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On defining the generalized rank weight

Ruud Pellikaanjoint work withRelinde Jurrius

Computational Aspects and Mathematical Methods for Finite Fieldsand their Applications in Information Theory

ACA 2015 Kalamata, 23 July 2015

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Content

1. Introduction

2. Generalized Hamming weight

3. Rank weight

4. Four spaces

5. Generalized rank weight

6. Alternative definitions

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Introduction

1. Error-correction, vectors in Fnq , Hamming weight

2. Network coding, matrices in Fm×nq , rank weight

3. Wire-tap channel, generalized rank weight

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Notation

Fq is the finite field with q elementsFqm is the finite field extension of Fq of degree m

An [n, k ] code over Fq is a subspace of Fnq of dimension k

The inner product on Fnq is defined by

x · y = x1y1 + · · · + xnyn

This inner product is bilinear, symmetric and non-degenerate

For an [n, k ] code C we define the dual or orthogonal code C⊥ as

C⊥ = { x ∈ Fnq | c · x = 0 for all c ∈ C }

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Support and weight

The support of x in Fnq is defined by

supp(x) = { j | xj 6= 0 }

The Hamming weight of x is defined by

wtH (x) = |supp(x)|

that is the number of nonzero entries of x

The support of subspace D of Fnq is defined by

supp(D ) = { j | xj 6= 0 for some x ∈ D }

The Hamming weight of D is defined by

wtH (D ) = |supp(D )|

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Generalized Hamming weight

Let C be an Fq -linear codeThen the minimum distance of C is

d (C ) = min{ wtH (c) | 0 6= c ∈ C }

The r-th generalized Hamming weight of C is

dr(C ) = min{ wtH (D ) | D subspace of C , dim(D ) = r }

So d1(C ) = d (C ).

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Network coding

Delsarte defined rank weightGabidulin applied to (network) coding

Choose a basis α1, . . . αm of Fqm as a vector space over Fq

Let C be an Fqm -linear code of length nLet c = (c1, . . . , cn) in CThen M (c) is the m × n matrix with entries cij :

cj =m∑i=1

cijαi

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Rank weight and distance

Let C be an Fqm -linear code of length n and c ∈ C

The rank weight of c iswtR (c) = rk(M (c))

The rank distance is defined by dR (x, y) = wtR (x− y)This defines a metric on Fn

qm

The rank distance of the code is

dR (C ) = min{ wtR (c) | 0 6= c ∈ C }

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Dictionary

The q-analogue of a finite set is a finite dimensional vector spaceWe list the q-analogues of some properties of subsets:

I , J subsets of {1, . . . , n} I , J subspaces of Fnq

∅ {0}I ∩ J intersection I ∩ J intersection

I ∪ J union I + J sum|I |, size of I dim(I ), dimension of I

Hamming distance on Fnq Rank distance on Fn

qm

Hamming weight on Fnq Rank weight on Fn

qm

supp(c)) Rsupp(c)) =?wtH (c)) = |supp(c)| wtR (c)) = dim(Rsupp(c))C an Fq -linear code C an Fqm -linear code

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Rank support

Let C be an Fqm -linear code of length n and c ∈ C

Rsupp(c) , the rank support of cis by definition the row space of M (c)Then

wtR (c) = rk(M (c)) = dim(Rsupp(c))

Let D be an Fqm -linear subcode of CRsupp(D ), the rank support of D isthe Fq -linear space generated by the Rsupp(d) with d ∈ DThe rank support weight of D is

wtR (D ) = dim Rsupp(D )

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Galois closure

The Frobenius map

ϕ : Fqm → Fqm with ϕ(x) = xq

is a field isomorphism that fixes Fq

The extension Fqm/Fq is Galois withcyclic Galois group generated by ϕExtend ϕ : Fn

qm → Fnqm component-wise

Let C be an Fqm -linear subspace of Fnqm

C ∗ is the Galois closure of C it is the smallest subspace of Fnqm

that contains C and that is closed under the action of ϕ

A subspace is called Galois closed if and only if it is equal to its ownGalois closure

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Restriction – extension

Let C ⊆ Fnqm be an Fqm -linear subspace

The restriction of C is defined by

C |Fq = C ∩ Fnq

Let D ⊆ Fnq be an Fq -linear subspace

D ⊗ Fqm is the extension of Dit is the Fqm -linear subspace of Fn

qm generated by D

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Trace

The trace mapTr : Fqm → Fq

is defined byx 7→ xq + · · · + xq

m

Extend Tr : Fnqm → Fn

q component-wise

Let C be an Fqm -linear subspace of Fnqm

Tr(C ) = { Tr(c) | c ∈ C }

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Summary

C ↪→ C ∗

↑ ↘ ↑ ⊗Fqm

C |Fq ↪→ Tr(C )

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Theorem (Giorgetti-Previtali 2010)

Let C be an Fqm -linear codeThen the following statements are equivalent:

I C is Galois closed: C = C ∗

I C is the extension of its restriction: C = (C |Fq)⊗ Fqm

I C has a basis in Fnq .

I The trace of C is equal to its restriction: Tr(C ) = C |Fq

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Rsupp is Trace

Let C be an Fqm -linear codeLet c ∈ C

Then the rows of the matrix M (c) areelements of the trace code Tr(C )

FurthermoreRsupp(C ) = Tr(C )

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Corollary

Let D be a subcode of the Fqm -linear code CThen

Rsupp(D ) = Tr(D )

and therefore

dR ,r(C ) = minD⊆C

dim(D )=r

wtR (D ) = minD⊆C

dim(D )=r

dim Tr(C ) = minD⊆C

dim(D )=r

dimD ∗

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Generalized rank weight

Several definitions are proposed for the generalized rank weight:

1. 2012 Oggier-Sboui

2. 2012 Kurihara-Matsumoto-Uyematsu

3. 2013 Ducoat

4. 2014 Jurrius-Pellikaan

5. 2015 Martínez-Peñas

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Definition Oggier-Sboui

C an Fqm -linear code

The r-th generalized rank weight of C

is defined by Oggier-Sboui as

minD⊆C

dim(D )=r

maxd∈D

wtR (d)

– “On the existence of generalized rank weights”IEEE Int. Symposium on Information Theory, pp. 406–410, 2012

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Definition Ducoat

C an Fqm -linear code

The r-th generalized rank weight of C

is defined by Ducoat as

minD⊆C

dim(D )=r

maxd∈D∗

wtR (d)

– “Generalized rank weights: a duality statement”Contemporary Mathematics, vol. 632, pp. 101–109, 2015

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Definition Kurihara-Matsumoto-Uyematsu

C an Fqm -linear code

The r-th generalized rank weight of C

is defined by Kurihara-Matsumoto-Uyematsu as

minV⊆Ln ,V=V∗dim(C∩V )≥r

dimV

– “New parameters of linear codes expressing security performance ofuniversal secure network coding”, Communication, Control, andComputing, 50th Annual Allerton Conference, pp. 533–540, 2012– “Relative generalized rank weight of linear codes and its applicationsto network coding”, arXiv:1301.5482v1, 2013

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Definition Jurrius-Pellikaan

C an Fqm -linear code

The r-th generalized rank weight of C

dR ,r(C ) = minD⊆C

dim(D )=r

wtR (D )

–“On defining generalized rank weights”arXiv:1506.02865

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Definition Martínez-Peñas

Let B = {b1, . . . ,bn} ⊆ Fnq be a basis of Fn

qm

Let ϕB : Fnqm → Fn

qm be defined byϕB (c) = x where c =

∑i xibi

Let D be an Fqm -linear code

wtR (D ) = min{ wtH (ϕB (D )) | B ⊆ Fnq a basis of Fn

qm }

Let C be an Fqm -linear code

dR ,r(C ) = minD⊆C

dim(D )=r

min{ wtH (ϕB (D )) | B ⊆ Fnq a basis of Fn

qm }

– “On the similarities between generalized rank and Hamming weightsand their applications to network coding”arXiv:1506.04036

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Conclusion

1. Oggier-Sboui

2. Ducoat

3. Kurihara-Matsumoto-Uyematsu

4. Jurrius-Pellikaan

5. Martínez-Peñas

If m ≥ n then,these definitions of the generalized rank weight are equivalent

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THANKS!

QUESTIONS?