Finite Spreads in Random Network Coding - UZHuser.math.uzh.ch/trautmann/spreads_in_RNC.pdf · 2012. 5. 3. · Finite Spreads in Random Network Coding Random Network Coding Deﬁnition

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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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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channel

sources sinks

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sources sinksinner nodes

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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



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



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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The codewords are rank-metric (matrix) codes and can bedecoded with the knowledge of the header.

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Random (linear) network coding II:

Inner nodes forward a random linear combination of theincoming information.

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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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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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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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Possible errors:

erasures (decrease in dimension)

insertions (increase in dimension)

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Possible errors:



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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Possible errors:



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 as Constant Dimension Codes





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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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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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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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.

2 In this vector space consider the trivial spread of allone-dimensional subspaces.

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.


3 Each of these lines over Fqk can now be considered as ak-dimensional subspace over Fq.

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.



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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.



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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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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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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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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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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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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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



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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3 This is the subcode of dimension n − k.

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4 Analogously one can define subcodes of dimensionn − 2k, n − 3k etc.

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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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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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Example

Consider G2(2, 4) and α2 + α + 1 = 0.


〈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)〉.


〈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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Decoding Spread Codes

Fqk

-representation decoder





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Fqk


Basic Idea of the Algorithm:

Divide each received vector R ∈ Fnq in blocks of length k,R1, ..., Rn

k.

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Fqk




k.

Find the first non-zero block from the left =: Rs andcompute φ(Rs) =: a.

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Fqk




k.


Store γ(R) := (φ(R1) · a−1, . . . , φ(Rn

k) · a−1).

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Fqk




k.


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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MGR-decoder





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MGR-decoder


1 Consider the received vector space as a matrix in reducedrow echelon form.

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MGR-decoder



2 Find the first k × k-block that has rank ≥ (k + 1)/2.

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MGR-decoder




3 This block will be the leading identity block of the codeword.

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MGR-decoder




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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Primitive orbit code decoder





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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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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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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)



2 If dim(R ∩ UP j) ≥ k+12

, decode to UP j.

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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)



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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Extended Reed-Solomon like decoder





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1 Find the leading identity block (as before) of R.

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2 This indicates the dimension of the subcode.

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3 Find Qx(x) = u1x and Qy(y) = u2y such thatQ(x, y) = Qx(x) + Qy(y) vanishes on a basis of R.

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4 Find φ(u) with the property that −Qx(x) ≡ Qy(φ(u)x).

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4 Find φ(u) with the property that −Qx(x) ≡ Qy(φ(u)x).

5 Output the information word u.

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Isometry and Automorphisms of Spread Codes





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Definition

Two constant dimension codes C1, C2 ⊆ Gq(k, n) are linearlyisometric if there exists A ∈ GLn

C1 = C2A.

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Definition


C1 = C2A.

Theorem

All Desarguesian spread codes are linearly isometric.

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Definition


C1 = C2A.

Theorem


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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Definition


C1 = C2A.

Theorem


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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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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Definition




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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Definition




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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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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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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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.

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Thank you.

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Random Network CodingFinite Spreads as Constant Dimension CodesDecoding Spread CodesFqk-representation decoderMGR-decoderPrimitive orbit code decoderExtended Reed-Solomon like decoder


Finite Spreads in Random Network Coding - UZHuser.math.uzh.ch/trautmann/spreads_in_RNC.pdf · 2012. 5. 3. · Finite Spreads in Random Network Coding Random Network Coding Deﬁnition

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