Generalized Gabidulin Codes and their Generator Matrices Alessandro - - PowerPoint PPT Presentation

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Generalized Gabidulin Codes and their Generator Matrices Alessandro - - PowerPoint PPT Presentation

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Generalized Gabidulin Codes and their Generator Matrices Alessandro Neri 23 July 2018 - LAWCI Alessandro Neri 11 June 2018 1 / 6 Generalized Reed-Solomon Codes Reed, Solomon (1960) F q [ x ] < k := { f ( x ) F q [ x ] | deg f < k }

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SLIDE 1

Generalized Gabidulin Codes and their Generator Matrices Alessandro Neri

23 July 2018 - LAWCI

Alessandro Neri 11 June 2018 1 / 6

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SLIDE 2

Generalized Reed-Solomon Codes

Reed, Solomon (1960) Fq[x]<k := {f (x) ∈ Fq[x] | deg f < k} .

Alessandro Neri 11 June 2018 1 / 6

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SLIDE 3

Generalized Reed-Solomon Codes

Reed, Solomon (1960) Fq[x]<k := {f (x) ∈ Fq[x] | deg f < k} . α1, . . . , αn ∈ Fq distinct elements b1, . . . , bn ∈ F∗

q

C = {(b1f (α1), b2f (α2), . . . , bnf (αn)) | f ∈ Fq[x]<k}

Alessandro Neri 11 June 2018 1 / 6

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SLIDE 4

Generalized Reed-Solomon Codes

Reed, Solomon (1960) Fq[x]<k := {f (x) ∈ Fq[x] | deg f < k} . α1, . . . , αn ∈ Fq distinct elements b1, . . . , bn ∈ F∗

q

C = {(b1f (α1), b2f (α2), . . . , bnf (αn)) | f ∈ Fq[x]<k} THEN C is the Generalized Reed-Solomon code GRSn,k(α, b)

Alessandro Neri 11 June 2018 1 / 6

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SLIDE 5

Generator matrix for GRS codes

Vk(α) =        1 1 . . . 1 α1 α2 . . . αn α2

1

α2

2

. . . α2

n

. . . . . . . . . αk−1

1

αk−1

2

. . . αk−1

n

       ,

Alessandro Neri 11 June 2018 2 / 6

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SLIDE 6

Generator matrix for GRS codes

Vk(α) =        1 1 . . . 1 α1 α2 . . . αn α2

1

α2

2

. . . α2

n

. . . . . . . . . αk−1

1

αk−1

2

. . . αk−1

n

       , Weighted Vandermonde (WV) matrix GRSn,k(α, b) = rowsp (Vk(α)diag(b)) =        b1 b2 . . . bn b1α1 b2α2 . . . bnαn b1α2

1

b2α2

2

. . . bnα2

n

. . . . . . . . . b1αk−1

1

b2αk−1

2

. . . bnαk−1

n

      

Alessandro Neri 11 June 2018 2 / 6

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SLIDE 7

Generalized Cauchy matrix

(1) x1, . . . , xr ∈ Fq pairwise distinct, (2) y1, . . . , ys ∈ Fq pairwise distinct, (3) y1, . . . , ys ∈ Fq \ {x1, . . . , xr}, (4) c1, . . . , cr, d1, . . . , ds ∈ F∗

q.

Alessandro Neri 11 June 2018 3 / 6

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SLIDE 8

Generalized Cauchy matrix

(1) x1, . . . , xr ∈ Fq pairwise distinct, (2) y1, . . . , ys ∈ Fq pairwise distinct, (3) y1, . . . , ys ∈ Fq \ {x1, . . . , xr}, (4) c1, . . . , cr, d1, . . . , ds ∈ F∗

q.

The matrix X ∈ Fr×s

q

defined by Xi,j = cidj

xi−yj is called Generalized Cauchy

(GC) Matrix. Theorem [Roth, Seroussi (1985)] There is a 1-1 correspondence between GC matrices and GRS codes codes given by X ← → rowsp( Ik | X ).

Alessandro Neri 11 June 2018 3 / 6

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SLIDE 9

Linearized Polynomials and Gabidulin Codes

Delsarte (1978), Gabidulin (1985), Kshevetskiy and Gabidulin (2005)

m−1

  • i=0

fix[i] a linearized polynomial over Fqm, [i] := qi. Gk,s :=

  • f0x + f1x[s] + . . . + fk−1x[s(k−1)] | fi ∈ Fqm

.

Alessandro Neri 11 June 2018 4 / 6

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SLIDE 10

Linearized Polynomials and Gabidulin Codes

Delsarte (1978), Gabidulin (1985), Kshevetskiy and Gabidulin (2005)

m−1

  • i=0

fix[i] a linearized polynomial over Fqm, [i] := qi. Gk,s :=

  • f0x + f1x[s] + . . . + fk−1x[s(k−1)] | fi ∈ Fqm

. g1, . . . , gn ∈ Fqm linearly independent over Fq s is integer coprime to m C = {(f (g1), f (g2), . . . , f (gn)) | f ∈ Gk,s}

Alessandro Neri 11 June 2018 4 / 6

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SLIDE 11

Linearized Polynomials and Gabidulin Codes

Delsarte (1978), Gabidulin (1985), Kshevetskiy and Gabidulin (2005)

m−1

  • i=0

fix[i] a linearized polynomial over Fqm, [i] := qi. Gk,s :=

  • f0x + f1x[s] + . . . + fk−1x[s(k−1)] | fi ∈ Fqm

. g1, . . . , gn ∈ Fqm linearly independent over Fq s is integer coprime to m C = {(f (g1), f (g2), . . . , f (gn)) | f ∈ Gk,s} THEN C is the Generalized Gabidulin code Gk,s(g1, . . . , gn) of parameter s

Alessandro Neri 11 June 2018 4 / 6

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SLIDE 12

Generator matrix for Gk,s(g1, . . . , gn)

Let σ : Fqm − → Fqm x − → xq be the q-Frobenius automorphism.

Alessandro Neri 11 June 2018 5 / 6

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SLIDE 13

Generator matrix for Gk,s(g1, . . . , gn)

Let σ : Fqm − → Fqm x − → xq be the q-Frobenius automorphism. s-Moore Matrix

Gk,s(g1, . . . , gn) = rowsp      g1 g2 . . . gn σs(g1) σs(g2) . . . σs(gn) . . . . . . . . . σs(k−1)(g1) σs(k−1)(g2) . . . σs(k−1)(gn)      ,

Alessandro Neri 11 June 2018 5 / 6

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SLIDE 14

Hamming vs Rank distance: Recap

dH dR

Alessandro Neri 11 June 2018 6 / 6

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SLIDE 15

Hamming vs Rank distance: Recap

dH GRS codes dR

Alessandro Neri 11 June 2018 6 / 6

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SLIDE 16

Hamming vs Rank distance: Recap

dH GRS codes

  • dR

Gabidulin codes

Alessandro Neri 11 June 2018 6 / 6

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SLIDE 17

Hamming vs Rank distance: Recap

dH GRS codes ← → WV matrices

  • dR

Gabidulin codes

Alessandro Neri 11 June 2018 6 / 6

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SLIDE 18

Hamming vs Rank distance: Recap

dH GRS codes ← → WV matrices

  • dR

Gabidulin codes ← → Moore matrices

Alessandro Neri 11 June 2018 6 / 6

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SLIDE 19

Hamming vs Rank distance: Recap

dH GRS codes ← → WV matrices ← → GC matrices

  • dR

Gabidulin codes ← → Moore matrices

Alessandro Neri 11 June 2018 6 / 6

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SLIDE 20

Hamming vs Rank distance: Recap

dH GRS codes ← → WV matrices ← → GC matrices

  • dR

Gabidulin codes ← → Moore matrices ← → ???

Alessandro Neri 11 June 2018 6 / 6

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SLIDE 21

Hamming vs Rank distance: Recap

dH GRS codes ← → WV matrices ← → GC matrices

  • dR

Gabidulin codes ← → Moore matrices ← → ??? The answer? It’s in the poster!

Alessandro Neri 11 June 2018 6 / 6