A new family of linear maximum rank distance codes

In this article we construct a new family of linear maximum rank distance (MRD) codes for all parameters. This family contains the only known family for general parameters, the Gabidulin codes, and contains codes inequivalent to the Gabidulin codes. This family also contains the well-known family of semifields known as Generalised Twisted Fields. We also calculate the automorphism group of these codes, including the automorphism group of the Gabidulin codes.


Preliminaries 1.Rank metric codes
Delsarte introduced rank metric codes in [8].A rank metric code C is a subset of a matrix space M = M m×n (F), F a field, equipped with the distance function d(X, Y ) := rank(X − Y ).A rank metric code is called F 0 -linear if it forms an F 0 -subspace of M for some subfield F 0 ≤ F. A maximum rank distance code (MRD-code) is a rank metric code over a finite field F q meeting the Singleton-like bound |C| ≤ q n(m−d+1) , where d is the minimum distance of C. If C is an F q0 -linear MRD code in M m×n (F q ) with d = m − k + 1 for a subfield F q0 of F q , we say that C has parameters [nm, nk, m − k + 1] q0 , with the subscript omitted when there is no ambiguity.
Delsarte [8] and Gabidulin [11] constructed linear MRD-codes over the finite field F q for every k, m and n.In the literature these are usually called (generalised) Gabidulin codes, although the first construction was by Delsarte.When n = m and k = 1, MRD-codes correspond to algebraic structures called quasifields, see Subsection 1.2.Cossidente-Marino-Pavese [6] recently constructed non-linear MRD-codes for n = m = 3, k = 2.When n = m and 1 < k < n − 1, no other linear MRD-codes were known.In this paper we will construct a new family of linear MRD-codes for each k, and we will show that they contain codes inequivalent to the generalised Gabidulin codes.

(Pre)semifields and quasifields
A finite presemifield is a division algebra in which multiplication is not necessarily associative; if a multiplicative identity element exists, it is called a semifield.We refer to [20] for background, definitions, and terminology.Presemifield are studied in equivalence classes known as isotopy classes.Presemifields of order q n with centre containing F q0 ≤ F q and left nucleus containing F q are in one-to-one correspondence [n 2 , n, n] q0 MRD-codes in M n (F q ) (i.e.F q0 -linear MRD-codes with k = 1).In the theory of semifields, such spaces are called semifield spread sets.Many constructions for finite semifields are known, see for example [18] and [20].There are two important operations defined on presemifields; the dual, which is the opposite algebra, and the transpose.These together form a chain of six (isotopy classes of) semifields, known as the Knuth orbit.
A quasifield is an algebraic structure satisfying the axioms of a division algebra, except perhaps left distributivity.Quasifields are in one-to-one correspondence with MRD-codes with k = 1 which are not necessarily linear (see [9]).Explicit statements of the correspondence between semifields, quasifields and MRD-codes can be found in [7].

Equivalence
There are different concepts of equivalence for rank metric codes, see for example [26].In this paper, two rank metric codes C, C ′ ⊂ M n (F q ) will be said to be equivalent if there exist invertible F q -linear transformations A, B and a field automorphism ρ ∈ Aut(F q ) such that C ′ = AC ρ B := {AX ρ B : X ∈ C} where X ρ is the matrix obtained from X by applying ρ to each entry.Note that each of these operations preserve the rank distance, and they form a group.The subgroup fixing C is called the automorphism group of C, and is denoted by Aut(C).
We call the set Ĉ := { X : X ∈ C} the adjoint of C, where X denotes the adjoint of X with respect to some non-degenerate symmetric bilinear form.This form is often chosen so that the adjoint is precisely matrix transposition, though we will not assume this.Note that taking the adjoint also preserves rank distance, and is often included in the definition of equivalence.However we find it more convenient to omit it, and if C is equivalent to Ĉ′ , we say that C and C ′ are adjoint-equivalent.
When k = 1 and C 1 , C 2 are linear, equivalence corresponds precisely to the presemifields associated to each code C i being isotopic, while adjoint-equivalence corresponds to one presemifield being isotopic to the transpose of the other.

Subspace codes
A subspace code is a set of subspaces of a vector space, with the distance function If all elements of the code have the same dimension, it is called a constant dimension code.These codes were introduced by Kotter and Kshischang [19], and have applications in random network coding.Rank metric codes define constant dimension subspace codes in the following way (see for example [13]).Given a matrix X we define the subspace S X = {(u, Xu) : u ∈ F n q } of F 2n q .Clearly, each S X is n-dimensional, and d s (S X , S Y ) = 2rank(X − Y ).Hence an MRD-code with minimum distance d defines a subspace code with minimum distance 2d.This is known as a lifted MRD-code [30].However, not every subspace code defines an MRD-code when d < n, see for example [17].Many of the best known constructions are constructed by perturbing a lifted Gabidulin code.In the case d = n, we have the wellknown correspondence between spreads and quasifields, and in the linear case between semifield spreads and semifields.where Tr denotes the matrix trace, and tr denotes the absolute trace from F q to F p , where p is prime and q = p e .Define the Delsarte dual C ⊥ of an F p -linear code C by

Delsarte's duality theorem
We choose the name Delsarte dual to distinguish from the notion of dual in semifield theory.Delsarte [8, Theorem 5.5] proved the following theorem, using the theory of association schemes.An elementary proof can be found in [28].
Note that when n = m = 2, k = 1, both C and C ⊥ are semifield spread sets (and correspond to rank two semifields, see [20]).In the context of semifields, this operation is known as the translation dual, see [22], and is a special case of the switching operation defined in [3].
It is clear that two codes C and C ′ are equivalent if and only if C ⊥ and C ′⊥ are equivalent; this was shown in [22] for semifield spread sets, the same proof works for this more general statement.Hence the classification of [n 2 , n 2 − n, 2] q0 -codes is equivalent to the classification of semifields of order q n with nucleus containing F q and centre containing F q0 up to isotopy.Furthermore, as semifields of order q 3 with centre containing F q have been fully classified by Menichetti [25], we have a full classification of all F q -linear MRD codes in M 3 (F q ): they are either the spread sets corresponding to a field or a generalised twisted field, or the Delsarte dual of one of these.Precise conditions for the equivalence of spread sets arising from generalised twisted fields can be found in [4].

Linearized polynomials, and properties of the Gabidulin code
Let us identify the vector space V (n, q) with the elements of F q n .It is well known that every F q -linear transformation from F q n to itself may be represented by a unique linearized polynomial of q-degree at most n − 1: that is, Recall that the q-degree is the maximum i such that f i = 0.These linearized polynomials form a ring isomorphic to M n (F q ), with the multiplication being composition modulo x q n − x (which we will denote by •, or simply by juxtaposition when the meaning is clear).
It is obvious that a linearized polynomial of q-degree k has rank at least n−k.In fact, a linearized polynomial of the form has rank at least n − k, for any s relatively prime to n.Let G k,s denote the set of linearized polynomials of this form, for a fixed s and k.These are the generalised Gabidulin codes [11], and are MRD-codes with dimension nk and minimum rank-distance n − k + 1.
Define G k = G k,1 , which is then the set of linearized polynomials of degree at most k − 1, i.e.
These were first constructed by Delsarte [8].Each code G 1,s is a semifield spread set corresponding to the field F q n .Remark 1. Rank metric codes are sometimes viewed as codes in (F q n ) n .The theories are basically identical (see e.g.[26]), with the main difference being that codes in this context are called linear if the corresponding set of linearized polynomials form an F q n -subspace of L n .Clearly the generalised Gabidulin codes are all F q n -linear.Note however that F q n -linearity is not preserved by the equivalence as defined in this paper.
The action of Aut(F q ) on M n (F q ) can be translated to an action on L n as follows.Given a linearized polynomial f and an automorphism ρ of F q , we define f ρ (x) = f (x ρ −1 ) ρ .If q = p e for p a prime, and Hence any automorphism of a code C can be written as (f •x p i , x p ne−i •g) for some linearized polynomials f, g.
The set S := {αx : α ∈ F × q n } is a subgroup of GL(n, q) isomorphic to F × q n , and is what is known as a Singer cycle.Then G k,s is fixed under the actions of S defined by f → (αx) • f and f → f • (αx), for α ∈ F × q n .It is also fixed by f → x p • f • x p ne−1 , and hence by the group {(αx p i , βx p ne−i ) : α, β ∈ F × q n , i ∈ {0, . . ., ne − 1}}.
We will show later that this is in fact the full stabiliser of G k for each k.
The adjoint of a linearized polynomial a = n−1 i=0 a i x q i with respect to the symmetric bilinear form (a, b) → Tr(ab) is given by â = n−1 i=0 a q i n−i x q i .It is easy to check that x q k • Ĝk = G k , and hence we have the following.
Lemma 1.Each Gabidulin code G k is equivalent to its adjoint Ĝk .
We define the bilinear form b on linearized polynomials by Note also that the Gabidulin codes form a chain: We now consider which subspaces of a Gabidulin code G k are equivalent to another Gabidulin code G r .
Proof.Clearly if f and g are invertible linearized polynomials satisfying the condition on degrees, then U is contained in G k , and isotopic to G r .
Consider f • αx q j • g, where α ∈ F q n .Then the coefficient of where indices are taken modulo n.If U is contained in G k , we must have that for each m ≥ k, j ≤ r − 1, a m (α) is zero for every α ∈ F q n .Hence for all m ≥ k, j ≤ r − 1 and i ∈ {0, . . ., n − 1} we have that As f 0 = 0, we get that g m = 0 for all m ≥ k.Let deg q (f ) = s, deg q (g) = t, and so f s g t = 0. Then g m−s−r+1 = 0 for all m ∈ {k, . . ., n − 1} = ∅, and hence t ≤ k − s − r, proving the claim.
In [26,Proposition IV.7] it was shown that the group {(αx, βx) : α, β ∈ F × q n } is a subgroup of the automorphism group of G k .Note that the result in [26,Theorem IV.4] refers to a different definition of equivalence to the definition used in this paper.We now give a complete description of the automorphism group of the Gabidulin codes.Theorem 3. The automorphism group of the Gabidulin code G k is given by {(αx p i , βx p ne−i ) : α, β ∈ F × q n , i ∈ {0, . . ., n − 1}}.
Proof.Clearly the given group is a subgroup of the automorphism group of G k .Suppose f G ρ k g = G k for some invertible linearized polynomials f, g and some ρ ∈ Aut(F q ).As G ρ k = G k for all ρ ∈ Aut(F q ), we may assume that ρ is the identity.Then f = f ′ • (αx q i ) for some α ∈ F × q n , i ∈ {0, . . ., n − 1}, where f ′ such that and by the proof of Theorem 2, we must have deg q (f ′ ) + deg q (g ′ ) = 0. Hence f ′ = x and g ′ = β q i x for some β ∈ F × q n , and so (f, g) = (αx q i , βx q n−i ), proving the claim.
Remark 2. An analogous proof shows that the automorphism group of any generalised Gabidulin code G k,s is equal to the automorphism group of G k .

Construction of new linear MRD-codes
The following Lemma is key to our construction.The result also follows from [15,Theorem 10].
Proof.Any k-dimensional F q -subspace U of F q n annihilated by a unique monic linearized polynomial of degree k, denoted by m U and called the minimal polynomial of U .Every linearized polynomial of degree k annihilating U is an F q n -multiple of m U , and hence it suffices to prove the result for any particular linearized polynomial of degree k annihilating U .
Then each H k (η, h) is an MRD-code with the same parameters as G k .
Proof.It is clear that , and the Delsarte dual satisfies Proof.This follows from a straightforward calculation.
Note that H k (0, h) = G k , so this family includes the Gabidulin codes.We now prove that there are new MRD-codes in this family.First we require the following lemma.
Lemma 4. Suppose η = 0. Then there is a unique subspace of Then by Theorem 2, U = f G k−1 g for some f, g with f 0 = 0 and deg(f This lemma allows us to calculate the automorphism group, and hence prove that the family H k (η, h) contains codes inequivalent to any generalised Gabidulin code, and therefore contains new MRD codes.
if and only if j = h and there exist α, β ∈ F q n such that ν = α 1−q h (β q k −q h ) p i η p i .
Proof.Suppose first that k = 2.By Lemma 4, there is a unique subspace of Hence any isomorphism from H k (η, h) to H k (ν, j) is also an automorphism of G k−1 .By Theorem 3, these are all of the form (αx p i , βx p ne−i ) for some α, β ∈ F × q n , i ∈ {0, . . ., ne − 1}.Now the coefficient of x in a generic element of (αx ) is equal to αβ p i f p i 0 , while the coefficient of x q k is αβ p ke+i (ηf q h 0 ) p i .Hence for this to lie in H k (ν, j) we must have αβ p ke+i (ηf q h 0 ) p i = ν(αβ p i f p i 0 ) q j for all f 0 ∈ F q n .This occurs if and only if j = h and αβ p ke+i η p i = ν(αβ p i ) q h , proving the last claim.Setting η = ν gives the automorphism group of H k (η, h).
It is clear that Aut(H k (η, h)) is strictly smaller than Aut(G s,k ), unless η = 0, proving that H k (η, h) is not equivalent to G k,s for k / ∈ {1, n − 1}.
When k = 1, the code H k (η, h) is a semifield spread set corresponding to a generalised twisted field, as introduced by Albert [1], i.e a presemifield with multiplication x • y = xy + ηx q h y q , where N (−η) = 1.
The equivalence and automorphisms of these follow from [4].By duality, we get the result for k = n − 1, completing the proof.
Note that Aut(H k (η, h)) contains the Singer cycle {(αx, x) : α ∈ F q n } if h = 0.In fact, H k (η, h) is F q nlinear in this case, showing that there exist F q n -linear MRD-codes which are not equivalent to generalised Gabidulin codes.This was also proved by computer calculation in [24] for n = 4, k = 2, q = 3.When h = k, Aut(H k (η, h)) contains the Singer cycle {(x, βx) : β ∈ F q n }.However this code is not F q n -linear.
Remark 3. As noted in the proof above, when k = 1, the code H k (η, h) is a semifield spread set corresponding to a generalised twisted field, see [1].For this reason we propose to name this family of codes twisted Gabidulin codes.
Remark 4. Note that when we view H k (η, h) as a code in (F q n ) n , it is F q n -linear if and only if h = 0.
and L(U g ) are equivalent, then C f and C g are equivalent, as For the converse, it can be shown that if X is an invertible linear map, then XC f is an F q n -subspace if and only X(x) = αx ρ for some α ∈ F q n and ρ ∈ Aut(F q ), whence As the identity is in C g , we get that Y = C f ρ (γf ρ + δx) −1 for some γ, δ ∈ F q n , and so C f is equivalent to C g if and only if C g = (C f ) ρ (γf ρ + δx) −1 .Then the argument from the preceding paragraph can be to show that U is equivalent to U g .Hence we have shown the following.
Theorem 7. Let f and g be scattered linearized polynomials.Then L(U f ) and L(U g ) are equivalent as linear sets if and only if C f and C g are equivalent as MRD-codes.
As far as the author is aware, there are only two known constructions for scattered linear sets of rank n in PG(1, q n ).They are f (x) = x q s , (n, s) = 1 (Blokhuis-Lavrauw [5]), and f (x) = x q + ηx q n−1 , with N (η) = 1 (Lunardon-Polverino [23]).The first family leads to generalised Gabidulin codes G 2,s , while the second lead to codes equivalent to H 2 (η, 1).These are the only F q n -linear codes in the family H 2 , and so we do not obtain any new scattered linear sets from this construction.

Assorted computational results
In [10] it was shown that the set of elements of minimum rank in an MRD-code is partitioned into constant rank subspaces of dimension n, each lying in the annihilator of a subspace of dimension k − 1. Hence an MRD-code with k = 2 must contain an n-dimensional constant rank n − 1 subspace of Ann(u) for each u.
Hence we can quickly classify linear MRD-codes with q = 2, 3, n = 3, and k = 2. First we calculate all constant rank 2 subspaces of Ann(u) up to equivalence for some chosen u, then extend them to MRD-codes and calculate the equivalence classes again.When q = 2, there is only the Gabidulin code G 2 .When q = 3, there are precisely two such codes, namely G 2 and H 2 .This in fact follows from Section 1.5.
When n = 4, q = 2, there is only one MRD-code with k = 2 containing a semifield spread set; the Gabidulin code, an 8-dimensional code of minimum distance 3. The only semifield spread sets it contains are all isotopic to F 16 .There are three isotopy classes of semifields of order 16, i.e.MRD-codes in M 4 (F q ) with minimum distance 4. One is maximal as a rank metric code of minimum distance 3.One is contained in a 5-dimensional code of minimum distance 3, but not a 6-dimensional code of minimum distance 3.
All calculations were carried out using the computer algebra package MAGMA.
Define the symmetric bilinear form b on M m,n (F) by b(X, Y ) := tr(Tr(XY T )),