on defining the generalized rank weightruudp/lectures/15-07-23-slides-kalamata.pdf · generalized...
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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?