/k Content 2/15 1. Introduction 2. Hamming weight 3. Rank weight - - PowerPoint PPT Presentation
On defining the generalized rank weight Ruud Pellikaan joint work with Relinde Jurrius Autonomous University Barcelona, 6 November 2014 /k Content 2/15 1. Introduction 2. Hamming weight 3. Rank weight 4. Extended rank weight enumerator
Ruud Pellikaan joint work with Relinde Jurrius Autonomous University Barcelona, 6 November 2014
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q, Hamming weight
q
, rank weight
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Some text S1
R1
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Fq is the finite field with q elements Fqm is the finite field extension of Fq of degree m An [n, k] code over Fq is a subspace of Fn
q of dimension k
The inner product on Fn
q 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 ∈ Fn
q | c · x = 0 for all c ∈ C }
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The support of x in Fn
q is defined by
supp(x) = { j | xj = 0 } The weight of x is defined by wt(x) = |supp(x)| that is the number of nonzero entries of x The support of subspace D of Fn
q is defined by
supp(D) = { j | xj = 0 for some x ∈ D } The weight of D is defined by wt(D) = |supp(D)|
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Let C be an Fq-linear code Then the minimum distance of C is d(C) = min{ wt(c) | 0 = c ∈ C } The r-th generalized distance of C is dr(C) = min{ wt(D) | D subspace of C, dim(D) = r } So d1(C) = d(C).
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Gabidulin defined rank weight Applications in network coding Choose a basis α1, . . . αm of Fqm as a vector space over Fq Let C be an Fqm-linear code of length n Let c = (c1, . . . , cn) in C Then M(c) is the m × n matrix with entries cij: cj =
m
cijαi
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Let C be an Fqm-linear code of length n and c ∈ C Rsupp(c) , the rank support of c is by definition the row space of M(c) The rank weight of c is wtR(c) = dim(Rsupp(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 = c ∈ C }
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The q-analogue of a finite set is a finite dimensional vector space We list the q-analogues of some properties of subsets: I, J subsets of {1, . . . , n} I, J subspaces of Fn
q
∅ {0} I ∩ J intersection I ∩ J intersection I ∪ J union I + J sum |I|, size of I dim(I), dimension of I n
k
n
k
Hamming distance on Fn
q
Rank distance on Fn
qm
C an Fq-linear code C an Fqm-linear code
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Let C be a linear code over Fq For a subset J of {1, 2, . . . , n} define C(J) = { c ∈ C | cj = 0 for all j ∈ J } l(J) = dim C(J) BJ(T) = T l(J) − 1 Bt(T) =
BJ(T) Note that BJ(qm) is the number of nonzero codewords in (C ⊗ Fqm)(J) Then WC(X, Y, T) = X n +
n
Bt(T)(X − Y)tY n−t
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This translation is sometimes ambiguous: I, J subsets of {1, . . . , n} I, J subspaces of Fn
q
C an Fq-linear code C an Fqm-linear code I ∩ J = ∅ I ∩ J = {0} J c complement of J J ⊥ orthoplement of J I ⊆ J c I ⊆ J ⊥ C(J) = {c ∈ C : cj = 0, ∀j ∈ J} ??? C(J) = {c ∈ C : supp(c) ∩ J = ∅} C(J) = {c ∈ C : Rsupp(c) ∩ J = {0}} C(J) = {c ∈ C : supp(c) ⊆ J c} C(J) = {c ∈ C : Rsupp(c) ⊆ J ⊥}
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Let C be an Fqm-linear code of length n For an Fq-linear subspace J of Fn
q define
C(J) = { c ∈ C : Rsupp(c) ⊆ J ⊥ } l(J) = dim C(J) B R
J (T)
= T m·l(J) B R
t (T)
=
BJ(T) Then B R
J (q) is the number of codewords in C(J)
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The extended rank weight enumerator is given by W R
C (X, Y, T) = n
A R
w(T)X n−wY w
Now A R
J (q) is the number of codewords in C of rank weight w
and A R
w(T) = n
(−1)t−n+wT(t−n+w
2 )
n − w
B R
t (T)
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