[PPT] - Some recent developments in the theory of linear MRD-codes Olga PowerPoint Presentation

SLIDE 1

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Some recent developments in the theory of linear MRD-codes

Olga Polverino

Università degli Studi della Campania, “Luigi Vanvitelli”

Finite Geometries 2017, Fifth Irsee Conference

Irsee

SLIDE 2

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Outline

1

Linear MRD-codes

2

Generalized Gabidulin Codes and linearized polynomials

3

Generalized Twisted Gabidulin Codes

4

MRD-codes-Maximum Scattered Spaces-Segre Variety

SLIDE 3

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

RD-codes Fm×n

q

SLIDE 4

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

RD-codes Fm×n

q

d(A, B) = rk (A − B) for A, B ∈ Fm×n

q

SLIDE 5

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

RD-codes Fm×n

q

d(A, B) = rk (A − B) for A, B ∈ Fm×n

q

C ⊆ Fm×n

q

(C, d)

SLIDE 6

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

RD-codes Fm×n

q

d(A, B) = rk (A − B) for A, B ∈ Fm×n

q

C ⊆ Fm×n

q

(C, d) Rank Distance Code (RD-code)

SLIDE 7

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

RD-codes Fm×n

q

d(A, B) = rk (A − B) for A, B ∈ Fm×n

q

C ⊆ Fm×n

q

(C, d) Rank Distance Code (RD-code) d(C) = min

A,B∈C, A=B{d(A, B)}

SLIDE 8

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

RD-codes Fm×n

q

d(A, B) = rk (A − B) for A, B ∈ Fm×n

q

C ⊆ Fm×n

q

(C, d) Rank Distance Code (RD-code) d(C) = min

A,B∈C, A=B{d(A, B)}

d(C) minimum distance of C

SLIDE 9

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linear RD-codes C → Fq-subspace of Fm×n

q

SLIDE 10

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linear RD-codes C → Fq-subspace of Fm×n

q

C Fq-linear RD-code

SLIDE 11

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linear RD-codes C → Fq-subspace of Fm×n

q

C Fq-linear RD-code t= dimq(C)

SLIDE 12

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linear RD-codes C → Fq-subspace of Fm×n

q

C Fq-linear RD-code t= dimq(C) C → [m × n, t, d]Fq

SLIDE 13

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linear RD-codes C → Fq-subspace of Fm×n

q

C Fq-linear RD-code t= dimq(C) C → [m × n, t, d]Fq t ≤ max{m, n}(min{m, n} − d + 1)

SLIDE 14

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linear RD-codes C → Fq-subspace of Fm×n

q

C Fq-linear RD-code t= dimq(C) C → [m × n, t, d]Fq t ≤ max{m, n}(min{m, n} − d + 1) Singleton-like bound

SLIDE 15

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linear RD-codes C → Fq-subspace of Fm×n

q

C Fq-linear RD-code t= dimq(C) C → [m × n, t, d]Fq t ≤ max{m, n}(min{m, n} − d + 1) Singleton-like bound

P. Delsarte: "Bilinear forms over a finite field, with applications to coding

theory", J. Comb. Theory Ser. A (1978)

SLIDE 16

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linear MRD-codes C linear RD-code, [m × n, t, d]Fq

SLIDE 17

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linear MRD-codes C linear RD-code, [m × n, t, d]Fq t = max{m, n}(min{m, n} − d + 1)

SLIDE 18

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linear MRD-codes C linear RD-code, [m × n, t, d]Fq t = max{m, n}(min{m, n} − d + 1) C → Maximum Rank Distance code (MRD-code)

SLIDE 19

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linear MRD-codes C linear RD-code, [m × n, t, d]Fq t = max{m, n}(min{m, n} − d + 1) C → Maximum Rank Distance code (MRD-code) d = 1 ⇒ t = m · n ⇒ C = Fm×n

q

SLIDE 20

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linear MRD-codes C linear RD-code, [m × n, t, d]Fq t = max{m, n}(min{m, n} − d + 1) C → Maximum Rank Distance code (MRD-code) d = 1 ⇒ t = m · n ⇒ C = Fm×n

q

max{m, n} | t → k =

t max{m,n}

SLIDE 21

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linear MRD-codes C linear RD-code, [m × n, t, d]Fq t = max{m, n}(min{m, n} − d + 1) C → Maximum Rank Distance code (MRD-code) d = 1 ⇒ t = m · n ⇒ C = Fm×n

q

max{m, n} | t → k =

t max{m,n}

k = min{m, n} − d + 1

SLIDE 22

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Adjoint and dual of MRD-codes C linear MRD-code [m × n, t, d]Fq

SLIDE 23

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Adjoint and dual of MRD-codes C linear MRD-code [m × n, t, d]Fq ⇓ CT linear MRD-code [n × m, t, d]Fq

SLIDE 24

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Adjoint and dual of MRD-codes C linear MRD-code [m × n, t, d]Fq ⇓ CT linear MRD-code [n × m, t, d]Fq the adjoint code of C

SLIDE 25

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Adjoint and dual of MRD-codes C linear MRD-code [m × n, t, d]Fq ⇓ CT linear MRD-code [n × m, t, d]Fq the adjoint code of C C linear MRD-code [m × n, t, d]Fq (d > 1)

SLIDE 26

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Adjoint and dual of MRD-codes C linear MRD-code [m × n, t, d]Fq ⇓ CT linear MRD-code [n × m, t, d]Fq the adjoint code of C C linear MRD-code [m × n, t, d]Fq (d > 1) C⊥ = {M ∈ Fm×n

q

: Tr(MNT) = 0 ∀N ∈ C}

SLIDE 27

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Adjoint and dual of MRD-codes C linear MRD-code [m × n, t, d]Fq ⇓ CT linear MRD-code [n × m, t, d]Fq the adjoint code of C C linear MRD-code [m × n, t, d]Fq (d > 1) C⊥ = {M ∈ Fm×n

q

: Tr(MNT) = 0 ∀N ∈ C} ⇓ C⊥ MRD-code [m × n, mn − t, min{m, n} − d + 2]Fq

SLIDE 28

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Adjoint and dual of MRD-codes C linear MRD-code [m × n, t, d]Fq ⇓ CT linear MRD-code [n × m, t, d]Fq the adjoint code of C C linear MRD-code [m × n, t, d]Fq (d > 1) C⊥ = {M ∈ Fm×n

q

: Tr(MNT) = 0 ∀N ∈ C} ⇓ C⊥ MRD-code [m × n, mn − t, min{m, n} − d + 2]Fq the Delsarte dual code of C

SLIDE 29

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Equivalence of RD-codes

SLIDE 30

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Equivalence of RD-codes C, C′ ⊂ Fm×n

q

linear RD-codes

SLIDE 31

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Equivalence of RD-codes C, C′ ⊂ Fm×n

q

linear RD-codes C, C′ equivalent

SLIDE 32

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Equivalence of RD-codes C, C′ ⊂ Fm×n

q

linear RD-codes C, C′ equivalent C′ = {ACσB : C ∈ C} A ∈ GL(m, Fq), B ∈ GL(n, Fq) and σ ∈ Aut(Fq)

SLIDE 33

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Equivalence of RD-codes C, C′ ⊂ Fm×n

q

linear RD-codes C, C′ equivalent C′ = {ACσB : C ∈ C} A ∈ GL(m, Fq), B ∈ GL(n, Fq) and σ ∈ Aut(Fq) Aut(C) Automorphism group of C

Aut(C) = {(A, B, σ) ∈ GL(m, q) × GL(n, q) × Aut(Fq) : ACσB = C}

SLIDE 34

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Left and Right idealiser of RD-codes C linear RD-code in Fm×n

q

SLIDE 35

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Left and Right idealiser of RD-codes C linear RD-code in Fm×n

q

L(C) = {Y ∈ Fm×m

q

: YC ∈ C for all C ∈ C} Left idealiser of C

SLIDE 36

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Left and Right idealiser of RD-codes C linear RD-code in Fm×n

q

L(C) = {Y ∈ Fm×m

q

: YC ∈ C for all C ∈ C} Left idealiser of C R(C) = {Z ∈ Fn×n

q

: CZ ∈ C for all C ∈ C} Right idealiser of C

SLIDE 37

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Left and Right idealiser of RD-codes C linear RD-code in Fm×n

q

L(C) = {Y ∈ Fm×m

q

: YC ∈ C for all C ∈ C} Left idealiser of C R(C) = {Z ∈ Fn×n

q

: CZ ∈ C for all C ∈ C} Right idealiser of C

D. Liebhold, G. Nebe: Automorphism groups of Gabidulin-like codes. Arch.
Math. (2016)

SLIDE 38

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Left and Right idealiser of RD-codes C linear RD-code in Fm×n

q

L(C) = {Y ∈ Fm×m

q

: YC ∈ C for all C ∈ C} Left idealiser of C R(C) = {Z ∈ Fn×n

q

: CZ ∈ C for all C ∈ C} Right idealiser of C L(C)∗ × R(C)∗ × {id} ⊆ Aut(C)

D. Liebhold, G. Nebe: Automorphism groups of Gabidulin-like codes. Arch.
Math. (2016)

SLIDE 39

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Left and Right idealiser of MRD-codes Lunardon-Trombetti-Zhou 2017, JACO

SLIDE 40

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Left and Right idealiser of MRD-codes Lunardon-Trombetti-Zhou 2017, JACO For two equivalent linear metric codes their right (resp. left) idealisers are also equivalent.

SLIDE 41

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Left and Right idealiser of MRD-codes Lunardon-Trombetti-Zhou 2017, JACO For two equivalent linear metric codes their right (resp. left) idealisers are also equivalent. Let C be a linear MRD code in Fm×n

q

with d > 1. If m ≤ n then L(C) is a finite field. Hence |L(C)| ≤ qm. If n ≤ m then R(C) is a finite field. Hence |R(C)| ≤ qn.

SLIDE 42

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Left and Right idealiser of MRD-codes Lunardon-Trombetti-Zhou 2017, JACO For two equivalent linear metric codes their right (resp. left) idealisers are also equivalent. Let C be a linear MRD code in Fm×n

q

with d > 1. If m ≤ n then L(C) is a finite field. Hence |L(C)| ≤ qm. If n ≤ m then R(C) is a finite field. Hence |R(C)| ≤ qn. If C is a linear MRD code in Fn×n

q

with d > 1 then the left and the right idealisers of C are both finite fields.

SLIDE 43

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Left and Right idealiser of MRD-codes Lunardon-Trombetti-Zhou 2017, JACO For two equivalent linear metric codes their right (resp. left) idealisers are also equivalent. Let C be a linear MRD code in Fm×n

q

with d > 1. If m ≤ n then L(C) is a finite field. Hence |L(C)| ≤ qm. If n ≤ m then R(C) is a finite field. Hence |R(C)| ≤ qn. If C is a linear MRD code in Fn×n

q

with d > 1 then the left and the right idealisers of C are both finite fields.

SLIDE 44

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Left and Right idealiser of MRD-codes Lunardon-Trombetti-Zhou 2017, JACO For two equivalent linear metric codes their right (resp. left) idealisers are also equivalent. Let C be a linear MRD code in Fm×n

q

with d > 1. If m ≤ n then L(C) is a finite field. Hence |L(C)| ≤ qm. If n ≤ m then R(C) is a finite field. Hence |R(C)| ≤ qn. If C is a linear MRD code in Fn×n

q

with d > 1 then the left and the right idealisers of C are both finite fields. dimL(C)(C) dimR(C)(C)

SLIDE 45

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

First Examples of linear MRD-codes

P. Delsarte: Bilinear forms over a finite field, with applications to coding

theory, J. Comb. Theory Ser. A (1978)

SLIDE 46

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

First Examples of linear MRD-codes

P. Delsarte: Bilinear forms over a finite field, with applications to coding

theory, J. Comb. Theory Ser. A (1978)

E. Gabidulin: Theory of codes with maximum rank distance, Probl. Inf.
Transm. (1985)

SLIDE 47

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

First Examples of linear MRD-codes

P. Delsarte: Bilinear forms over a finite field, with applications to coding

theory, J. Comb. Theory Ser. A (1978)

E. Gabidulin: Theory of codes with maximum rank distance, Probl. Inf.
Transm. (1985)
A. Kshevetskiy, E. Gabidulin: The new construction of rank codes,

Proceedings ISIT, (2005)

SLIDE 48

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Gabidulin MRD-codes Gk = {f(x) = a0x+a1xq+. . . ak−1xq(k−1) : a0, a1, . . . , ak−1 ∈ Fqn}, n, k ∈ Z+ k < n

SLIDE 49

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Gabidulin MRD-codes Gk = {f(x) = a0x+a1xq+. . . ak−1xq(k−1) : a0, a1, . . . , ak−1 ∈ Fqn}, n, k ∈ Z+ k < n dimFqKer f ≤ k − 1

SLIDE 50

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Gabidulin MRD-codes Gk = {f(x) = a0x+a1xq+. . . ak−1xq(k−1) : a0, a1, . . . , ak−1 ∈ Fqn}, n, k ∈ Z+ k < n dimFqKer f ≤ k − 1 ⇒ dimFqIm f ≥ n − k + 1

SLIDE 51

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Gabidulin MRD-codes Gk = {f(x) = a0x+a1xq+. . . ak−1xq(k−1) : a0, a1, . . . , ak−1 ∈ Fqn}, n, k ∈ Z+ k < n dimFqKer f ≤ k − 1 ⇒ dimFqIm f ≥ n − k + 1 B Fq-basis of Fqn → Mf ∈ Fn×n

q

matrix associated with f w.r.t. B

SLIDE 52

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Gabidulin MRD-codes Gk = {f(x) = a0x+a1xq+. . . ak−1xq(k−1) : a0, a1, . . . , ak−1 ∈ Fqn}, n, k ∈ Z+ k < n dimFqKer f ≤ k − 1 ⇒ dimFqIm f ≥ n − k + 1 B Fq-basis of Fqn → Mf ∈ Fn×n

q

matrix associated with f w.r.t. B CGk = {Mf : f ∈ Gk} ⊆ Fn×n

q

,

SLIDE 53

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Gabidulin MRD-codes Gk = {f(x) = a0x+a1xq+. . . ak−1xq(k−1) : a0, a1, . . . , ak−1 ∈ Fqn}, n, k ∈ Z+ k < n dimFqKer f ≤ k − 1 ⇒ dimFqIm f ≥ n − k + 1 B Fq-basis of Fqn → Mf ∈ Fn×n

q

matrix associated with f w.r.t. B CGk = {Mf : f ∈ Gk} ⊆ Fn×n

q

, CGk Gabidulin MRD-code [n × n, kn, n − k + 1]Fq

SLIDE 54

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Gabidulin MRD-codes Gk = {f(x) = a0x+a1xq+· · ·+ak−1xq(k−1) : a0, a1, . . . , ak−1 ∈ Fqn}, n, k ∈ Z+ k < n

SLIDE 55

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Gabidulin MRD-codes Gk = {f(x) = a0x+a1xq+· · ·+ak−1xq(k−1) : a0, a1, . . . , ak−1 ∈ Fqn}, n, k ∈ Z+ k < n U ⊆ Fqn, dimFqU = m ≤ n, BU basis of U

SLIDE 56

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Gabidulin MRD-codes Gk = {f(x) = a0x+a1xq+· · ·+ak−1xq(k−1) : a0, a1, . . . , ak−1 ∈ Fqn}, n, k ∈ Z+ k < n U ⊆ Fqn, dimFqU = m ≤ n, BU basis of U CGk(U) = {Mf|U : f ∈ Gk} ⊆ Fn×m

q

,

SLIDE 57

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Gabidulin MRD-codes Gk = {f(x) = a0x+a1xq+· · ·+ak−1xq(k−1) : a0, a1, . . . , ak−1 ∈ Fqn}, n, k ∈ Z+ k < n U ⊆ Fqn, dimFqU = m ≤ n, BU basis of U CGk(U) = {Mf|U : f ∈ Gk} ⊆ Fn×m

q

, CGk(U) Gabidulin MRD-code [n × m, kn, n − k + 1]Fq

SLIDE 58

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Gabidulin MRD-codes Gk = {f(x) = a0x+a1xq+· · ·+ak−1xq(k−1) : a0, a1, . . . , ak−1 ∈ Fqn}, n, k ∈ Z+ k < n U ⊆ Fqn, dimFqU = m ≤ n, BU basis of U CGk(U) = {Mf|U : f ∈ Gk} ⊆ Fn×m

q

, CGk(U) Gabidulin MRD-code [n × m, kn, n − k + 1]Fq CGk(U)T Gabidulin MRD-code [m × n, kn, n − k + 1]Fq

SLIDE 59

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Generalised Gabidulin MRD-codes Gk,s = {f(x) = a0x+a1xqs+· · ·+ak−1xqs(k−1) : a0, a1, . . . , ak−1 ∈ Fqn}, n, k, s ∈ Z+, k < n, gcd(n, s) = 1

SLIDE 60

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Generalised Gabidulin MRD-codes Gk,s = {f(x) = a0x+a1xqs+· · ·+ak−1xqs(k−1) : a0, a1, . . . , ak−1 ∈ Fqn}, n, k, s ∈ Z+, k < n, gcd(n, s) = 1 CGk,s = {Mf : f ∈ Gk} ⊆ Fn×n

q

,

SLIDE 61

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Generalised Gabidulin MRD-codes Gk,s = {f(x) = a0x+a1xqs+· · ·+ak−1xqs(k−1) : a0, a1, . . . , ak−1 ∈ Fqn}, n, k, s ∈ Z+, k < n, gcd(n, s) = 1 CGk,s = {Mf : f ∈ Gk} ⊆ Fn×n

q

,

CGk,s Generalised Gabidulin MRD-code [n × n, kn, n − k + 1]Fq

SLIDE 62

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Generalised Gabidulin MRD-codes Gk,s = {f(x) = a0x+a1xqs+· · ·+ak−1xqs(k−1) : a0, a1, . . . , ak−1 ∈ Fqn}, n, k, s ∈ Z+, k < n, gcd(n, s) = 1 CGk,s = {Mf : f ∈ Gk} ⊆ Fn×n

q

,

CGk,s Generalised Gabidulin MRD-code [n × n, kn, n − k + 1]Fq CGk,s(U) Generalised Gabidulin MRD-code [n × m, kn, n − k + 1]Fq CGk,s(U)T Generalised Gabidulin MRD-code [m × n, kn, n − k + 1]Fq

SLIDE 63

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linearized polynomials Fn×n

q

SLIDE 64

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linearized polynomials Fn×n

q

↓ End(Fqn, Fq)

SLIDE 65

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linearized polynomials Fn×n

q

↓ End(Fqn, Fq) ↓ Ln,q = {f(x) = a0x + a1xq + . . . an−1xq(n−1) : a0, a1, . . . , an−1 ∈ Fqn}

SLIDE 66

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linearized polynomials Fn×n

q

↓ End(Fqn, Fq) ↓ Ln,q = {f(x) = a0x + a1xq + . . . an−1xq(n−1) : a0, a1, . . . , an−1 ∈ Fqn} (Ln,q, +, ◦) ≃ (Fn×n

q

, +, ·)

SLIDE 67

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linearized polynomials Fn×n

q

↓ End(Fqn, Fq) ↓ Ln,q = {f(x) = a0x + a1xq + . . . an−1xq(n−1) : a0, a1, . . . , an−1 ∈ Fqn} (Ln,q, +, ◦) ≃ (Fn×n

q

, +, ·)

→

composition modulo xqn − x

SLIDE 68

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linearized polynomials C, C′ ⊆ Ln,q equivalent MRD-codes

SLIDE 69

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linearized polynomials C, C′ ⊆ Ln,q equivalent MRD-codes C′ = {g ◦ f σ ◦ h : f ∈ C} f, g permutation q-polynomials, σ ∈ Aut(Fq)

SLIDE 70

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linearized polynomials C, C′ ⊆ Ln,q equivalent MRD-codes C′ = {g ◦ f σ ◦ h : f ∈ C} f, g permutation q-polynomials, σ ∈ Aut(Fq) C ⊆ Ln,q MRD-code

SLIDE 71

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linearized polynomials C, C′ ⊆ Ln,q equivalent MRD-codes C′ = {g ◦ f σ ◦ h : f ∈ C} f, g permutation q-polynomials, σ ∈ Aut(Fq) C ⊆ Ln,q MRD-code L(C) = {g ∈ Ln,q : g ◦ f ∈ C for all f ∈ C} Left idealiser of C

SLIDE 72

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linearized polynomials C, C′ ⊆ Ln,q equivalent MRD-codes C′ = {g ◦ f σ ◦ h : f ∈ C} f, g permutation q-polynomials, σ ∈ Aut(Fq) C ⊆ Ln,q MRD-code L(C) = {g ∈ Ln,q : g ◦ f ∈ C for all f ∈ C} Left idealiser of C R(C) = {h ∈ Ln,q : f ◦ h ∈ C for all f ∈ C} Right idealiser of C

SLIDE 73

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linearized polynomials

Gk,s = {f(x) = a0x + a1xqs + . . . ak−1xqs(k−1) : a0, a1, . . . , ak−1 ∈ Fqn} n, k, s ∈ Z+, k < n, gcd(n, s) = 1 Gk,s Generalised Gabidulin MRD-code [n × n, kn, n − k + 1]Fq

SLIDE 74

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linearized polynomials

Gk,s = {f(x) = a0x + a1xqs + . . . ak−1xqs(k−1) : a0, a1, . . . , ak−1 ∈ Fqn} n, k, s ∈ Z+, k < n, gcd(n, s) = 1 Gk,s Generalised Gabidulin MRD-code [n × n, kn, n − k + 1]Fq

SLIDE 75

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linearized polynomials

Gk,s = {f(x) = a0x + a1xqs + . . . ak−1xqs(k−1) : a0, a1, . . . , ak−1 ∈ Fqn} n, k, s ∈ Z+, k < n, gcd(n, s) = 1 Gk,s Generalised Gabidulin MRD-code [n × n, kn, n − k + 1]Fq Fqn = {τα(x) = αx : α ∈ Fqn}

SLIDE 76

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linearized polynomials

Gk,s = {f(x) = a0x + a1xqs + . . . ak−1xqs(k−1) : a0, a1, . . . , ak−1 ∈ Fqn} n, k, s ∈ Z+, k < n, gcd(n, s) = 1 Gk,s Generalised Gabidulin MRD-code [n × n, kn, n − k + 1]Fq Fqn = {τα(x) = αx : α ∈ Fqn} f ◦ τα ∈ Gk,s, τα ◦ f ∈ Gk,s for all f ∈ Gk,s, τα ∈ Fqn

SLIDE 77

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linearized polynomials

Gk,s = {f(x) = a0x + a1xqs + . . . ak−1xqs(k−1) : a0, a1, . . . , ak−1 ∈ Fqn} n, k, s ∈ Z+, k < n, gcd(n, s) = 1 Gk,s Generalised Gabidulin MRD-code [n × n, kn, n − k + 1]Fq Fqn = {τα(x) = αx : α ∈ Fqn} f ◦ τα ∈ Gk,s, τα ◦ f ∈ Gk,s for all f ∈ Gk,s, τα ∈ Fqn L(Gk,s) = Fqn R(Gk,s) = Fqn

SLIDE 78

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linearized polynomials

Gk,s = {f(x) = a0x + a1xqs + . . . ak−1xqs(k−1) : a0, a1, . . . , ak−1 ∈ Fqn} n, k, s ∈ Z+, k < n, gcd(n, s) = 1 Gk,s Generalised Gabidulin MRD-code [n × n, kn, n − k + 1]Fq Fqn = {τα(x) = αx : α ∈ Fqn} f ◦ τα ∈ Gk,s, τα ◦ f ∈ Gk,s for all f ∈ Gk,s, τα ∈ Fqn L(Gk,s) = Fqn R(Gk,s) = Fqn

SLIDE 79

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Linearized polynomials

Gk,s = {f(x) = a0x + a1xqs + . . . ak−1xqs(k−1) : a0, a1, . . . , ak−1 ∈ Fqn} n, k, s ∈ Z+, k < n, gcd(n, s) = 1 Gk,s Generalised Gabidulin MRD-code [n × n, kn, n − k + 1]Fq Fqn = {τα(x) = αx : α ∈ Fqn} f ◦ τα ∈ Gk,s, τα ◦ f ∈ Gk,s for all f ∈ Gk,s, τα ∈ Fqn L(Gk,s) = Fqn R(Gk,s) = Fqn Gk,s Fqn-linear MRD-code

SLIDE 80

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Finite presemifields

J. De La Cruz, Kiermaier, Wassermann, Willem: Algebraic structures of

MRD codes, Adv. Math. Commun. (2016)

SLIDE 81

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Finite presemifields

J. De La Cruz, Kiermaier, Wassermann, Willem: Algebraic structures of

MRD codes, Adv. Math. Commun. (2016)

J. Sheekey: A new family of linear maximum rank distance codes, Adv.
Math. Commun. (2016)

SLIDE 82

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Finite presemifields

J. De La Cruz, Kiermaier, Wassermann, Willem: Algebraic structures of

MRD codes, Adv. Math. Commun. (2016)

J. Sheekey: A new family of linear maximum rank distance codes, Adv.
Math. Commun. (2016)

C Fq-linear MRD-code, [n × n, n, n]Fq

SC presemifield of order qn with left nucleus containing Fq

SLIDE 83

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Finite presemifields

J. De La Cruz, Kiermaier, Wassermann, Willem: Algebraic structures of

MRD codes, Adv. Math. Commun. (2016)

J. Sheekey: A new family of linear maximum rank distance codes, Adv.
Math. Commun. (2016)

C Fq-linear MRD-code, [n × n, n, n]Fq

πSC semifield plane of order qn

SLIDE 84

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Finite presemifields

J. De La Cruz, Kiermaier, Wassermann, Willem: Algebraic structures of

MRD codes, Adv. Math. Commun. (2016)

J. Sheekey: A new family of linear maximum rank distance codes, Adv.
Math. Commun. (2016)

C Fq-linear MRD-code, [n × n, n, n]Fq

πSC semifield plane of order qn

G1,s → SG1,s isotopic to Fqn

SLIDE 85

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Finite presemifields

J. De La Cruz, Kiermaier, Wassermann, Willem: Algebraic structures of

MRD codes, Adv. Math. Commun. (2016)

J. Sheekey: A new family of linear maximum rank distance codes, Adv.
Math. Commun. (2016)

C Fq-linear MRD-code, [n × n, n, n]Fq

πSC semifield plane of order qn

G1,s → πSG1,s Desarguesian plane

SLIDE 86

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Generalized Twisted Gabidulin Codes Sheekey, 2016

Hk,s(η, h) = {f(x) = a0x + a1xqs + · · · + ak−1xqs(k−1) + ηaqh

0 xqsk : ai ∈ Fqn}

n, k, s ∈ Z+, k < n, gcd(n, s) = 1, Nq(η) = (−1)nk

SLIDE 87

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Generalized Twisted Gabidulin Codes Sheekey, 2016

Hk,s(η, h) = {f(x) = a0x + a1xqs + · · · + ak−1xqs(k−1) + ηaqh

0 xqsk : ai ∈ Fqn}

n, k, s ∈ Z+, k < n, gcd(n, s) = 1, Nq(η) = (−1)nk

SLIDE 88

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Generalized Twisted Gabidulin Codes Sheekey, 2016

Hk,s(η, h) = {f(x) = a0x + a1xqs + · · · + ak−1xqs(k−1) + ηaqh

0 xqsk : ai ∈ Fqn}

n, k, s ∈ Z+, k < n, gcd(n, s) = 1, Nq(η) = (−1)nk Hk,s(η, h) MRD-code [n × n, kn, n − k + 1]Fq

SLIDE 89

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Generalized Twisted Gabidulin Codes Sheekey, 2016

Hk,s(η, h) = {f(x) = a0x + a1xqs + · · · + ak−1xqs(k−1) + ηaqh

0 xqsk : ai ∈ Fqn}

n, k, s ∈ Z+, k < n, gcd(n, s) = 1, Nq(η) = (−1)nk Hk,s(η, h) MRD-code [n × n, kn, n − k + 1]Fq Hk,s(0, h) = Gk,s

SLIDE 90

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Generalized Twisted Gabidulin Codes Sheekey, 2016

Hk,s(η, h) = {f(x) = a0x + a1xqs + · · · + ak−1xqs(k−1) + ηaqh

0 xqsk : ai ∈ Fqn}

n, k, s ∈ Z+, k < n, gcd(n, s) = 1, Nq(η) = (−1)nk Hk,s(η, h) MRD-code [n × n, kn, n − k + 1]Fq Hk,s(0, h) = Gk,s H1,s(η, h) → a0x + ηaqh

0 xqs

Generalized twisted field

SLIDE 91

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Generalized Twisted Gabidulin Codes Sheekey, 2016

Hk,s(η, h) = {f(x) = a0x + a1xqs + · · · + ak−1xqs(k−1) + ηaqh

0 xqsk : ai ∈ Fqn}

n, k, s ∈ Z+, k < n, gcd(n, s) = 1, Nq(η) = (−1)nk Hk,s(η, h) MRD-code [n × n, kn, n − k + 1]Fq Hk,s(0, h) = Gk,s H1,s(η, h) → a0x + ηaqh

0 xqs

Generalized twisted field

H2(η, 1)

K. Otal, F. Özbudak: Explicit Construction of Some Non-Gabidulin Linear

Maximum Rank Distance Codes, Adv. Math. Commun. (2016)

SLIDE 92

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Generalized Twisted Gabidulin Codes Sheekey 2016/ Lunardon-Trombetti-Zhou, ArXiv (2015)

SLIDE 93

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Generalized Twisted Gabidulin Codes Sheekey 2016/ Lunardon-Trombetti-Zhou, ArXiv (2015) If η = 0, then Hk,s(η, h) ≃ Gk,s unless k ∈ {1, n − 1} and h ∈ {0, 1}.

SLIDE 94

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Generalized Twisted Gabidulin Codes Sheekey 2016/ Lunardon-Trombetti-Zhou, ArXiv (2015) If η = 0, then Hk,s(η, h) ≃ Gk,s unless k ∈ {1, n − 1} and h ∈ {0, 1}. Lunardon-Trombetti-Zhou (2017),JACO

SLIDE 95

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Generalized Twisted Gabidulin Codes Sheekey 2016/ Lunardon-Trombetti-Zhou, ArXiv (2015) If η = 0, then Hk,s(η, h) ≃ Gk,s unless k ∈ {1, n − 1} and h ∈ {0, 1}. Lunardon-Trombetti-Zhou (2017),JACO Hk,s(η, h)

SLIDE 96

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Generalized Twisted Gabidulin Codes Sheekey 2016/ Lunardon-Trombetti-Zhou, ArXiv (2015) If η = 0, then Hk,s(η, h) ≃ Gk,s unless k ∈ {1, n − 1} and h ∈ {0, 1}. Lunardon-Trombetti-Zhou (2017),JACO Hk,s(η, h) If η = 0, then Hk,s(0, h) = Gk,s and L(Gk,s) = R(Gk,s) ≃ Fqn.

SLIDE 97

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Generalized Twisted Gabidulin Codes Sheekey 2016/ Lunardon-Trombetti-Zhou, ArXiv (2015) If η = 0, then Hk,s(η, h) ≃ Gk,s unless k ∈ {1, n − 1} and h ∈ {0, 1}. Lunardon-Trombetti-Zhou (2017),JACO Hk,s(η, h) If η = 0, then Hk,s(0, h) = Gk,s and L(Gk,s) = R(Gk,s) ≃ Fqn. If η = 0 and 1 < k < n − 1, then L(Hk,s(η, h)) ≃ Fqgcd(n,h) and R(Hk,s(η, h)) ≃ Fqgcd(n,sk−h).

SLIDE 98

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Subfamilies of H H = {Hk,s(η, h) ⊆ Ln,q}

SLIDE 99

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Subfamilies of H H = {Hk,s(η, h) ⊆ Ln,q} G = {Gk,s ⊆ Ln,q}

SLIDE 100

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Subfamilies of H H = {Hk,s(η, h) ⊆ Ln,q} G = {Gk,s ⊆ Ln,q} HL = {C ∈ H : L(C) ≃ Fqn} HR = {C ∈ H : R(C) ≃ Fqn}

SLIDE 101

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Subfamilies of H H = {Hk,s(η, h) ⊆ Ln,q} G = {Gk,s ⊆ Ln,q} HL = {C ∈ H : L(C) ≃ Fqn} HR = {C ∈ H : R(C) ≃ Fqn} G = HR ∩ HL

SLIDE 102

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Subfamilies of H H = {Hk,s(η, h) ⊆ Ln,q} G = {Gk,s ⊆ Ln,q} HL = {C ∈ H : L(C) ≃ Fqn} HR = {C ∈ H : R(C) ≃ Fqn} G = HR ∩ HL G ⊂ HL ⊂ H G ⊂ HR ⊂ H

SLIDE 103

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Subfamilies of M M = {C ⊂ Ln,q : C MRD − code, d > 1}

SLIDE 104

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Subfamilies of M M = {C ⊂ Ln,q : C MRD − code, d > 1} ML,R = {C ∈ M : L(C), R(C) ≃ Fqn} ML = {C ∈ M : L(C) ≃ Fqn} MR = {C ∈ M : R(C) ≃ Fqn} ML,R ⊂ ML ⊂ M

SLIDE 105

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Subfamilies of M M = {C ⊂ Ln,q : C MRD − code, d > 1} ML,R = {C ∈ M : L(C), R(C) ≃ Fqn} ML = {C ∈ M : L(C) ≃ Fqn} MR = {C ∈ M : R(C) ≃ Fqn} ML,R ⊂ ML ⊂ M

SLIDE 106

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

The family ML ML ↔ MRD-codes Fqn-linear

SLIDE 107

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

The family ML ML ↔ MRD-codes Fqn-linear C ∈ ML = ⇒ ∃ C′ ≃ C such that L(C′) = Fqn

SLIDE 108

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

The family ML ML ↔ MRD-codes Fqn-linear C ∈ ML = ⇒ ∃ C′ ≃ C such that L(C′) = Fqn G ⊂ HL ⊆ ML

SLIDE 109

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

The family ML ML ↔ MRD-codes Fqn-linear C ∈ ML = ⇒ ∃ C′ ≃ C such that L(C′) = Fqn G ⊂ HL ⊆ ML

Horlemann-Trautmann, Marshall: New Criteria for MRD and Gabidulin Codes and some Rank-Metric Code Constructions, arXiv:1507.08641v3 (2016)

SLIDE 110

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and maximum scattered spaces Sheekey, AMC (2016)

SLIDE 111

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and maximum scattered spaces Sheekey, AMC (2016) C ∈ ML, k = 2

SLIDE 112

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and maximum scattered spaces Sheekey, AMC (2016) C ∈ ML, k = 2 C MRD-code, [n × n, 2n, n − 1]Fq, L(C) = Fqn C ≃ Cf = [x, f(x)]Fqn

SLIDE 113

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and maximum scattered spaces Sheekey, AMC (2016) C ∈ ML, k = 2 C MRD-code, [n × n, 2n, n − 1]Fq, L(C) = Fqn C ≃ Cf = [x, f(x)]Fqn

SLIDE 114

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and maximum scattered spaces Sheekey, AMC (2016) C ∈ ML, k = 2 C MRD-code, [n × n, 2n, n − 1]Fq, L(C) = Fqn C ≃ Cf = [x, f(x)]Fqn

Φf(x) = f(x)

x

SLIDE 115

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and maximum scattered spaces Sheekey, AMC (2016) C ∈ ML, k = 2 C MRD-code, [n × n, 2n, n − 1]Fq, L(C) = Fqn C ≃ Cf = [x, f(x)]Fqn

Φf(x) = f(x)

x

− → |Im(Φf)| = qn−1 + qn−2 + · · · + q + 1

SLIDE 116

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and maximum scattered spaces Sheekey, AMC (2016) C ∈ ML, k = 2 C MRD-code, [n × n, 2n, n − 1]Fq, L(C) = Fqn C ≃ Cf = [x, f(x)]Fqn

Uf = {(x, f(x)) : x ∈ Fqn} ⊂ V = Fqn × Fqn

SLIDE 117

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and maximum scattered spaces Sheekey, AMC (2016) C ∈ ML, k = 2 C MRD-code, [n × n, 2n, n − 1]Fq, L(C) = Fqn C ≃ Cf = [x, f(x)]Fqn

Uf = {(x, f(x)) : x ∈ Fqn} ⊂ V = Fqn × Fqn

Maximum Scattered Space of V

SLIDE 118

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Maximum scattered spaces-Maximum scattered linear sets U Fq-subspace of V = V(r, qn), with Fq-dimension k

SLIDE 119

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Maximum scattered spaces-Maximum scattered linear sets U Fq-subspace of V = V(r, qn), with Fq-dimension k Definition

SLIDE 120

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Maximum scattered spaces-Maximum scattered linear sets U Fq-subspace of V = V(r, qn), with Fq-dimension k Definition U is a scattered Fq-subspace of V if dimFq(U ∩ [v]Fqn) ≤ 1 for all v ∈ V.

SLIDE 121

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Maximum scattered spaces-Maximum scattered linear sets U Fq-subspace of V = V(r, qn), with Fq-dimension k Definition U is a scattered Fq-subspace of V if dimFq(U ∩ [v]Fqn) ≤ 1 for all v ∈ V. U is a Fq-scattered subspace if |LU| = qk−1 + qk−2 + · · · + q + 1 where LU ⊂ PG(V)

SLIDE 122

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Maximum scattered spaces-Maximum scattered linear sets U Fq-subspace of V = V(r, qn), with Fq-dimension k Definition U is a scattered Fq-subspace of V if dimFq(U ∩ [v]Fqn) ≤ 1 for all v ∈ V. U is a Fq-scattered subspace if |LU| = qk−1 + qk−2 + · · · + q + 1 where LU ⊂ PG(V) LU scattered Fq-linear set

SLIDE 123

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Maximum scattered spaces-Maximum scattered linear sets U Fq-subspace of V = V(r, qn), with Fq-dimension k Definition U is a scattered Fq-subspace of V if dimFq(U ∩ [v]Fqn) ≤ 1 for all v ∈ V. U is a Fq-scattered subspace if |LU| = qk−1 + qk−2 + · · · + q + 1 where LU ⊂ PG(V) LU scattered Fq-linear set U is a Maximum Scattered Fq-subspace of V if dimFqU is the highest possible dimension of a scattered Fq-subspace

f V

SLIDE 124

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Maximum scattered spaces-Maximum scattered linear sets U Fq-subspace of V = V(r, qn), with Fq-dimension k Definition U is a scattered Fq-subspace of V if dimFq(U ∩ [v]Fqn) ≤ 1 for all v ∈ V. U is a Fq-scattered subspace if |LU| = qk−1 + qk−2 + · · · + q + 1 where LU ⊂ PG(V) LU scattered Fq-linear set U is a Maximum Scattered Fq-subspace of V if dimFqU is the highest possible dimension of a scattered Fq-subspace

f V
A. Blokhuis, M. Lavrauw: Scattered spaces with respect to a spread

in PG(n,q), Geom. Dedicata (2000)

SLIDE 125

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Maximum Scattered Spaces Blokhuis-Lavrauw 2000

SLIDE 126

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Maximum Scattered Spaces Blokhuis-Lavrauw 2000 U ⊂ V = V(r, qn) scattered Fq-subspace ⇒ dimFqU ≤ rn

2

SLIDE 127

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Maximum Scattered Spaces Blokhuis-Lavrauw 2000 U ⊂ V = V(r, qn) scattered Fq-subspace ⇒ dimFqU ≤ rn

2

U ⊂ V = V(r, qn) scattered Fq-subspace, dimFqU = rn

2

⇓

SLIDE 128

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Maximum Scattered Spaces Blokhuis-Lavrauw 2000 U ⊂ V = V(r, qn) scattered Fq-subspace ⇒ dimFqU ≤ rn

2

U ⊂ V = V(r, qn) scattered Fq-subspace, dimFqU = rn

2

⇓ LU is a two-intersection set (w.r.t. hyperplanes) in PG(V, Fqn) = PG(r − 1, qn).

SLIDE 129

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Maximum Scattered Spaces Blokhuis-Lavrauw 2000 U ⊂ V = V(r, qn) scattered Fq-subspace ⇒ dimFqU ≤ rn

2

U ⊂ V = V(r, qn) scattered Fq-subspace, dimFqU = rn

2

⇓ LU is a two-intersection set (w.r.t. hyperplanes) in PG(V, Fqn) = PG(r − 1, qn). Two-weight linear codes Two-weight linear codes ← → Projective two-intersection sets

SLIDE 130

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Maximum Scattered Spaces Blokhuis-Lavrauw 2000 U ⊂ V = V(r, qn) scattered Fq-subspace ⇒ dimFqU ≤ rn

2

U ⊂ V = V(r, qn) scattered Fq-subspace, dimFqU = rn

2

⇓ LU is a two-intersection set (w.r.t. hyperplanes) in PG(V, Fqn) = PG(r − 1, qn). Two-weight linear codes Two-weight linear codes ← → Projective two-intersection sets

SLIDE 131

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Maximum Scattered Spaces Blokhuis-Lavrauw 2000 U ⊂ V = V(r, qn) scattered Fq-subspace ⇒ dimFqU ≤ rn

2

U ⊂ V = V(r, qn) scattered Fq-subspace, dimFqU = rn

2

⇓ LU is a two-intersection set (w.r.t. hyperplanes) in PG(V, Fqn) = PG(r − 1, qn). Two-weight linear codes Two-weight linear codes ← → Projective two-intersection sets

R. Calderbank, W.M. Kantor: The geometry of two-weight codes, Bull.

London Math. Soc. (1986)

SLIDE 132

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and Maximum Scattered Spaces Sheekey, AMC (2016) C ∈ ML, k = 2 C MRD-code [n × n, 2n, n − 1]Fq, L(C) = Fqn C ≃ Cf = [x, f(x)]Fqn

Uf = {(x, f(x)) : x ∈ Fqn} ⊂ V = Fqn × Fqn

Maximum Scattered Fq-subspace of V

SLIDE 133

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and Maximum Scattered Spaces Sheekey, AMC (2016) C ∈ ML, k = 2 C MRD-code [n × n, 2n, n − 1]Fq, L(C) = Fqn C ≃ Cf = [x, f(x)]Fqn ⊂ Ln,q = V(n2, qn)

Uf = {(x, f(x)) : x ∈ Fqn} ⊂ Fqn × Fqn = V(2, qn)

Maximum Scattered Fq-subspace of V(2, Fqn)

SLIDE 134

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and Maximum Scattered Spaces Sheekey, AMC (2016)

SLIDE 135

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and Maximum Scattered Spaces Sheekey, AMC (2016) Uf ≃ Ug w.r.t. ΓL(V(2, qn)) ⇐ ⇒ Cf ≃ Cg as MRD-codes

SLIDE 136

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and Maximum Scattered Spaces Sheekey, AMC (2016) Uf ≃ Ug w.r.t. ΓL(V(2, qn)) ⇐ ⇒ Cf ≃ Cg as MRD-codes Known examples

SLIDE 137

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and Maximum Scattered Spaces Sheekey, AMC (2016) Uf ≃ Ug w.r.t. ΓL(V(2, qn)) ⇐ ⇒ Cf ≃ Cg as MRD-codes Known examples U1 = {(x, xqs): x ∈ Fqn}, 1 ≤ s ≤ n − 1, gcd(s, n) = 1. U2 = {(x, δxqs + xqn−s): x ∈ Fqn}, Nqn/q(δ) = 0, 1, gcd(s, n) = 1.

SLIDE 138

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and Maximum Scattered Spaces Sheekey, AMC (2016) Uf ≃ Ug w.r.t. ΓL(V(2, qn)) ⇐ ⇒ Cf ≃ Cg as MRD-codes Known examples U1 = {(x, xqs): x ∈ Fqn}, 1 ≤ s ≤ n − 1, gcd(s, n) = 1.

Blokhuis-Lavrauw: Scattered spaces with respect to a spread in PG(n, q), Geom. Dedicata (2000)

U2 = {(x, δxqs + xqn−s): x ∈ Fqn}, Nqn/q(δ) = 0, 1, gcd(s, n) = 1.

Lunardon-O.P.: Blocking sets and derivable partial spreads, J. Algebr.

Comb. (2001)

SLIDE 139

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and maximum scattered spaces Sheekey, AMC (2016) Uf ≃ Ug w.r.t. ΓL(V) ⇐ ⇒ Cf ≃ Cg as MRD-codes Known examples U1 ← → C1 ∈ G

SLIDE 140

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and maximum scattered spaces Sheekey, AMC (2016) Uf ≃ Ug w.r.t. ΓL(V) ⇐ ⇒ Cf ≃ Cg as MRD-codes Known examples U1 ← → C1 ∈ G U2 ← → C2 ∈ HL

SLIDE 141

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

New examples U3 = {(x, δxqs + xqs+n/2): x ∈ Fqn}, Nqn/qn/2(δ) / ∈ {0, 1}, gcd(s, n/2) = 1,

SLIDE 142

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

New examples U3 = {(x, δxqs + xqs+n/2): x ∈ Fqn}, Nqn/qn/2(δ) / ∈ {0, 1}, gcd(s, n/2) = 1,

Csajbók-Marino-O.P.-Zanella: A new family of MRD-codes, https://arxiv.org/abs/1707.08487 (2017)

SLIDE 143

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

New examples U3 = {(x, δxqs + xqs+n/2): x ∈ Fqn}, Nqn/qn/2(δ) / ∈ {0, 1}, gcd(s, n/2) = 1, n = 6 and q > 4, n = 8 and q odd

Csajbók-Marino-O.P.-Zanella: A new family of MRD-codes, https://arxiv.org/abs/1707.08487 (2017)

SLIDE 144

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

New examples U3 = {(x, δxqs + xqs+n/2): x ∈ Fqn}, Nqn/qn/2(δ) / ∈ {0, 1}, gcd(s, n/2) = 1, n = 6 and q > 4, n = 8 and q odd

Csajbók-Marino-O.P.-Zanella: A new family of MRD-codes, https://arxiv.org/abs/1707.08487 (2017)

U4 = {(x, xq + xq3 + δxq5): x ∈ Fq6}, with δ2 + δ = 1, q ≡ 0, ±1 (mod 5)

SLIDE 145

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

New examples U3 = {(x, δxqs + xqs+n/2): x ∈ Fqn}, Nqn/qn/2(δ) / ∈ {0, 1}, gcd(s, n/2) = 1, n = 6 and q > 4, n = 8 and q odd

Csajbók-Marino-O.P.-Zanella: A new family of MRD-codes, https://arxiv.org/abs/1707.08487 (2017)

U4 = {(x, xq + xq3 + δxq5): x ∈ Fq6}, with δ2 + δ = 1, q ≡ 0, ±1 (mod 5)

Csajbók-Marino-Zullo: New scattered linear sets of the projective line, ArXiv (2017)

SLIDE 146

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

New examples U3 = {(x, δxqs + xqs+n/2): x ∈ Fqn}, Nqn/qn/2(δ) / ∈ {0, 1}, gcd(s, n/2) = 1, n = 6 and q > 4, n = 8 and q odd

Csajbók-Marino-O.P.-Zanella: A new family of MRD-codes, https://arxiv.org/abs/1707.08487 (2017)

U4 = {(x, xq + xq3 + δxq5): x ∈ Fq6}, with δ2 + δ = 1, q ≡ 0, ±1 (mod 5)

Csajbók-Marino-Zullo: New scattered linear sets of the projective line, ArXiv (2017)

k = 2, n = 6, 8 ⇒ HL ⊂ ML

SLIDE 147

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and Maximum Scattered Spaces Classification results C MRD-code, [3 × 3, 6, 2]Fq, Fq3-linear

SLIDE 148

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and Maximum Scattered Spaces Classification results C MRD-code, [3 × 3, 6, 2]Fq, Fq3-linear C ∈ G

SLIDE 149

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and Maximum Scattered Spaces Classification results C MRD-code, [3 × 3, 6, 2]Fq, Fq3-linear C ∈ G

Lavrauw-Van De Voorde: On linear sets on a projective line,Des. Codes Cryptogr. (2010)

SLIDE 150

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and Maximum Scattered Spaces Classification results C MRD-code, [3 × 3, 6, 2]Fq, Fq3-linear C ∈ G

Lavrauw-Van De Voorde: On linear sets on a projective line,Des. Codes Cryptogr. (2010)

C MRD-code, [4 × 4, 8, 3]Fq, Fq4-linear

SLIDE 151

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and Maximum Scattered Spaces Classification results C MRD-code, [3 × 3, 6, 2]Fq, Fq3-linear C ∈ G

Lavrauw-Van De Voorde: On linear sets on a projective line,Des. Codes Cryptogr. (2010)

C MRD-code, [4 × 4, 8, 3]Fq, Fq4-linear C ∈ G C ∈ HL

SLIDE 152

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and Maximum Scattered Spaces Classification results C MRD-code, [3 × 3, 6, 2]Fq, Fq3-linear C ∈ G

Lavrauw-Van De Voorde: On linear sets on a projective line,Des. Codes Cryptogr. (2010)

C MRD-code, [4 × 4, 8, 3]Fq, Fq4-linear C ∈ G C ∈ HL

Bonoli-O.P.: Fq-linear blocking sets in PG(2, q4),Innovations in Incidence Geometry (2005)

SLIDE 153

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and Maximum Scattered Spaces Classification results C MRD-code, [3 × 3, 6, 2]Fq, Fq3-linear C ∈ G

Lavrauw-Van De Voorde: On linear sets on a projective line,Des. Codes Cryptogr. (2010)

C MRD-code, [4 × 4, 8, 3]Fq, Fq4-linear C ∈ G C ∈ HL

Bonoli-O.P.: Fq-linear blocking sets in PG(2, q4),Innovations in Incidence Geometry (2005) Csajbók-Zanella: Maximum scattered linear sets of PG(1, q4), Discrete Mathematics (2017)

SLIDE 154

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and Maximum Scattered Spaces Classification results C MRD-code, [3 × 3, 6, 2]Fq, Fq3-linear C ∈ G

Lavrauw-Van De Voorde: On linear sets on a projective line,Des. Codes Cryptogr. (2010)

C MRD-code, [4 × 4, 8, 3]Fq, Fq4-linear C ∈ G C ∈ HL

Bonoli-O.P.: Fq-linear blocking sets in PG(2, q4),Innovations in Incidence Geometry (2005) Csajbók-Zanella: Maximum scattered linear sets of PG(1, q4), Discrete Mathematics (2017) For q = 2- Horlemann-Trautmann, Marshall: New criteria for MRD and Gabidulin codes and some rank-metric code constructions,arXiv:1507.08641v3 (2015)

SLIDE 155

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and the Segre Variety Lunardon, JCTA (2017) End(Fqn, Fq) = V(n2, q) − → P = PG(n2 − 1, q)

SLIDE 156

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and the Segre Variety Lunardon, JCTA (2017) End(Fqn, Fq) = V(n2, q) − → P = PG(n2 − 1, q) rank 1 linear maps − → Sn,n ⊂ P = PG(n2 − 1, q)

SLIDE 157

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and the Segre Variety Lunardon, JCTA (2017) End(Fqn, Fq) = V(n2, q) − → P = PG(n2 − 1, q) rank 1 linear maps − → Sn,n ⊂ P = PG(n2 − 1, q) rank ≤ h linear maps − → Sh

n,n ⊂ P = PG(n2 − 1, q)

SLIDE 158

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and the Segre Variety Lunardon, JCTA (2017) End(Fqn, Fq) = V(n2, q) − → P = PG(n2 − 1, q) rank 1 linear maps − → Sn,n ⊂ P = PG(n2 − 1, q) rank ≤ h linear maps − → Sh

n,n ⊂ P = PG(n2 − 1, q)

C ⊂ End(Fqn, Fq) linear MRD-code, [n × n, kn, n − k + 1]Fq

SLIDE 159

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and the Segre Variety Lunardon, JCTA (2017) End(Fqn, Fq) = V(n2, q) − → P = PG(n2 − 1, q) rank 1 linear maps − → Sn,n ⊂ P = PG(n2 − 1, q) rank ≤ h linear maps − → Sh

n,n ⊂ P = PG(n2 − 1, q)

C ⊂ End(Fqn, Fq) linear MRD-code, [n × n, kn, n − k + 1]Fq

SLIDE 160

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and the Segre Variety Lunardon, JCTA (2017) End(Fqn, Fq) = V(n2, q) − → P = PG(n2 − 1, q) rank 1 linear maps − → Sn,n ⊂ P = PG(n2 − 1, q) rank ≤ h linear maps − → Sh

n,n ⊂ P = PG(n2 − 1, q)

C ⊂ End(Fqn, Fq) linear MRD-code, [n × n, kn, n − k + 1]Fq

P(C) ⊂ P

dim P(C) = nk − 1 such that

SLIDE 161

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and the Segre Variety Lunardon, JCTA (2017) End(Fqn, Fq) = V(n2, q) − → P = PG(n2 − 1, q) rank 1 linear maps − → Sn,n ⊂ P = PG(n2 − 1, q) rank ≤ h linear maps − → Sh

n,n ⊂ P = PG(n2 − 1, q)

C ⊂ End(Fqn, Fq) linear MRD-code, [n × n, kn, n − k + 1]Fq

P(C) ⊂ P

dim P(C) = nk − 1 such that P(C) ∩ Sn−k

n,n

= ∅

SLIDE 162

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and the Segre Variety Lunardon, JCTA (2017) End(Fqn, Fq) = V(n2, q) − → P = PG(n2 − 1, q) rank 1 linear maps − → Sn,n ⊂ P = PG(n2 − 1, q) rank ≤ h linear maps − → Sh

n,n ⊂ P = PG(n2 − 1, q)

C ⊂ End(Fqn, Fq) linear MRD-code, [n × n, kn, n − k + 1]Fq

P(C)

⇒ P(C)Φ = P(CT)

SLIDE 163

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

MRD-codes and the Segre Variety Lunardon, JCTA (2017) End(Fqn, Fq) = V(n2, q) − → P = PG(n2 − 1, q) rank 1 linear maps − → Sn,n ⊂ P = PG(n2 − 1, q) rank ≤ h linear maps − → Sh

n,n ⊂ P = PG(n2 − 1, q)

C ⊂ End(Fqn, Fq) linear MRD-code, [n × n, kn, n − k + 1]Fq

P(C)

⇒ P(C)Φ = P(CT) P(C) ⇒ P(C)τ = P(C⊥)

SLIDE 164

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Open problems Open problems

SLIDE 165

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Open problems Open problems Understand whether G = ML,R

SLIDE 166

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Open problems Open problems Understand whether G = ML,R k = 1, k = n − 1 k = 2, k = n − 2

SLIDE 167

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Open problems Open problems Understand whether G = ML,R k = 1, k = n − 1 k = 2, k = n − 2 Find new examples in the family ML \ HL existing for an infinite number of values of q and n.

SLIDE 168

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Open problems Open problems Understand whether G = ML,R k = 1, k = n − 1 k = 2, k = n − 2 Find new examples in the family ML \ HL existing for an infinite number of values of q and n. Generalize the connection found by Sheekey between linear sets and MRD-codes for the whole family ML.

SLIDE 169

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim

Open problems Open problems Understand whether G = ML,R k = 1, k = n − 1 k = 2, k = n − 2 Find new examples in the family ML \ HL existing for an infinite number of values of q and n. Generalize the connection found by Sheekey between linear sets and MRD-codes for the whole family ML. Understand the relationship between Sheekey’s construction and Lunardon’s approach.

SLIDE 170

Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim