Let be a prime power, be a positive integer, and denote by the finite field with elements. Rank metric codes can be seen as sets of -linear endomorphisms of equipped with the rank distance, that is the distance between two elements is defined as the (linear algebraic) rank of their difference. Since the -algebra of the -linear endomorphisms of and the -algebra of -linearized polynomials of -degree smaller than are isomorphic, each rank metric code can be also seen as a subset of . For any , we define and . In this context, a rank metric code with minimum distance achieving the equality in the Singleton-like bound
is called maximum rank distance (MRD) code. Rank metric codes and in particular MRD codes have been introduced several times [delsarte1978bilinear, gabidulin1985theory] and have been widely investigated in the last few years, due to applications in network coding [MR2450762] and cryptography [MR3678916]. Two rank metric codes are said to be equivalent if there exist two invertible -linearized polynomials and a field automorphism such that
where if and the composition has to be considered modulo . In order to study the equivalence between two rank metric codes, one can make use of the idealisers. They have been introduced in [liebhold2016automorphism], where the left and right idealisers of a rank metric code are defined respectively as
Such objects have also been investigated in [lunardon2018nuclei], where they have been called respectively middle and right nuclei.
In this paper we are interested in -linear MRD codes, that is MRD codes such that is equivalent to ; see [sheekeysurvey, Definition 12]. Thus, every -linear MRD code is equivalent to an -subspace of and we will always consider MRD codes which are -subspaces of . The MRD condition for -linear rank metric codes reads as follows. By (1), an -linear rank metric code , with and minimum distance , is an MRD code if and only if , or equivalently,
This paper is devoted to the investigation of -linear MRD codes which are exceptional. An -linear MRD code is an exceptional MRD code if the rank metric code
is an MRD code for infinitely many . Only two families of exceptional -linear MRD codes are known:
, with , see [delsarte1978bilinear, gabidulin1985theory, kshevetskiy2005new];
, with and , see [MR3543528, lunardon2018generalized].
The first family is known as generalized Gabidulin codes and the second one as generalized twisted Gabidulin codes.
Although the definition of exceptional -linear MRD codes appears in this paper for the first time, it has been already studied in particular subcases in different contexts.
In the case , exceptional MRD codes have been considered via so-called exceptional scattered polynomials. Let and be a nonnegative integer . Then is said to be scattered of index if for every
The term scattered arises from a geometric framework; see [blokhuis2000scattered]. Indeed, is scattered of index if and only the -subspace
has the property that
for every nonzero vector, that is is scattered with respect to the Desarguesian spread . Sheekey in [MR3543528], taking into account (2), pointed out the following connection between scattered polynomials and -linear MRD codes: is scattered of index if and only if is an MRD code with . The polynomial is said to be exceptional scattered of index if it is scattered of index as a polynomial in , for infinitely many ; see [MR3812212]. Taking into account (2), a polynomial is exceptional scattered of index if and only if the corresponding MRD code is exceptional. While several families of scattered polynomials have been constructed in recent years [MR3543528, lunardon2018generalized, lunardon2000blocking, zanella2019condition, MR4173668, longobardi2021linear, longobardi2021large, NPZ, zanella2020vertex, csajbok2018new, csajbok2018new2, marino2020mrd, blokhuis2000scattered], only two families of exceptional ones are known:
of index , with (polynomials of so-called pseudoregulus type);
of index , with and (so-called LP polynomials).
Such two families correspond to the known exceptional -linear MRD codes (G) and (T).
Several tools have already been proposed in the study of exceptional scattered polynomials, related to algebraic curves or Galois extensions of function fields; see [Bartoli:2020aa4, MR3812212, MR4190573, MR4163074, Bartoli:2021]. However, their classification is still unknown when the index is greater than .
For , the only known families of -subspaces of corresponding to MRD codes are (G) and (T) as described above and Delsarte dual codes of the MRD codes associated with scattered polynomials. In [MR4110235] it has been shown that the only exceptional -linear MRD codes spanned by monomials are the codes (G), in connection with so-called Moore exponent sets.
It is therefore natural to investigate exceptional -linear MRD codes not generated by monomials. To this aim, we generalize the notion of Moore exponent set; see Section 3.
Using a connection between the generators of -linear rank metric codes and certain algebraic hypersurfaces , we obtain a partial classification of exceptional -linear MRD codes. Tools from intersection theory (see Section 2) yield sufficient conditions on the generators for to be MRD, via the existence of -rational absolutely irreducible components of .
Our main results can be summarized as follows.
Let be an exceptional -dimensional -linear MRD code containing at least a separable polynomial and a monomial. If , assume also that . Let be the minimum integer such that .
If and , with and , then is a generalized Gabidulin code.
If and , with for each , then is exceptional scattered of index .
When contains a separable polynomial and a monomial, we call the non-negative integer of Main Theorem the index of .
2. Preliminaries on algebraic varieties
An algebraic hypersurface is an algebraic variety that can be defined by a single polynomial equation. An algebraic hypersurface defined over a field is absolutely irreducible if the associated polynomial is irreducible over every algebraic extension of . An absolutely irreducible -rational component of a hypersurface , defined by the polynomial , is simply an absolutely irreducible hypersurface which is associated to a factor of defined over . For a finite field , let denote its algebraic closure. Also, (resp. ) denotes the -dimensional projective (resp. affine) space over the field .
We recall some known results on algebraic hypersurfaces of which our approach will make use.
[MR2648536, Lemma 2.1] Let be an absolutely irreducible hypersurface and be an -rational hypersurface of . If has a non-repeated -rational absolutely irreducible component, then has a non-repeated -rational absolutely irreducible component.
With the symbol we denote the intersection multiplicity of two plane curves in at a point . Classical results on such an integer can be found in most of the textbooks on algebraic curves. For other concepts related to algebraic varieties we refer to [MR0463157]. For the special case of curves, a good reference is [MR1042981].
[MR1359909, Proposition 2] Let be such that for some . Let be a point in the affine plane and write
where is zero or homogeneous of degree and . Suppose that for some linear polynomial such that . Then , where and are the curves defined by and respectively.
[MR3239294, Lemma 4.3], [MR4110235, Lemma 2.5] Let be such that for some . Let be a point in the affine plane and write
where is zero or homogeneous of degree and . Suppose that for some linear polynomial such that and . Then or , where and are the curves defined by and respectively.
[MR1042981, Section 3.3] Let be such that for some . Let be a point in the affine plane and write
where is zero or homogeneous of degree and . Suppose that factors into distinct linear factors in . Then .
where is zero or homogeneous of degree and . If or contains a non-repeated absolutely irreducible -rational factor, then contains a non-repeated absolutely irreducible -rational factor.
Let be the non-repeated absolutely irreducible -rational factor in (resp. ). Consider the unique absolutely irreducible factor of such that (resp. ). If were not defined over , then there would exist another absolutely irreducible factor of satisfying (resp. ), where is the -Frobenius automorphism of , whence (resp. ), a contradiction. ∎
In the sequel we will investigate hypersurfaces connected with Moore polynomial sets; see Definition 3.3. In particular, we are interested in getting information on the existence of -rational absolutely irreducible components of curves contained in such hypersurfaces.
The approach that we follow has been used for the first time by Janwa, McGuire and Wilson [MR1359909]
to classify functions onthat are almost perfect nonlinear for infinitely many , in particular for monomial functions. It can be summarized by the following theorem.
[MR3326175, Lemma 2] Let be a curve of degree and let be the set of its singular points. Also, let denote the maximum possible intersection multiplicity of two putative components of at . If
then possesses at least one absolutely irreducible component defined over .
3. Moore polynomial sets and MRD codes
Let be a prime power and be a positive integer. Consider -linearly independent polynomials and denote by the -tuple . Define
For any , define .
If are -linearly dependent, then .
Without loss of generality, suppose that and with . Then
so that the first row of is a linear combination of the remaining rows. Then . ∎
The converse of Lemma 3.1 is not true in general, the following being a counterexample.
Let and be positive integers with even and . Consider . Let be a subset of . Then is the Moore matrix
and if and only if the elements are -linearly independent; see [MR3087321, Corollary 2.1.95]. Therefore, if are -linearly independent elements in and , then even though are -linearly independent.
The following definition identifies the tuples for which the converse of Lemma 3.1 holds and it will be crucial in our investigation for exceptional MRD codes.
Let , where is a positive integer and . We say that is a Moore polynomial set for and if, for any ,
If is a Moore polynomial set for and for infinitely many , we say that is an exceptional Moore polynomial set for and .
Moore polynomial sets can be characterized in terms of MRD codes as follows.
Let and be positive integers with , and denote by the -tuple , where are -linearly independent. The -linear rank metric code
is an MRD code if and only if is a Moore polynomial set for and .
Suppose that is singular for some , that is, there exist such that , for every . This means that is contained in the kernel of . Since is an MRD code, it follows that , and hence are -linearly dependent.
Conversely, suppose that is a Moore polynomial set for and . Assume by contradiction that there exists with and write with . Let where are -linearly independent. Then is singular because its columns are -linearly dependent through for all . Therefore, is not a Moore polynomial set for and . ∎
As a natural consequence, a characterization of the exceptionality property is obtained.
Let be an -linear rank metric code. The following are equivalent:
is an exceptional MRD code.
Every -basis of defines an exceptional Moore polynomial set for and .
There exists an -basis of for which is an exceptional Moore polynomial set for and .
We will investigate exceptional MRD codes by means of exceptional Moore polynomial sets.
4. Moore polynomial sets and varieties over finite fields
In this section we study exceptional -linear MRD codes of dimension under the assumption that contains a monomial. Up to equivalence, we can assume that contains a separable polynomial. We denote by the smallest non-negative integer such that .
If is an -linear MRD code in , then contains an invertible map (see [lunardon2018nuclei, Lemma 2.1] and[ravagnani2016rank, Lemma 52]), and hence contains the identity . If is exceptional, then does not depend on the infinitely many ’s for which is MRD. On the contrary, may depend on , so that may not be exceptional.
On the other hand, the assumption that contains a separable polynomial does not affect the exceptionality of , since decreases by .
Note that there exist such that the following hold:
are monic and -linearly independent;
are all distinct;
are all distinct, and for some ;
for any , if is a monomial then .
Therefore, by Corollary 3.5, we investigate Moore polynomial sets as in the following definition.
A Moore polynomial set satisfying Assumptions 4.2 is said to be a Moore polynomial set for and of index .
A key tool in our approach is a link between Moore polynomial sets ans algebraic hypersurface. To this aim, we introduce the following -rational hypersurfaces: is the hypersurface defined by the affine equation
and is the hypersurface . Note that
with a suitable choice of the representative for the points . Since are -linearized, the polynomial divides , so that is a component of . Therefore we can define the -rational variety with affine equation
The link between Moore polynomial sets and algebraic hypersurfaces is straightforward.
The -tuple is a Moore polynomial set for and if and only if all the affine -rational points of lie on .
For any , the condition is equivalent to being an affine -rational point of , while the condition is equivalent to being a point of . The claim follows. ∎
In the case when are monomials, Theorem 4.4 was already noticed (using a slightly different terminology) and used in [MR4110235] to prove the following result.
[MR4110235, Theorems 1.1, 3.2, 4.1] Let be a set of non-negative integers with such that is not in arithmetic progression. Suppose that one of the following holds:
, and .
Then is not a Moore polynomial set for and .
In the sequel, we will use the following notation: for any , write and .
4.1. Moore polynomial sets of index 0
In this section we investigate Moore polynomial sets of index , so that . Without loss of restriction, we assume .
Suppose that one of the following holds:
, and .
If is a Moore polynomial set for and of index , then is in arithmetic progression and is in arithmetic progression for some with .
In order to prove the claim on the ’s, consider the intersection between
and the hyperplane at infinity. Note that is defined by
Suppose that is not in arithmetic progression. Then it has been shown in [MR4110235, Theorems 3.1 and 4.2] that contains an -rational non-repeated absolutely irreducible component . It follows by Lemma 2.5 that has an -rational non-repeated absolutely irreducible component. Then, as shown in [MR4110235] (in page 9 for , and in page 17 for ), there exists an affine -rational point in . Thus, is not a Moore polynomial set for and by Theorem 4.4.
Now suppose that is not in arithmetic progression for any . Consider the tangent variety of at the origin . Then
Now the same arguments as above show that has an -rational non-repeated absolutely irreducible component. Then has an -rational non-repeated absolutely irreducible component by Lemma 2.5. Therefore, as in [MR4110235], has an affine -rational point not in , so that is not a Moore polynomial set for and . ∎
In the rest of this subsection, is a Moore polynomial set for and of index , satisfying the assumptions of Theorem 4.6, so that