Maximum rank distance codes: constructions, classifications, and - - PowerPoint PPT Presentation

Maximum rank distance codes: constructions, classifications, and applications

John Sheekey

UCD, Dublin, Ireland

Dagstuhl, August 2016

Rank metric codes

A rank metric code is a set C ⊂ Mm×n(Fq) of m × n matrices (m ≤ n) over Fq with the distance function d(X, Y) := rank(X − Y).

Rank metric codes

A rank metric code is a set C ⊂ Mm×n(Fq) of m × n matrices (m ≤ n) over Fq with the distance function d(X, Y) := rank(X − Y). A code is F-linear if it is a subspace over F ≤ Fq.

MRD codes

Singleton-like bound: A subspace of matrices where every nonzero element has rank at least d has dimension at most n(m − d + 1).

MRD codes

Singleton-like bound: A subspace of matrices where every nonzero element has rank at least d has dimension at most n(m − d + 1). A code meeting this bound is said to be a Maximum Rank Distance (MRD) code.

MRD codes

Singleton-like bound: A subspace of matrices where every nonzero element has rank at least d has dimension at most n(m − d + 1). A code meeting this bound is said to be a Maximum Rank Distance (MRD) code. If C is an MRD-code which is linear over Fq with minimum rank-distance d, we say it has parameters [m × n, n(m − d + 1), d]q.

MRD codes

Singleton-like bound: A subspace of matrices where every nonzero element has rank at least d has dimension at most n(m − d + 1). A code meeting this bound is said to be a Maximum Rank Distance (MRD) code. If C is an MRD-code which is linear over Fq with minimum rank-distance d, we say it has parameters [m × n, n(m − d + 1), d]q. Delsarte (1978), and later Gabidulin (1985), constructed examples for all parameters using linearized polynomials.

Applications

Rank-metric codes have seen renewed interest in recent years due to new potential applications.

Applications

Rank-metric codes have seen renewed interest in recent years due to new potential applications.

◮ Code-based cryptography; ◮ Subspace codes (random network coding); ◮ Index coding; ◮ ...?

Applications

Rank-metric codes have seen renewed interest in recent years due to new potential applications.

◮ Code-based cryptography; ◮ Subspace codes (random network coding); ◮ Index coding; ◮ ...?

For this reason we would like to find more examples of MRD-codes.

Equivalence of rank metric codes

Two codes C1 and C2 are said to be equivalent if there exist invertible matrices A and B, a matrix D, and an automorphism ρ

• f Fq, such that

C2 = {AX ρB + D : X ∈ C1}

Equivalence of rank metric codes

Two codes C1 and C2 are said to be equivalent if there exist invertible matrices A and B, a matrix D, and an automorphism ρ

• f Fq, such that

C2 = {AX ρB + D : X ∈ C1} Clearly operations of this form preserve rank distance.

Equivalence of rank metric codes

Two codes C1 and C2 are said to be equivalent if there exist invertible matrices A and B, a matrix D, and an automorphism ρ

• f Fq, such that

C2 = {AX ρB + D : X ∈ C1} Clearly operations of this form preserve rank distance. Can also include the transpose operation when n = m.

Duality

There is also a notion of duality, which preserves the MRD property: b(X, Y) = Tr(XY T).

Duality

There is also a notion of duality, which preserves the MRD property: b(X, Y) = Tr(XY T). Delsarte showed that if C is MRD with minimum distance d, then C⊥ is MRD with minimum distance m − d + 2.

Duality

There is also a notion of duality, which preserves the MRD property: b(X, Y) = Tr(XY T). Delsarte showed that if C is MRD with minimum distance d, then C⊥ is MRD with minimum distance m − d + 2. See paper of Ravagnani (2015) for an elementary proof.

Linearized polynomials

A linearized polynomial is a polynomial in Fqn[x] of the form f(x) = f0x + f1xq + · · · + fn−1xqn−1.

Linearized polynomials

A linearized polynomial is a polynomial in Fqn[x] of the form f(x) = f0x + f1xq + · · · + fn−1xqn−1. Each such polynomial is an Fq-linear map from Fqn to itself.

Linearized polynomials

A linearized polynomial is a polynomial in Fqn[x] of the form f(x) = f0x + f1xq + · · · + fn−1xqn−1. Each such polynomial is an Fq-linear map from Fqn to itself. Linearized polynomials ⇔ Mn(Fq) Composition mod xqn − x ⇔ Matrix multiplication

Linearized polynomials

A linearized polynomial is a polynomial in Fqn[x] of the form f(x) = f0x + f1xq + · · · + fn−1xqn−1. Each such polynomial is an Fq-linear map from Fqn to itself. Linearized polynomials ⇔ Mn(Fq) Composition mod xqn − x ⇔ Matrix multiplication With this representation, we can talk of a code being Fqn-linear.

Linearized polynomials

A linearized polynomial is a polynomial in Fqn[x] of the form f(x) = f0x + f1xq + · · · + fn−1xqn−1. Each such polynomial is an Fq-linear map from Fqn to itself. Linearized polynomials ⇔ Mn(Fq) Composition mod xqn − x ⇔ Matrix multiplication With this representation, we can talk of a code being Fqn-linear. In this setting, the dual operation becomes very easy to use.

Delsarte/Gabidulin codes

The Delsarte/Gabidulin code Gk is a the set of linearized polynomials of q-degree at most k − 1, i.e. Gk := {f0x + f1xq + · · · + fk−1xqk−1 : fi ∈ Fqn}.

Delsarte/Gabidulin codes

The Delsarte/Gabidulin code Gk is a the set of linearized polynomials of q-degree at most k − 1, i.e. Gk := {f0x + f1xq + · · · + fk−1xqk−1 : fi ∈ Fqn}. Clearly each element of Gk has at most qk−1 roots, and hence rank at least n − k + 1 =: d.

Delsarte/Gabidulin codes

The Delsarte/Gabidulin code Gk is a the set of linearized polynomials of q-degree at most k − 1, i.e. Gk := {f0x + f1xq + · · · + fk−1xqk−1 : fi ∈ Fqn}. Clearly each element of Gk has at most qk−1 roots, and hence rank at least n − k + 1 =: d. Gk has dimension nk = n(n − d + 1) over Fq.

Delsarte/Gabidulin codes

The Delsarte/Gabidulin code Gk is a the set of linearized polynomials of q-degree at most k − 1, i.e. Gk := {f0x + f1xq + · · · + fk−1xqk−1 : fi ∈ Fqn}. Clearly each element of Gk has at most qk−1 roots, and hence rank at least n − k + 1 =: d. Gk has dimension nk = n(n − d + 1) over Fq. In fact, it is linear over Fqn.

Delsarte/Gabidulin codes

The Delsarte/Gabidulin code Gk is a the set of linearized polynomials of q-degree at most k − 1, i.e. Gk := {f0x + f1xq + · · · + fk−1xqk−1 : fi ∈ Fqn}. Clearly each element of Gk has at most qk−1 roots, and hence rank at least n − k + 1 =: d. Gk has dimension nk = n(n − d + 1) over Fq. In fact, it is linear over Fqn. Hence Gk is an Fqn-linear MRD-code with parameters [n × n, nk, n − k + 1]q.

Delsarte/Gabidulin codes

The Delsarte/Gabidulin code Gk is a the set of linearized polynomials of q-degree at most k − 1, i.e. Gk := {f0x + f1xq + · · · + fk−1xqk−1 : fi ∈ Fqn}. Clearly each element of Gk has at most qk−1 roots, and hence rank at least n − k + 1 =: d. Gk has dimension nk = n(n − d + 1) over Fq. In fact, it is linear over Fqn. Hence Gk is an Fqn-linear MRD-code with parameters [n × n, nk, n − k + 1]q. Can replace xqi with xqsi for any s with (n, s) = 1, and define Gk,s (generalised Gabidulin codes).

Non-square case

We can produce MRD-codes in Mm×n from an MRD code in Mm×n by either shortening or puncturing; CU = {f : f ∈ C | f(U) = 0}, dim(U) = n − m; Ch := {f ◦ h : f ∈ C}, rank(h) = m.

Non-square case

We can produce MRD-codes in Mm×n from an MRD code in Mm×n by either shortening or puncturing; CU = {f : f ∈ C | f(U) = 0}, dim(U) = n − m; Ch := {f ◦ h : f ∈ C}, rank(h) = m.

◮ When are different codes obtained by

puncturing/shortening an MRD code equivalent?

◮ Are there examples of MRD codes which cannot be

• btained by puncturing/shortening?

Alternative formulation

We can also think of Rank-Metric codes as sets of vectors in (Fqn)m.

Alternative formulation

We can also think of Rank-Metric codes as sets of vectors in (Fqn)m. The rank weight of a vector (v0, . . . , vm−1) is given by dimv0, . . . , vm−1Fq.

Alternative formulation

We can also think of Rank-Metric codes as sets of vectors in (Fqn)m. The rank weight of a vector (v0, . . . , vm−1) is given by dimv0, . . . , vm−1Fq. We go from one setting to the other by fixing an Fq-basis for Fqn (say, {e0, . . . en−1}), and identifying f ↔ (f(e0), . . . , f(em−1)).

Alternative formulation

We can also think of Rank-Metric codes as sets of vectors in (Fqn)m. The rank weight of a vector (v0, . . . , vm−1) is given by dimv0, . . . , vm−1Fq. We go from one setting to the other by fixing an Fq-basis for Fqn (say, {e0, . . . en−1}), and identifying f ↔ (f(e0), . . . , f(em−1)). Using this representation, Horlemann-Trautmann and Marshall gave very useful criteria for identifying when a code is equivalent to a (generalised) Gabidulin code.

Case n = m = d

With d = n = m, an Fq-linear [n × n, n, n] MRD-code is a subspace of matrices where every nonzero is invertible.

Case n = m = d

With d = n = m, an Fq-linear [n × n, n, n] MRD-code is a subspace of matrices where every nonzero is invertible. We can find an example by taking a representation of Fqn in Mn(Fq).

Linear [n, n, n]-codes

◮ Denote by Ry the map of multiplication by y, i.e.

Ry(x) = xy.

Linear [n, n, n]-codes

◮ Denote by Ry the map of multiplication by y, i.e.

Ry(x) = xy.

◮ Let C(Fqn) be the set of all such maps.

Linear [n, n, n]-codes

◮ Denote by Ry the map of multiplication by y, i.e.

Ry(x) = xy.

◮ Let C(Fqn) be the set of all such maps. ◮ Each map is a linear map (matrix), because

(x + z)y = xy + zy;

Linear [n, n, n]-codes

◮ Denote by Ry the map of multiplication by y, i.e.

Ry(x) = xy.

◮ Let C(Fqn) be the set of all such maps. ◮ Each map is a linear map (matrix), because

(x + z)y = xy + zy;

◮ The set is a subspace, because x(y + z) = xy + xz;

Linear [n, n, n]-codes

◮ Denote by Ry the map of multiplication by y, i.e.

Ry(x) = xy.

◮ Let C(Fqn) be the set of all such maps. ◮ Each map is a linear map (matrix), because

(x + z)y = xy + zy;

◮ The set is a subspace, because x(y + z) = xy + xz; ◮ Each non-zero element is invertible, because

xy = 0 ⇔ x = 0 or y = 0.

Linear [n, n, n]-codes

◮ Denote by Ry the map of multiplication by y, i.e.

Ry(x) = xy.

◮ Let C(Fqn) be the set of all such maps. ◮ Each map is a linear map (matrix), because

(x + z)y = xy + zy;

◮ The set is a subspace, because x(y + z) = xy + xz; ◮ Each non-zero element is invertible, because

xy = 0 ⇔ x = 0 or y = 0. In fact C(Fqn) is precisely the Gabidulin code G1.

Linear [n, n, n]-codes

◮ Denote by Ry the map of multiplication by y, i.e.

Ry(x) = xy.

◮ Let C(Fqn) be the set of all such maps. ◮ Each map is a linear map (matrix), because

(x + z)y = xy + zy;

◮ The set is a subspace, because x(y + z) = xy + xz; ◮ Each non-zero element is invertible, because

xy = 0 ⇔ x = 0 or y = 0. In fact C(Fqn) is precisely the Gabidulin code G1. Note that nowhere have we used the fact that multiplication is associative.

Semifields and rank metric codes

Fq-linear [n × n, n, n] MRD-code correspond precisely to semifields, i.e. nonassociative division algebras.

Semifields and rank metric codes

Fq-linear [n × n, n, n] MRD-code correspond precisely to semifields, i.e. nonassociative division algebras. The first non-trivial semifield was constructed by Dickson (1906).

Semifields and rank metric codes

Fq-linear [n × n, n, n] MRD-code correspond precisely to semifields, i.e. nonassociative division algebras. The first non-trivial semifield was constructed by Dickson (1906). Semifields are well-studied in finite geometry, in particular as they correspond to a particular class of projective planes.

Semifields and rank metric codes

Fq-linear [n × n, n, n] MRD-code correspond precisely to semifields, i.e. nonassociative division algebras. The first non-trivial semifield was constructed by Dickson (1906). Semifields are well-studied in finite geometry, in particular as they correspond to a particular class of projective planes. There are many constructions, and some classifications in small cases, but there are many open problems still.

Semifields and rank metric codes

Fq-linear [n × n, n, n] MRD-code correspond precisely to semifields, i.e. nonassociative division algebras. The first non-trivial semifield was constructed by Dickson (1906). Semifields are well-studied in finite geometry, in particular as they correspond to a particular class of projective planes. There are many constructions, and some classifications in small cases, but there are many open problems still. If we remove the requirement of linearity, we get a correspondence with quasifields.

Example

Albert’s Generalised Twisted Fields (1965): x ◦ y = xy − cxqiyqk, where x, y, c ∈ Fqn, N(c) = 1.

New construction for general k

Define Hk(a, h) to be the set of linearized polynomials of degree at most k satisfying fk = af qh

0 , with N(a) = (−1)nk.

Hk(a, h) := {f0x + f1xq + · · · + fk−1xqk−1

New construction for general k

Define Hk(a, h) to be the set of linearized polynomials of degree at most k satisfying fk = af qh

0 , with N(a) = (−1)nk.

Hk(a, h) := {f0x + f1xq + · · · + fk−1xqk−1+af qh

0 xqk : fi ∈ Fqn}.

New construction for general k

Define Hk(a, h) to be the set of linearized polynomials of degree at most k satisfying fk = af qh

0 , with N(a) = (−1)nk.

Hk(a, h) := {f0x + f1xq + · · · + fk−1xqk−1+af qh

0 xqk : fi ∈ Fqn}.

Theorem (S.)

Hk(a, h) is an MRD-code with parameters [n × n, nk, n − k + 1]q. Furthermore, Hk(a, h) is not equivalent to Gk unless k ∈ {1, n − 1} and h ∈ {0, 1}.

New construction for general k

Define Hk(a, h) to be the set of linearized polynomials of degree at most k satisfying fk = af qh

0 , with N(a) = (−1)nk.

Hk(a, h) := {f0x + f1xq + · · · + fk−1xqk−1+af qh

0 xqk : fi ∈ Fqn}.

Theorem (S.)

Hk(a, h) is an MRD-code with parameters [n × n, nk, n − k + 1]q. Furthermore, Hk(a, h) is not equivalent to Gk unless k ∈ {1, n − 1} and h ∈ {0, 1}. Again, we can replace xqi with xqsi, and the same construction works: Hk,s(a, h).

Minimum polynomial of a subspace

For any k-dimensional subspace U of Fqn, there exists a unique monic linearized polynomial of degree k whose roots are precisely the elements of U. mU(x) =

• u∈U

(x − u) =

• x

xq · · · xqk α1 αq

1

· · · αqk

1

. . . . . . ... . . . αk αq

k

· · · αqk

k

• ,

where U = α1, . . . , αk.

New construction - properties

Hk,s(a, h) := {f0x +f1xqs +· · ·+fk−1xqs(k−1) +af qh

0 xqsk : fi ∈ Fqn}. ◮ The inequivalence to (generalised) Gabidulin codes is

proven by calculating the automorphism groups.

New construction - properties

Hk,s(a, h) := {f0x +f1xqs +· · ·+fk−1xqs(k−1) +af qh

0 xqsk : fi ∈ Fqn}. ◮ The inequivalence to (generalised) Gabidulin codes is

proven by calculating the automorphism groups.

◮ The full equivalence problem for this family is known

(s = 1: S., s > 1: Lunardon-Trombetti-Zhou).

New construction - properties

Hk,s(a, h) := {f0x +f1xqs +· · ·+fk−1xqs(k−1) +af qh

0 xqsk : fi ∈ Fqn}. ◮ The inequivalence to (generalised) Gabidulin codes is

proven by calculating the automorphism groups.

◮ The full equivalence problem for this family is known

(s = 1: S., s > 1: Lunardon-Trombetti-Zhou).

◮ Choosing a = 0 returns the (generalised) Gabidulin codes.

New construction - properties

Hk,s(a, h) := {f0x +f1xqs +· · ·+fk−1xqs(k−1) +af qh

0 xqsk : fi ∈ Fqn}. ◮ The inequivalence to (generalised) Gabidulin codes is

proven by calculating the automorphism groups.

◮ The full equivalence problem for this family is known

(s = 1: S., s > 1: Lunardon-Trombetti-Zhou).

◮ Choosing a = 0 returns the (generalised) Gabidulin codes. ◮ Choosing h = 0 gives Fqn-linear codes.

New construction - properties

Hk,s(a, h) := {f0x +f1xqs +· · ·+fk−1xqs(k−1) +af qh

0 xqsk : fi ∈ Fqn}. ◮ The inequivalence to (generalised) Gabidulin codes is

proven by calculating the automorphism groups.

◮ The full equivalence problem for this family is known

(s = 1: S., s > 1: Lunardon-Trombetti-Zhou).

◮ Choosing a = 0 returns the (generalised) Gabidulin codes. ◮ Choosing h = 0 gives Fqn-linear codes. ◮ Choosing k = 1 gives a semifield known as a generalised

twisted field (Albert).

New construction - properties

Hk,s(a, h) := {f0x +f1xqs +· · ·+fk−1xqs(k−1) +af qh

0 xqsk : fi ∈ Fqn}. ◮ The inequivalence to (generalised) Gabidulin codes is

proven by calculating the automorphism groups.

◮ The full equivalence problem for this family is known

(s = 1: S., s > 1: Lunardon-Trombetti-Zhou).

◮ Choosing a = 0 returns the (generalised) Gabidulin codes. ◮ Choosing h = 0 gives Fqn-linear codes. ◮ Choosing k = 1 gives a semifield known as a generalised

twisted field (Albert).

◮ We call these twisted Gabidulin codes.

Other properties

◮ The codes Hk are contained in a Gabidulin code Gk+1, ◮ and contain a Gabidulin code Gk−1.

Other properties

◮ The codes Hk are contained in a Gabidulin code Gk+1, ◮ and contain a Gabidulin code Gk−1.

Do one/both of these properties characterise these codes? If not, can we construct other examples with one of these properties?

Other constructions

Horlemann-Trautmann and Marshall have constructed various families of non-Gabidulin MRD codes for certain fields and parameters.

Other constructions

Horlemann-Trautmann and Marshall have constructed various families of non-Gabidulin MRD codes for certain fields and parameters. Are these families inequivalent to twisted Gabidulin codes?

Classifications

Recent work (Neri, Horlemann-Trautmann, Randrianarisoa, Rosenthal) suggests that MRD-codes are quite abundant, even if we restrict to the Fqn-linear case.

Classifications

Recent work (Neri, Horlemann-Trautmann, Randrianarisoa, Rosenthal) suggests that MRD-codes are quite abundant, even if we restrict to the Fqn-linear case. Even so, there are not many constructions or classifications known, except in the case d = n = m.

Classifications

Recent work (Neri, Horlemann-Trautmann, Randrianarisoa, Rosenthal) suggests that MRD-codes are quite abundant, even if we restrict to the Fqn-linear case. Even so, there are not many constructions or classifications known, except in the case d = n = m. Classification in general appears to be a very difficult problem. However restricting to Fqn-linearity may make the problem tractable.

3 × 3

◮ All linear codes are classified, and are (twisted) Gabidulin

(due to a result of Menichetti (1977)).

◮ All non-linear codes are classified for d = 3, q = 2, 3. ◮ There are constructions for non-linear codes with d = 2

(Cossidente-Marino-Pavese (2015)).

3 × 4

◮ All linear codes are classified for q = 2

(Honold-Kiermaier-Kurz).

◮ All non-linear codes are classified for d = 3, q = 2. ◮ The case d = 2, q = 2 is open.

4 × 4

◮ All linear codes are classified for d = 4 (and hence d = 2),

q = 2, 3, 4, 5.

◮ All non-linear codes are classified for d = 4, q = 2. ◮ All linear codes are classified for d = 3, q = 2 - there is a

unique code, which is Gabidulin.

◮ All Fqn-linear codes are (probably) classified for all q; all

are (twisted) Gabidulin.

Larger sizes

◮ 5 × 5: All linear codes are classified for d = 5 q = 2, 3. ◮ 5 × 5: Fqn-linear, d = 4 is still open (and probably doable). ◮ 6 × 6: All linear codes are classified for d = 6, q = 2. ◮ p × p: p prime, d = p, q large enough: all linear codes are

(twisted) Gabidulin (Menichetti);

◮ n × n: Lots of constructions for linear d = n (semifields) ◮ n × n: Recent constructions for non-linear, d = n − 1

(Durante-Sicilliano, Ozbudak et.al). Semifield classifications by Knuth; Walker; Dempwolff; Rua-Combarro-Ranilla.

Structure of an arbitrary MRD code

The weight enumerator of an MRD code is completely determined (Delsarte).

Structure of an arbitrary MRD code

The weight enumerator of an MRD code is completely determined (Delsarte). Some things are known about the structure of the non-invertible elements (e.g Dumas-Gow-McGuire-S.).

Structure of an arbitrary MRD code

The weight enumerator of an MRD code is completely determined (Delsarte). Some things are known about the structure of the non-invertible elements (e.g Dumas-Gow-McGuire-S.). However there are still some open questions such as:

Structure of an arbitrary MRD code

The weight enumerator of an MRD code is completely determined (Delsarte). Some things are known about the structure of the non-invertible elements (e.g Dumas-Gow-McGuire-S.). However there are still some open questions such as: Does a linear MRD-code of minimum distance d necessarily contain a linear MRD-code of minimum distance n? Which linear MRD-codes can be extended to MRD-codes of larger dimension?

Subspace Codes

We can “lift” a rank metric code to a subspace code: SA := (x, xA) : x ∈ Fn

q ≤ V(2n, q).

Subspace Codes

We can “lift” a rank metric code to a subspace code: SA := (x, xA) : x ∈ Fn

q ≤ V(2n, q).

Then a rank-metric code C gives a subspace code with the same cardinality and minimum distance 2d.

Subspace Codes

We can “lift” a rank metric code to a subspace code: SA := (x, xA) : x ∈ Fn

q ≤ V(2n, q).

Then a rank-metric code C gives a subspace code with the same cardinality and minimum distance 2d. The automorphism group of the lifted code consists of elements

• f the form

A−1 A−1D B

• .

Subspace Codes

We can “lift” a rank metric code to a subspace code: SA := (x, xA) : x ∈ Fn

q ≤ V(2n, q).

Then a rank-metric code C gives a subspace code with the same cardinality and minimum distance 2d. The automorphism group of the lifted code consists of elements

• f the form

A−1 A−1D B

• .

If C is linear, then the automorphism group contains the subgroup I A I

• : A ∈ C
• .

Subspace Codes

MRD codes do not correspond to optimal subspace codes

Subspace Codes

MRD codes do not correspond to optimal subspace codes (unless d = n = m: correspondence between quasifields and spreads).

Subspace Codes

MRD codes do not correspond to optimal subspace codes (unless d = n = m: correspondence between quasifields and spreads). However some of the best constructions for subspace codes come from lifting an MRD code, extending it, and then modifying slightly.

Subspace Codes

MRD codes do not correspond to optimal subspace codes (unless d = n = m: correspondence between quasifields and spreads). However some of the best constructions for subspace codes come from lifting an MRD code, extending it, and then modifying slightly. See http://subspacecodes.uni-bayreuth.de/ (Heinlein-Kiermaier-Kurz-Wasserman).

Subspace Codes

MRD codes do not correspond to optimal subspace codes (unless d = n = m: correspondence between quasifields and spreads). However some of the best constructions for subspace codes come from lifting an MRD code, extending it, and then modifying slightly. See http://subspacecodes.uni-bayreuth.de/ (Heinlein-Kiermaier-Kurz-Wasserman). Can twisted Gabidulin codes do any better than Gabidulin codes using the above idea?

Further questions

◮ Decoding algorithms for twisted Gabidulin? ◮ Criteria for recognising twisted Gabidulin codes? ◮ Twisted Gabidulin are (probably) bad for crypto; can we

find codes with less structure?

Further questions

◮ Decoding algorithms for twisted Gabidulin? ◮ Criteria for recognising twisted Gabidulin codes? ◮ Twisted Gabidulin are (probably) bad for crypto; can we

find codes with less structure? Dankeschön! Go raibh míle maith agaibh!