in Random Network Coding
Random Network Coding
Possible errors:
erasures (decrease in dimension)
insertions (increase in dimension)
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Finite Spreads in Random Network Coding
Random Network Coding
Possible errors:
erasures (decrease in dimension)
insertions (increase in dimension)
Definition
Subspace metric:
dS(U ,V) = dim(U + V) − dim(U ∩ V)
Injection metric:
dI(U ,V) = max(dimU ,dimV) − dim(U ∩ V)
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Finite Spreads in Random Network Coding
Random Network Coding
Possible errors:
erasures (decrease in dimension)
insertions (increase in dimension)
Definition
Subspace metric:
dS(U ,V) = dim(U + V) − dim(U ∩ V)
Injection metric:
dI(U ,V) = max(dimU ,dimV) − dim(U ∩ V)
=⇒ equivalent for constant dimension codes!
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
1 Random Network Coding
2 Finite Spreads as Constant Dimension Codes
3 Decoding Spread CodesFqk-representation decoderMGR-decoderPrimitive orbit code decoderExtended Reed-Solomon like decoder
4 Isometry and Automorphisms of Spread Codes
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
Definition
A k-spread of Fnq is a set of subspaces of dimension k such thatthey pairwise intersect only trivially and they cover the wholevector space Fnq .
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
Definition
A k-spread of Fnq is a set of subspaces of dimension k such thatthey pairwise intersect only trivially and they cover the wholevector space Fnq .
A spread exists if and only if k|n. As a constant dimensioncode, a spread has cardinality (qn − 1)/(qk − 1) and minimumsubspace distance 2k.
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
Fqk-representations of Fqn :
1 Consider Fqn as an extension field of Fqk of degree l := n/k,
which is isomorphic to the vector space Flqk
.
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
Fqk-representations of Fqn :
1 Consider Fqn as an extension field of Fqk of degree l := n/k,
which is isomorphic to the vector space Flqk
.
2 In this vector space consider the trivial spread of allone-dimensional subspaces.
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
Fqk-representations of Fqn :
1 Consider Fqn as an extension field of Fqk of degree l := n/k,
which is isomorphic to the vector space Flqk
.
2 In this vector space consider the trivial spread of allone-dimensional subspaces.
3 Each of these lines over Fqk can now be considered as ak-dimensional subspace over Fq.
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
Fqk-representations of Fqn :
1 Consider Fqn as an extension field of Fqk of degree l := n/k,
which is isomorphic to the vector space Flqk
.
2 In this vector space consider the trivial spread of allone-dimensional subspaces.
3 Each of these lines over Fqk can now be considered as ak-dimensional subspace over Fq.
4 Since the lines of Flqk
intersect only trivially and with asimple counting argument it follows that the correspondingk-dimensional subspaces of Fnq form a spread.
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
Fqk-representations of Fqn :
1 Consider Fqn as an extension field of Fqk of degree l := n/k,
which is isomorphic to the vector space Flqk
.
2 In this vector space consider the trivial spread of allone-dimensional subspaces.
3 Each of these lines over Fqk can now be considered as ak-dimensional subspace over Fq.
4 Since the lines of Flqk
intersect only trivially and with asimple counting argument it follows that the correspondingk-dimensional subspaces of Fnq form a spread.
We call “such” spreads Desarguesian.
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
The matrix point of view:Let p ∈ Fq[x] irreducible of degree k and P its companionmatrix. Then the set
{rs[
I B1 B2 . . . Bnk−1
]
| Bi ∈ Fq[P ]}
∪ {rs[
0 I B2 . . . Bnk−1
]
| Bi ∈ Fq[P ]}
...
∪ {rs[
0 0 0 . . . 0 I]
}
is a spread code.
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
Primitive orbit construction:
1 Choose a primitive element α of Fqn .
2 Let c := (qn − 1)/(qk − 1).
3 Then Fqk∼= 〈αc〉 ∪ {0}.
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
Primitive orbit construction:
1 Choose a primitive element α of Fqn .
2 Let c := (qn − 1)/(qk − 1).
3 Then Fqk∼= 〈αc〉 ∪ {0}.
4 The set{
αi · Fqk | i = 0, . . . , c − 1}
represents a spreadcode in Fqn ∼= F
nq .
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
Primitive orbit construction:
1 Choose a primitive element α of Fqn .
2 Let c := (qn − 1)/(qk − 1).
3 Then Fqk∼= 〈αc〉 ∪ {0}.
4 The set{
αi · Fqk | i = 0, . . . , c − 1}
represents a spreadcode in Fqn ∼= F
nq .
This spread code is an orbit code UG for some G ≤ GLn, whereG = 〈P 〉 and P is the companion matrix of the minimalpolynomial of α.
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
Example
Over the binary field let p(x) := x6 + x + 1 be primitive, α aroot of p(x) and P its companion matrix. For the 3-dimensionalspread compute c = 63
7= 9 and construct a basis for the
starting point of the orbit:
u1 = φ−1(α0) = φ−1(1) = (100000)
u2 = φ−1(αc) = φ−1(α9) = φ−1(α4 + α3) = (000110)
u3 = φ−1(α2c) = φ−1(α18) = φ−1(α3 + α2 + α + 1) = (111100)
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
Example
Over the binary field let p(x) := x6 + x + 1 be primitive, α aroot of p(x) and P its companion matrix. For the 3-dimensionalspread compute c = 63
7= 9 and construct a basis for the
starting point of the orbit:
u1 = φ−1(α0) = φ−1(1) = (100000)
u2 = φ−1(αc) = φ−1(α9) = φ−1(α4 + α3) = (000110)
u3 = φ−1(α2c) = φ−1(α18) = φ−1(α3 + α2 + α + 1) = (111100)
The starting point is
U = rs
1 0 0 0 0 00 0 0 1 1 01 1 1 1 0 0
= rs
1 0 0 0 0 00 1 1 0 1 00 0 0 1 1 0
and the orbit of the group generated by P on U is a spread code.13 / 29
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
Extended Reed-Solomon like construction:
1 Choose a primitive element α of Fqn−k .
2 An information word u ∈ Fn−kq is encoded into thecodeword 〈φ−1(αi, φ(u)αi) | i = n − 2k, . . . , n − k − 1〉.
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
Extended Reed-Solomon like construction:
1 Choose a primitive element α of Fqn−k .
2 An information word u ∈ Fn−kq is encoded into thecodeword 〈φ−1(αi, φ(u)αi) | i = n − 2k, . . . , n − k − 1〉.
3 This is the subcode of dimension n − k.
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
Extended Reed-Solomon like construction:
1 Choose a primitive element α of Fqn−k .
2 An information word u ∈ Fn−kq is encoded into thecodeword 〈φ−1(αi, φ(u)αi) | i = n − 2k, . . . , n − k − 1〉.
3 This is the subcode of dimension n − k.
4 Analogously one can define subcodes of dimensionn − 2k, n − 3k etc.
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
Extended Reed-Solomon like construction:
1 Choose a primitive element α of Fqn−k .
2 An information word u ∈ Fn−kq is encoded into thecodeword 〈φ−1(αi, φ(u)αi) | i = n − 2k, . . . , n − k − 1〉.
3 This is the subcode of dimension n − k.
4 Analogously one can define subcodes of dimensionn − 2k, n − 3k etc.
5 The union of all subcodes is a spread code.
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
Example
Consider G2(2, 4) and α2 + α + 1 = 0.
The message (1, 0) is encoded into
〈φ−1(1, φ(u)), φ−1(α, φ(u)α)〉
= 〈φ−1(1, 1), φ−1(α,α)〉
= 〈(1, 0, 1, 0), (0, 1, 0, 1)〉.
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Finite Spreads in Random Network Coding
Finite Spreads as Constant Dimension Codes
Example
Consider G2(2, 4) and α2 + α + 1 = 0.
The message (1, 0) is encoded into
〈φ−1(1, φ(u)), φ−1(α, φ(u)α)〉
= 〈φ−1(1, 1), φ−1(α,α)〉
= 〈(1, 0, 1, 0), (0, 1, 0, 1)〉.
The message (0, 1) is encoded into
〈φ−1(1, φ(u)), φ−1(α, φ(u)α)〉
= 〈φ−1(1, α), φ−1(α,α + 1)〉
= 〈(1, 0, 0, 1), (0, 1, 1, 1)〉.
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Finite Spreads in Random Network Coding
Decoding Spread Codes
Fqk
-representation decoder
1 Random Network Coding
2 Finite Spreads as Constant Dimension Codes
3 Decoding Spread CodesFqk-representation decoderMGR-decoderPrimitive orbit code decoderExtended Reed-Solomon like decoder
4 Isometry and Automorphisms of Spread Codes
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Finite Spreads in Random Network Coding
Decoding Spread Codes
Fqk
-representation decoder
Basic Idea of the Algorithm:
Divide each received vector R ∈ Fnq in blocks of length k,R1, ..., Rn
k.
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Finite Spreads in Random Network Coding
Decoding Spread Codes
Fqk
-representation decoder
Basic Idea of the Algorithm:
Divide each received vector R ∈ Fnq in blocks of length k,R1, ..., Rn
k.
Find the first non-zero block from the left =: Rs andcompute φ(Rs) =: a.
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Finite Spreads in Random Network Coding
Decoding Spread Codes
Fqk
-representation decoder
Basic Idea of the Algorithm:
Divide each received vector R ∈ Fnq in blocks of length k,R1, ..., Rn
k.
Find the first non-zero block from the left =: Rs andcompute φ(Rs) =: a.
Store γ(R) := (φ(R1) · a−1, . . . , φ(Rn
k) · a−1).
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Finite Spreads in Random Network Coding
Decoding Spread Codes
Fqk
-representation decoder
Basic Idea of the Algorithm:
Divide each received vector R ∈ Fnq in blocks of length k,R1, ..., Rn
k.
Find the first non-zero block from the left =: Rs andcompute φ(Rs) =: a.
Store γ(R) := (φ(R1) · a−1, . . . , φ(Rn
k) · a−1).
If you found ≥ ⌈k+12
⌉ linearly independent elements R withthe same γ(R), decode to
φ−1(Fqk · γ(R)).
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Finite Spreads in Random Network Coding
Decoding Spread Codes
MGR-decoder
1 Random Network Coding
2 Finite Spreads as Constant Dimension Codes
3 Decoding Spread CodesFqk-representation decoderMGR-decoderPrimitive orbit code decoderExtended Reed-Solomon like decoder
4 Isometry and Automorphisms of Spread Codes
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Finite Spreads in Random Network Coding
Decoding Spread Codes
MGR-decoder
Basic Idea of the Algorithm:
1 Consider the received vector space as a matrix in reducedrow echelon form.
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Finite Spreads in Random Network Coding
Decoding Spread Codes
MGR-decoder
Basic Idea of the Algorithm:
1 Consider the received vector space as a matrix in reducedrow echelon form.
2 Find the first k × k-block that has rank ≥ (k + 1)/2.
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Finite Spreads in Random Network Coding
Decoding Spread Codes
MGR-decoder
Basic Idea of the Algorithm:
1 Consider the received vector space as a matrix in reducedrow echelon form.
2 Find the first k × k-block that has rank ≥ (k + 1)/2.
3 This block will be the leading identity block of the codeword.
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Finite Spreads in Random Network Coding
Decoding Spread Codes
MGR-decoder
Basic Idea of the Algorithm:
1 Consider the received vector space as a matrix in reducedrow echelon form.
2 Find the first k × k-block that has rank ≥ (k + 1)/2.
3 This block will be the leading identity block of the codeword.
4 Find the correct other blocks by examining certain minorsof the matrix.
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Finite Spreads in Random Network Coding
Decoding Spread Codes
Primitive orbit code decoder
1 Random Network Coding
2 Finite Spreads as Constant Dimension Codes
3 Decoding Spread CodesFqk-representation decoderMGR-decoderPrimitive orbit code decoderExtended Reed-Solomon like decoder
4 Isometry and Automorphisms of Spread Codes
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Finite Spreads in Random Network Coding
Decoding Spread Codes
Primitive orbit code decoder
Code:C = {UP i | i = 0, . . . , c − 1}
Sent word:
V = UP j ∼= {φ(u1), ..., φ(uqk−1)} · αj ∪ {0}
=⇒ ∃v ∈ V : φ(u1)αj = φ(v)
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Finite Spreads in Random Network Coding
Decoding Spread Codes
Primitive orbit code decoder
Code:C = {UP i | i = 0, . . . , c − 1}
Sent word:
V = UP j ∼= {φ(u1), ..., φ(uqk−1)} · αj ∪ {0}
=⇒ ∃v ∈ V : φ(u1)αj = φ(v)
Idea of the algorithm:
1 For some u ∈ U and r ∈ R compute j := logα(φ(r)/φ(u)).
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Finite Spreads in Random Network Coding
Decoding Spread Codes
Primitive orbit code decoder
Code:C = {UP i | i = 0, . . . , c − 1}
Sent word:
V = UP j ∼= {φ(u1), ..., φ(uqk−1)} · αj ∪ {0}
=⇒ ∃v ∈ V : φ(u1)αj = φ(v)
Idea of the algorithm:
1 For some u ∈ U and r ∈ R compute j := logα(φ(r)/φ(u)).
2 If dim(R ∩ UP j) ≥ k+12
, decode to UP j.
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Finite Spreads in Random Network Coding
Decoding Spread Codes
Primitive orbit code decoder
Code:C = {UP i | i = 0, . . . , c − 1}
Sent word:
V = UP j ∼= {φ(u1), ..., φ(uqk−1)} · αj ∪ {0}
=⇒ ∃v ∈ V : φ(u1)αj = φ(v)
Idea of the algorithm:
1 For some u ∈ U and r ∈ R compute j := logα(φ(r)/φ(u)).
2 If dim(R ∩ UP j) ≥ k+12
, decode to UP j.
3 Otherwise choose a new r ∈ R and restart.
4 If no decoding possible, choose a new u ∈ U and restart.
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Finite Spreads in Random Network Coding
Decoding Spread Codes
Extended Reed-Solomon like decoder
1 Random Network Coding
2 Finite Spreads as Constant Dimension Codes
3 Decoding Spread CodesFqk-representation decoderMGR-decoderPrimitive orbit code decoderExtended Reed-Solomon like decoder
4 Isometry and Automorphisms of Spread Codes
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Finite Spreads in Random Network Coding
Decoding Spread Codes
Extended Reed-Solomon like decoder
Basic Idea of the Algorithm:
1 Find the leading identity block (as before) of R.
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Finite Spreads in Random Network Coding
Decoding Spread Codes
Extended Reed-Solomon like decoder
Basic Idea of the Algorithm:
1 Find the leading identity block (as before) of R.
2 This indicates the dimension of the subcode.
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Finite Spreads in Random Network Coding
Decoding Spread Codes
Extended Reed-Solomon like decoder
Basic Idea of the Algorithm:
1 Find the leading identity block (as before) of R.
2 This indicates the dimension of the subcode.
3 Find Qx(x) = u1x and Qy(y) = u2y such thatQ(x, y) = Qx(x) + Qy(y) vanishes on a basis of R.
23 / 29
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Finite Spreads in Random Network Coding
Decoding Spread Codes
Extended Reed-Solomon like decoder
Basic Idea of the Algorithm:
1 Find the leading identity block (as before) of R.
2 This indicates the dimension of the subcode.
3 Find Qx(x) = u1x and Qy(y) = u2y such thatQ(x, y) = Qx(x) + Qy(y) vanishes on a basis of R.
4 Find φ(u) with the property that −Qx(x) ≡ Qy(φ(u)x).
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Finite Spreads in Random Network Coding
Decoding Spread Codes
Extended Reed-Solomon like decoder
Basic Idea of the Algorithm:
1 Find the leading identity block (as before) of R.
2 This indicates the dimension of the subcode.
3 Find Qx(x) = u1x and Qy(y) = u2y such thatQ(x, y) = Qx(x) + Qy(y) vanishes on a basis of R.
4 Find φ(u) with the property that −Qx(x) ≡ Qy(φ(u)x).
5 Output the information word u.
23 / 29
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Finite Spreads in Random Network Coding
Isometry and Automorphisms of Spread Codes
1 Random Network Coding
2 Finite Spreads as Constant Dimension Codes
3 Decoding Spread CodesFqk-representation decoderMGR-decoderPrimitive orbit code decoderExtended Reed-Solomon like decoder
4 Isometry and Automorphisms of Spread Codes
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Finite Spreads in Random Network Coding
Isometry and Automorphisms of Spread Codes
Definition
Two constant dimension codes C1, C2 ⊆ Gq(k, n) are linearlyisometric if there exists A ∈ GLn
C1 = C2A.
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Finite Spreads in Random Network Coding
Isometry and Automorphisms of Spread Codes
Definition
Two constant dimension codes C1, C2 ⊆ Gq(k, n) are linearlyisometric if there exists A ∈ GLn
C1 = C2A.
Theorem
All Desarguesian spread codes are linearly isometric.
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Finite Spreads in Random Network Coding
Isometry and Automorphisms of Spread Codes
Definition
Two constant dimension codes C1, C2 ⊆ Gq(k, n) are linearlyisometric if there exists A ∈ GLn
C1 = C2A.
Theorem
All Desarguesian spread codes are linearly isometric.
Proof: Since there is only one spread of lines in Flqk
, differentDesarguesian spreads of Fnq can only arise from the different
isomorphisms between Fqk and Fkq .
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Finite Spreads in Random Network Coding
Isometry and Automorphisms of Spread Codes
Definition
Two constant dimension codes C1, C2 ⊆ Gq(k, n) are linearlyisometric if there exists A ∈ GLn
C1 = C2A.
Theorem
All Desarguesian spread codes are linearly isometric.
Proof: Since there is only one spread of lines in Flqk
, differentDesarguesian spreads of Fnq can only arise from the different
isomorphisms between Fqk and Fkq .
Theorem
All spreads generated as primitive orbit codes are linearlyisometric.
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Finite Spreads in Random Network Coding
Isometry and Automorphisms of Spread Codes
Definition
For a given code C ⊆ P(Fnq ),
Aut(C) = {A ∈ GLn|CA = C}
is called the (linear) automorphism group of the code.
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Finite Spreads in Random Network Coding
Isometry and Automorphisms of Spread Codes
Definition
For a given code C ⊆ P(Fnq ),
Aut(C) = {A ∈ GLn|CA = C}
is called the (linear) automorphism group of the code.
Theorem
The linear automorphism group of a Desarguesian spread codeC ⊆ Gq(k, n) is isomorphic to GLn
k(qk) × Aut(Fqk).
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Finite Spreads in Random Network Coding
Isometry and Automorphisms of Spread Codes
Definition
For a given code C ⊆ P(Fnq ),
Aut(C) = {A ∈ GLn|CA = C}
is called the (linear) automorphism group of the code.
Theorem
The linear automorphism group of a Desarguesian spread codeC ⊆ Gq(k, n) is isomorphic to GLn
k(qk) × Aut(Fqk).
Proof: Let l := n/k. We know that PGLl(qk) is the group of all
Fqk-linear bijections of Pl−1(Fqk) and that Aut(Fqk) is the set of
all automorphisms of Fqk that stabilize Fq. Thus,
PGLl(qk) × Aut(Fqk) is the set of all Fq-linear bijections of
Pl−1(Fqk).
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Finite Spreads in Random Network Coding
Isometry and Automorphisms of Spread Codes
Corollary
The automorphism group of a Desarguesian spread code inGq(k, n) is generated by all elements in GLn where thek × k-blocks are elements of Fq[P ] and block diagonal matriceswhere the blocks represent an automorphism of Fqk .
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Finite Spreads in Random Network Coding
Isometry and Automorphisms of Spread Codes
Corollary
The automorphism group of a Desarguesian spread code inGq(k, n) is generated by all elements in GLn where thek × k-blocks are elements of Fq[P ] and block diagonal matriceswhere the blocks represent an automorphism of Fqk .
Another point of view: the generator matrices of the codewords are of the type
U =[
B1 B2 . . . Bl]
where the blocks Bi are an element of Fq[P ]. To stay inside thisstructure (i.e. to apply an automorphism) we can permute theblocks, do block-wise multiplications or do block-wise additionswith elements from Fq[P ]. This coincides with the structure ofthe automorphism groups from before.
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Finite Spreads in Random Network Coding
Isometry and Automorphisms of Spread Codes
Example
Consider G3(2, 4) and the irreducible polynomialp(x) = x2 + x + 2, i.e.
P =
(
0 11 2
)
C = rs[
I 0]
∪ {rs[
I P i]
| i = 0, . . . , 7} ∪ rs[
0 I]
Its automorphism group has 11520 elements:
Aut(C) =
〈(
II
)
,
(
IP
)
,
(
I PI
)
,
(
QQ
)〉
where Q =
(
1 02 2
)
∈ GL2. Here Q represents the only
non-trivial automorphism of F32, i.e. x 7→ x3.
28 / 29
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Finite Spreads in Random Network Coding
Isometry and Automorphisms of Spread Codes
Thank you.
29 / 29
Random Network CodingFinite Spreads as Constant Dimension CodesDecoding Spread CodesFqk-representation decoderMGR-decoderPrimitive orbit code decoderExtended Reed-Solomon like decoder
Isometry and Automorphisms of Spread Codes