Keywords. Sum-rank-metric codes, Linearized Reed–Solomon codes, Rational normal curve, Hilbert sequence.

MSC classification. 11T71, 14H45, 51E20, 94B27.

Introduction↩︎

The connection between linear codes endowed with the Hamming metric and projective algebraic varieties is well known and has proved to be highly fruitful. It links problems that may appear to belong to quite different areas: for example the MDS conjecture and the existence of large arcs [1], [2], minimal codes and strong blocking sets [3], covering problems and saturating sets [4]. The basic idea is that equivalence classes of linear codes can be associated with projective equivalence classes of point sets in projective space, and many coding-theoretic properties can be interpreted geometrically. A particularly significant example is given by Reed–Solomon codes, which correspond to sets of points lying on the so-called rational normal curve. An important feature of this identification is the following: it is well known that the rational normal curve is an intersection of quadrics. As a consequence, the points defined by a Reed–Solomon code lie on many more quadrics than a random set of points would. This property can be reinterpreted in terms of the Schur square of the code and is by now well understood [5]. In this paper, we aim to extend these connections to codes in another metric, namely the sum-rank metric, and to obtain analogous results for the optimal codes in this setting, namely the so-called linearized Reed–Solomon codes.

Sum-rank metric codes have recently attracted increasing attention due to their wide range of applications, including space-time coding, multishot network coding, and distributed storage (see [6] for a complete overview of these applications). Moreover, due to the existence of families of sum-rank metric codes achieving optimal parameters, they are also emerging in cryptography, where they are being explored as potential building blocks for post-quantum code-based primitives [7], [8]. In this setting, the goal is to design schemes that aim to resist known structural attacks on constructions based on Hamming and rank-metric codes, while keeping key sizes relatively small. Since, in code-based cryptographic systems, security often relies on the difficulty of identifying the algebraic structure of the underlying code, the distinguishability problem, i.e., the ability to distinguish structured codes from random ones, is emerging as a central issue. If such distinguishers exist, they may undermine security proofs and potentially lead to practical attacks.

In the last few years, sum-rank metric codes have been extensively studied in a series of papers (see for instance  [9]–[16]). In this setting, codes with optimal parameters are called maximum sum-rank distance (MSRD) codes. Among the few known constructions of MSRD codes, the family of linearized Reed–Solomon codes, introduced in [12], is the most prominent and extensively studied so far. Recently, the equivalence problem for linearized Reed–Solomon codes has been investigated in [17], [18], while [19], [20] proposed the first attempts at an algebraic characterization of this family. However, these results are not fully general since [19] requires partial knowledge of the defining parameters of the code, while [20] applies only to some parameters choices.

Linearized Reed–Solomon codes naturally extend both Reed–Solomon and Gabidulin codes. The former being optimal for the Hamming metric and so are the latter for the rank metric, this suggests that linearized Reed–Solomon codes may inherit structural features that characterize these families. Reed–Solomon codes have been fully characterized by means of a Schur-product-based distinguisher [5] (see also [21]). As we mentioned above, this is related to the geometric interpretation of Reed–Solomon codes as subsets of rational normal curve (see for example [22]). In the rank-metric setting, Gabidulin codes were first algebraically characterized through Overbeck’s distinguisher [23]. Similarly to the above, Gabidulin codes can also be described geometrically in terms of linear sets. This perspective was used in [24] to identify Gabidulin codes via a Schur-product-based invariant, inspired by that introduced in the Hamming-metric setting.

In the theory of linearized Reed–Solomon codes, at least two fundamental problems still have only partial answers. Firstly, even though they generalize both Reed–Solomon and Gabidulin codes, the known distinguishers for the latter have not yet been fully adapted to the sum-rank metric setting. Secondly, although the geometry of sum-rank metric codes has been investigated in [10], [25], a geometric characterization of linearized Reed–Solomon codes is still lacking.

In this paper, we aim to address these gaps by investigating linearized Reed–Solomon codes from a geometric perspective. Motivated by the connection between Reed–Solomon codes and the rational normal curve, we consider a \(q\)-analogue of the latter, previously studied in [26] in the context of rank-metric codes, and we show that it captures the structure of the geometric object associated with linearized Reed–Solomon codes. In particular, we prove that linearized Reed–Solomon codes induce a partition of the set of \(\mathbb{F}_{q^m}\)-rational points of this \(q\)-analogue of the rational normal curve into subsets with a remarkable combinatorial structure, namely scattered linear sets, and a strong interplay among them. Some of these properties were previously established in [25] using different techniques based on viewing linearized Reed–Solomon codes as \((k-1)\)-designs and our result contributes to a deeper understanding of these objects. This description yields a geometric characterization of linearized Reed–Solomon codes for certain parameter choices. Similarly to the case of the rational normal curve which is an intersection of quadrics, our \(q\)-analogue is an intersection of degree \((q+1)\) hypersurfaces: actually, it can be described as a determinantal variety. Hence, the points defined by such linearized Reed–Solomon codes lie on many more hypersurfaces of degree \(q+1\) than a random set of points would. By relating sum-rank metric codes with a suitable generalization of the Hamming-metric code associated with rank-metric codes (see [27]), we show that the method proposed here extends the Schur-product-based invariant introduced in [24] for Gabidulin codes. Even though the behavior of homogeneous forms of degree \(q+1\) suffices to characterize linearized Reed–Solomon codes, it is natural to extend our investigation to higher degree forms, in order to obtain a finer invariant. This leads to the study of the Hilbert function of the coordinate ring associated with the set of projective points defined by our \(q\)-analogue of the rational normal curve. In particular, we obtain a comprehensive description of the behavior of this sequence and identify its points of regularity. Finally, by specializing the results obtained in the last part of the paper, we provide a deeper understanding of the geometric structure of Gabidulin codes, thereby addressing some of the open questions left in [24].

Outline↩︎

The paper is structured as follows. Section 1 introduces the necessary background, including the notions of Schur powers and Hilbert function in the context of coding theory, followed by the notion of sum-rank-metric codes together with an overview of their geometric interpretation, and concluding with the definition of linearized Reed–Solomon codes. In Section 2, we develop the geometric interpretation of linearized Reed–Solomon codes, focusing in particular on their connection with a \(q\)-analogue of the rational normal curve, and identify the geometric properties that characterize this family. Finally, Section 3 is dedicated to the Hilbert function of linearized Reed–Solomon codes, and shows how our approach can be adapted to better understand the geometric interpretation of Gabidulin codes.

1 sec:Preliminaries↩︎

1.1 Schur Powers and Hilbert Function↩︎

Let \(q\) be a prime power, \(\mathbb{F}_q\) be a finite field with \(q\) elements, and \(\overline{\mathbb{F}}_q\) be its algebraic closure. Throughout this paper, we only consider linear codes: a \(k\)-dimensional code \(\mathcal{C}\) over \(\mathbb{F}_q^n\) is a \(k\)-dimensional linear subspace of \(\mathbb{F}_q^n\) endowed with a metric. For \(\mathbf{v}=(v_1,\ldots,v_n),\,\mathbf{w}=(w_1,\ldots,w_n)\in\mathbb{F}_q^n\), we denote by \(\star\) the standard componentwise product \(\mathbf{v}\star\mathbf{w}\mathrel{\vcenter{:}}=(v_1w_1,\ldots,v_nw_n)\). The Schur product \(\star\) of two codes \(\mathcal{C}_1,\mathcal{C}_2\subseteq\mathbb{F}_q^n\) is defined as \[\mathcal{C}_1\star\mathcal{C}_2\mathrel{\vcenter{:}}=\langle c_1\star c_2\,:\,c_1,c_2\in\mathcal{C}\rangle_{\mathbb{F}_q}.\] Then, for \(i\geq1\), it is natural to recursively define the \(i\)-th Schur power of a code \(\mathcal{C}\subseteq\mathbb{F}_q^n\) as \[\mathcal{C}^{(i)} \mathrel{\vcenter{:}}= \mathcal{C}\star\mathcal{C}^{(i-1)},\] where \(\mathcal{C}^{(0)}\mathrel{\vcenter{:}}=\langle(1,\dots,1)\rangle_{\mathbb{F}_q}\) is the code generated by the all-ones vector of length \(n\). For a comprehensive discussion on the Schur product of codes, we refer to [28]. For the purposes of this work, we only recall that \(\dim\left(C^{(i)}\right)\leq\dim\left(C^{(i+1)}\right)\) for all \(i\geq0\). This property motivates the following definition.

The Hilbert sequence and the Castelnuovo–Mumford regularity are closely related to the corresponding notions in commutative algebra from which they derive their names. Let \(\mathcal{C}\) be a linear code of dimension \(k\) in \(\mathbb{F}_q^n\) and \(G\in\mathbb{F}_q^{k\times n}\) be a generator matrix for \(\mathcal{C}\). Suppose that \(\mathcal{C}\) has full support, i.e. \(G\) has no zero column. For \(i\in\{1,\dots,n\}\), let \([\mathbf{g}_i]\) be the point in the projective space \(\mathbb{P}^{k-1}(\overline{\mathbb{F}}_q)\) that corresponds to the \(i\)-th column \(\mathbf{g}_i\) of \(G\). Then, we define the set of projective points associated with \(\mathcal{C}\) as \[\Pi_G\mathrel{\vcenter{:}}=\left\{[\mathbf{g}_i]:1\leq i\leq n\right\}\subseteq\mathbb{P}^{k-1}(\overline{\mathbb{F}}_q)\,.\] Notice that \(\Pi_G\) is a set, and hence its points are considered without multiplicity. In particular, \(\lvert\Pi_G\rvert=n\) if and only if the columns of \(G\) are pairwise linearly independent. While the set \(\Pi_G\) depends on the chosen matrix \(G\), this dependence is only up to projective equivalence, since changing the generator matrix induces a linear projective transformation of \(\Pi_G\). As a result, the geometric properties of \(\Pi_G\) are independent of the specific choice of \(G\).

Given a variety \(\mathcal{V}\subseteq\mathbb{P}^{k-1}(\mathbb{F}_q)\), let \(I(\mathcal{V})\subseteq\mathbb{F}_q[x_1,\dots,x_k]\) be its vanishing ideal (i.e., the homogeneous ideal of polynomials that evaluate to zero on \(\mathcal{V}\)) and \(S_\mathcal{V}\mathrel{\vcenter{:}}=\mathbb{F}_q[x_1,\dots,x_k]/I(\mathcal{V})\) be its homogeneous coordinate ring. We denote by \((S_\mathcal{V})_d\) the homogeneous part of degree \(d\) of \(S_\mathcal{V}\). For \(s\geq0\), \[\mathop{\mathrm{HF}}_{S_\mathcal{V}}\,:\,s \longmapsto \dim\left(\mathbb{F}_q[x_1,\dots,x_k]/I(\mathcal{V})\right)_s\] is the Hilbert function of \(S_\mathcal{V}\). It is well known that there exists a unique polynomial \(\mathop{\mathrm{HP}}_{S_\mathcal{V}}\), called the Hilbert polynomial of \(S_\mathcal{V}\), whose evaluation sequence at non-negative integers agrees with the Hilbert function for all sufficiently large \(s\). The smallest integer \(\bar s\) for which \(\mathop{\mathrm{HF}}_{S_\mathcal{V}}(s)=\mathop{\mathrm{HP}}_{S_\mathcal{V}}(s)\), for all \(s \geq\bar s\), is called the Hilbert regularity \(\mathop{\mathrm{H_{reg}}}(S_\mathcal{V})\) of \(S_\mathcal{V}\). Note, in particular, that the Castelnuovo–Mumford regularity (as it is defined in the context of commutative algebra) and the Hilbert regularity of \(S_\mathcal{V}\) coincide if the ideal \(I(\mathcal{V})\) is 0-dimensional. We refer to [30] for further details. The following proposition illustrates the aforementioned relationship between Hilbert functions and dimension sequences.

1.2 Sum-Rank-Metric Codes and their Geometric Interpretation↩︎

Let \(t,m,n_1,\dots,n_t\) be positive integers such that \(k\leq n\mathrel{\vcenter{:}}=\sum_{i=1}^t n_i\). We say that \(n_1,\dots,n_t\) define a partition of the vector \(\mathbf{n}\mathrel{\vcenter{:}}=(n_1,\dots,n_t)\) and we set \[\mathbb{F}_{q^m}^{\mathbf{n}}\mathrel{\vcenter{:}}=\bigoplus_{i=1}^t \mathbb{F}_{q^m}^{n_i}.\]

It immediately follows that the sum-rank weight reduces to the rank weight over \(\mathbb{F}_{q^m}^n\) if \(t=1\), while it coincides with the Hamming weight over \(\mathbb{F}_{q^m}^n\) if \(n_1=\dots=n_t=1\). Furthermore, by evaluating the distance between two vectors as the weight of their difference, both the rank and the sum-rank weight naturally induce a metric over \(\mathbb{F}_{q^m}^n\) and \(\mathbb{F}_{q^m}^{\mathbf{n}}\), respectively.

In what follows, we restrict ourselves to reporting only what is strictly relevant for our purposes. For further information on sum-rank-metric codes, we refer the reader to [6], [9]. To keep track of the original partition defined by \(\mathbf{n}\), we will often represent a generator matrix \(G\) of \(\mathcal{C}\subseteq\mathbb{F}_{q^m}^{\mathbf{n}}\) as the block matrix \((G_1|\cdots|G_t)\), where \(G_i\in\mathbb{F}_{q^m}^{k\times n_i}\) for all \(i\in\{1,\dots,t\}\). We say that \(\mathcal{C}\) is non-degenerate if the columns of each submatrix \(G_i\) are linearly independent over \(\mathbb{F}_q\). In the following, we will always assume codes to be non-degenerate. Two \([\mathbf{n},k,d]_{q^m/q}\) codes are said to be equivalent if there exists a self-isometry of the ambient space that maps one onto the other. The following theorem characterizes the possible isometries of \(\mathbb{F}_{q^m}^{\mathbf{n}}\) with respect to the sum-rank metric.

Note that, since the multiplication on the right by invertible matrices over \(\mathbb{F}_q\) is a sum-rank-metric isometry, two isometric \(\mathbb{F}_{q^m}\)–linear sum-rank metric codes \(\mathcal{C}, \mathcal{C}'\) with respective generator matrices \(G,G'\) might be associated with sets of points \(\Pi_G, \Pi_{G'}\) that are not projectively equivalent. In light of Proposition 2, this implies that the dimension sequence of codes is not invariant under sum-rank equivalence and therefore does not provide meaningful structural information about the underlying code. As a consequence, in the sum-rank-metric setting, the above defined set \(\Pi_G\) is not the most suitable geometric object that a code can be associated with. The same problem already holds for rank-metric codes, and it was overcome in [27], [32] by defining a different geometric interpretation of codes that classifies them in terms of their equivalence classes. In a similar way, in [10] the authors associate sum-rank-metric codes with an analogous geometric object whose properties are deeply connected with structural properties of the associated family of codes and can therefore be studied to reveal them. In the following, we work with a similar, but slightly modified, version of the geometric object introduced in [10].
Let \(\mathcal{C}\) be an \([\mathbf{n},k]_{q^m/q}\) code with generator matrix \(G=(G_1|\dots|G_t)\). For \(i\in\{1,\dots,t\}\), consider the column space \(\mathcal{U}_i\mathrel{\vcenter{:}}=\mathop{\mathrm{colsp}}_{\mathbb{F}_q}(G_i)\) of \(G_i\). Similarly to the classical case described in the previous subsection, we aim to derive an associated projective variety. For this reason, for every \(i\in\{1,\dots,t\}\), we consider the projective set \[\mathcal{L}_{\mathcal{U}_i}\mathrel{\vcenter{:}}=\left\{[\mathbf{u}]:\mathbf{u}\in\mathcal{U}_i\setminus\{\mathbf{0}\}\right\}\subseteq\mathbb{P}^{k-1}(\overline{\mathbb{F}}_{q^m}).\] Finally, we define the set of points associated with \(\mathcal{C}\) with respect to the generator matrix \(G\) as \[\mathcal{L}_{G}\mathrel{\vcenter{:}}=\mathcal{L}_{\mathcal{U}_1}\cup\cdots\cup \mathcal{L}_{\mathcal{U}_t}\subseteq\mathbb{P}^{k-1}(\overline{\mathbb{F}}_{q^m}).\] We point out that we are always working set theoretically, and therefore up to multiplicity, as in the classical case. Regarding the cardinality of \(\mathcal{L}_G\), we immediately obtain the following inequality \[\lvert \mathcal{L}_G\rvert\leq\sum_{i=1}^t\frac{q^{n_i}-1}{q-1}.\]

The next proposition follows immediately from the definition of \(\mathcal{L}_G\) and Theorem 3.

In order to fully extend the geometric interpretation of codes introduced in the previous subsection to the sum-rank-metric setting, we conclude this paragraph by introducing the following notion (see [10], [27]) which is designed so that its associated set of projective points faithfully captures the geometric object of a sum-rank-metric code. Let \(\sim_{\mathbb{F}_q}\) be the proportionality relation over \(\mathbb{F}_{q^m}^k\) such that for all \(u,u'\in\mathbb{F}_{q^m}^k\), \(u\sim_{\mathbb{F}_q}u'\) if and only if \(u=\lambda u'\), for some \(\lambda\in\mathbb{F}_q^*\).

In particular, note that \(\mathcal{C}^\mathrm{H}\) is a \(k\)-dimensional code over \(\mathbb{F}_{q^m}^L\), with \(L\mathrel{\vcenter{:}}= L_1+\dots+L_t\), and, as its name suggests, it is considered to be endowed with the Hamming metric. Moreover, it was proved in [10] that equivalent codes with respect to the sum-rank metric are mapped into equivalent associated Hamming-metric codes, with respect to the Hamming metric (to be precise, our definition is the projective version of the associated code, without repeated columns, but the result in [10] implies the equivalence in our case).

Consider the projective system \(\Pi_{G^\mathrm{H}}\subseteq\mathbb{P}^{k-1}(\mathbb{F}_{q^m})\) associated with \(\mathcal{C}^\mathrm{H}\), as it was described in Remark 1. Then, the relevance of the associated Hamming-metric code is immediately evident when realizing that \(\Pi_{G^\mathrm{H}}\) coincides precisely with the set of projective points \(\mathcal{L}_G\) associated with its underlying sum-rank-metric code \(\mathcal{C}=\mathop{\mathrm{rowsp}}_{\mathbb{F}_{q^m}}(G)\). Therefore, in view of Proposition 2, in the sum-rank-metric setting it is natural to introduce the following definition, whose notation aligns with that adopted in [24] in the context of rank-metric codes.

The choice of introducing this definition is further supported by the fact that equivalent \([\mathbf{n},k]_{q^m/q}\) codes are associated with projectively equivalent sets of points in \(\mathbb{P}^{k-1}(\mathbb{F}_{q^m})\), as stated in Proposition 5. As a result, the structural properties of a code \(\mathcal{C}=\mathop{\mathrm{rowsp}}_{\mathbb{F}_{q^m}}(G)\) which are reflected in its associated set of projective points can be studied through its corresponding \((q^m/q)\)-dimension sequence. In particular, the definition of \((q^m/q)\)-Hilbert sequence and Proposition 2 immediately imply that \[\begin{align} \label{eq:32Fq-Hilbert32seq} \dim{\left(\mathcal{C}^{\textrm{H}}\right)}^{(i)}&=\dim\big(\mathbb{F}_{q^m}[x_1,\dots,x_k]/I(\mathcal{L}_G)\big)_i \nonumber \\ &= \binom{k+i-1}{i}-\dim({I(\mathcal{L}_G)}_i). \end{align}\tag{1}\]

1.3 Linearized Reed–Solomon Codes↩︎

As in the cases of linear block codes and rank-metric codes, a Singleton-like bound also holds in the sum-rank-metric setting.

Codes attaining the Singleton-like bound are called maximum sum-rank distance separable codes, or shortly MSRD codes. In this section, we focus on a subfamily of MSRD codes known as linearized Reed–Solomon codes. These codes were introduced by Martínez-Peñas in [12] and can be viewed as a special class of evaluation codes. To define them, we first recall the notion of skew polynomials, originally introduced in a more general setting by Ore in [33].
Let \(\sigma\) be a generator of the Galois group \(\mathop{\mathrm{Gal}}(\mathbb{F}_{q^m}/\mathbb{F}_q)\). The skew polynomial ring \(\mathbb{F}_{q^m}[x;\sigma]\) is the set of formal polynomials \[\bigg\{f=\sum_i f_ix^i\,:\,f_i\in\mathbb{F}_{q^m}\text{ and finitely many }f_i\text{'s are different from }0 \bigg\},\] equipped with the standard componentwise polynomial addition \[f+g=\sum_i(f_i+g_i)x^i\] and the multiplication rule \[x\alpha=\sigma(\alpha)x,\] which is extended to polynomials by associativity and distributivity. Note that \(\mathbb{F}_{q^m}[x;\sigma]\) trivially reduces to the standard polynomial ring \(\mathbb{F}_{q^m}[x]\) if \(\sigma\) is the identity map, \(\mathop{\mathrm{id}}_{\mathbb{F}_{q^m}}\), over \(\mathbb{F}_{q^m}\).
The degree of a non-zero skew polynomial \(f=\sum_i f_ix^i\in\mathbb{F}_{q^m}[x;\sigma]\) is defined as \(\max\{i\,:\,f_i\neq0\}\), while we set \(-\infty\) to be the degree of the zero polynomial. We will denote by \(\mathbb{F}_{q^m}[x;\sigma]_{<k}\) the set of skew polynomials of degree less than \(k\).

Standard polynomial evaluation is not meaningful when considering Ore polynomials, as it does not define a ring homomorphism. Hence, it is needed to define an alternative evaluation map for the ring \(\mathbb{F}_{q^m}[x;\sigma]\). Here, we adopt the approach commonly referred to as generalized operator evaluation. For \(a,b\in\mathbb{F}_{q^m}\), we define the evaluation map \[\mathop{\mathrm{ev}}_{a,b}:\mathbb{F}_{q^m}[x;\sigma]\longrightarrow\mathbb{F}_{q^m}, \;\sum_i f_ix^i\longmapsto \sum_i f_i\left(\sigma^i(b)N_i(a)\right),\] where the \(N_i\)’s denote the \(i\)-th truncated norm operators, namely, \[\label{eq:partial95norms} \forall a \in \mathbb{F}_{q^m},\quad N_0(a)\mathrel{\vcenter{:}}= 1 \qquad \text{and}\qquad \forall i >0,\;\; N_i(a)\mathrel{\vcenter{:}}=\prod_{j=0}^{i-1}\sigma^j(a).\tag{2}\] In particular, \(N_m\) is nothing but the usual norm on \(\mathbb{F}_{q^m}/\mathbb{F}_q\) operator: \[\label{eq:norm} N_m : \mathbb{F}_{q^m}\longrightarrow \mathbb{F}_q,\;\;a \longmapsto \prod_{i=0}^{m-1} \sigma^i(a).\tag{3}\] To introduce linearized Reed–Solomon codes, we use a multi-point evaluation version of the map above. Let \(\mathbf{a}=(a_1,\dots,a_t)\in\mathbb{F}_{q^m}^t\) and \(\mathbf{b}=(\mathbf{b}_1,\dots,\mathbf{b}_t)\in\mathbb{F}_{q^m}^{\mathbf{n}}\), with \(\mathbf{b}_i=(b_{i,1},\dots,b_{i,n_i})\in\mathbb{F}_{q^m}^{n_i}\) for \(i\in\{1,\dots,t\}\). Then, we define \[\begin{align} \mathop{\mathrm{ev}}_{\mathbf{a},\mathbf{b}} :\;\mathbb{F}_{q^m}[x;\sigma]&\longrightarrow \mathbb{F}_{q^m}^\mathbf{n}\\ f &\longmapsto \left(\mathop{\mathrm{ev}}_{a_1,b_{1,1}}(f),\dots,\mathop{\mathrm{ev}}_{a_1,b_{1,n_1}}(f)\,\big|\,\dots\,\big|\,\mathop{\mathrm{ev}}_{a_t,b_{t,1}}(f),\dots,\mathop{\mathrm{ev}}_{a_t,b_{t,n_t}}(f)\right). \end{align}\] In order to use the map \(\mathop{\mathrm{ev}}_{\mathbf{a},\mathbf{b}}\) for our goal, we need to impose some further conditions on the evaluation parameters \(\mathbf{a}\) and \(\mathbf{b}\). Aligning with the notation introduced in [10], we will say that \((\mathbf{a},\mathbf{b})\in\mathbb{F}_{q^m}^t\times\mathbb{F}_{q^m}^\mathbf{n}\) is an evaluation pair if the norms over \(\mathbb{F}_q\) of the components of \(\mathbf{a}\) are non-zero and pairwise distinct, and the elements of each component of \(\mathbf{b}\) are linearly independent over \(\mathbb{F}_q\). In particular, note that the last condition is equivalent to require that \(w_{\mathop{\mathrm{srk}}}(\mathbf{b})=n\).

Note that the conditions for \((\mathbf{a},\mathbf{b})\in\mathbb{F}_{q^m}^t\times\mathbb{F}_{q^m}^\mathbf{n}\) to form an evaluation pair imply that the maximum number of blocks of a linearized Reed–Solomon is \(t=q-1\) and that the length of each block must satisfy \(n_i\leq m\), for all \(i\in\{1,\dots,t\}\).

It is immediate to see that a generator matrix for \(\mathop{\mathrm{LRS}}_k(\mathbf{a},\mathbf{b})\) is defined as the block matrix \(G=(G_1|\cdots|G_t)\), with \[\begin{align} \label{eq:32gen32matrix32LRS} G_i &=\left(\begin{array}{ccc} b_{i,1} & \dots & b_{i,n_i}\\ \sigma(b_{i,1})N_1(a_i) & \cdots & \sigma(b_{i,n_i})N_1(a_i) \\ \sigma^2(b_{i,1})N_2(a_i) & \cdots & \sigma^2(b_{i,n_i})N_2(a_i) \\ \vdots & & \vdots \\ \sigma^{k-1}(b_{i,1})N_{k-1}(a_i) & \cdots & \sigma^{k-1}(b_{i,n_i})N_{k-1}(a_i) \\ \end{array}\right) \nonumber \\ &= \underbrace{\left(\begin{array}{cccc} 1 & & & \\ & N_1(a_i) & & \\ & & \ddots & \\ & & & N_{k-1}(a_i) \end{array}\right)}_{\mathrel{\vcenter{:}}= D_i} \;\underbrace{\left(\begin{array}{ccc} b_{i,1} & \cdots & b_{i,n_i}\\ \sigma(b_{i,1}) & \cdots & \sigma(b_{i,n_i}) \\ \vdots & & \vdots \\ \sigma^{k-1}(b_{i,1}) & \cdots & \sigma^{k-1}(b_{i,n_i}) \\ \end{array}\right)}_{\mathrel{\vcenter{:}}= M_i}, \end{align}\tag{4}\] for \(i\in\{1\dots,t\}\).

In [10] the notion of linearized Reed–Solomon codes was further generalized by extending the evaluation of skew polynomials at the points \(0\) and \(\infty\).

Note once again that because of the constraints imposed by the definition of an evaluation pair, other than the last two one-length blocks, a doubly-extended linearized Reed–Solomon can have at most \(q-1\) blocks of length \(m\).

2 The Geometry of Linearized Reed–Solomon Codes↩︎

It is well known that a \(k\)-dimensional Reed–Solomon code over \(\mathbb{F}_q^n\) corresponds to a set of points in \(\mathbb{P}^{k-1}(\mathbb{F}_q)\) lying on a rational normal curve (see  [22]). In the rank-metric setting, MRD codes over \(\mathbb{F}_{q^m}^n\) can be similarly associated with finite geometric objects, namely linear sets scattered with respect to hyperplanes in \(\mathbb{P}^{k-1}(\mathbb{F}_{q^m})\) (see [35]). However, up to now, no specific geometric characterization of Gabidulin codes was known, except for the work of [24]. In this section, we investigate the geometry of linearized Reed–Solomon codes, developing a generalization of the connection existing in the Hamming-metric setting and providing a foundation for a deeper geometric understanding of Gabidulin codes, which will be developed further in the next section, specifically in Subsection 3.2.

Let \(j\) be a natural number coprime with \(m\). Consider the vectors \(\boldsymbol{\alpha}\mathrel{\vcenter{:}}=(\alpha_1,\alpha_2,\dots, \alpha_k)\), \(\boldsymbol{\beta}\mathrel{\vcenter{:}}=(\beta_1,\beta_2,\dots,\beta_k)\in\mathbb{Z}^k\), where \[\alpha_i\mathrel{\vcenter{:}}=\frac{q^{(i-1)j}-1}{q^j-1}\text{ and }\beta_i\mathrel{\vcenter{:}}=\frac{q^{(k-1)j}-1}{q^j-1}-\alpha_i\] for all \(i\in\{1,\dots,k\}\).

Let \(\sigma:\overline{\mathbb{F}}_{q^m}\longrightarrow\overline{\mathbb{F}}_{q^m}\) be the automorphism given by \(\sigma(a)=a^{q^j}\) for all \(a\in\overline{\mathbb{F}}_{q^m}\). Note that the restriction of \(\sigma\) to \(\mathbb{F}_{q^m}\) is a generator of the Galois group \(\mathop{\mathrm{Gal}}(\mathbb{F}_{q^m}/\mathbb{F}_q)\). Using the notion of truncated norm introduced above, the curve \(X_{q^j}\) can be equivalently described as the set \[X_{q^j}=\left\{\left[N_0(a):N_1(a):\cdots:N_{k-1}(a)\right]:a\in \overline{\mathbb{F}}_{q^m}\right\} \cup \{P_\infty\},\] where \(P_\infty\mathrel{\vcenter{:}}=[0:\cdots:0:1]\) and the \(N_i\)’s are the truncated norms defined in (2 ) and (3 ).

The \(q^j\)-rational normal curve can be realized as a monomial projection of the rational normal curve of degree \(\alpha_k\). However, this property is shared by a large family of monomial curves and therefore does not uniquely characterize it. The terminology is instead justified by the theorem below, which provides a determinantal representation of the curve. This representation is given in terms of a matrix of forms that acts as a natural \(q^j\)-analogue of the classical Hankel matrix defining standard rational normal curves.

Recall that an \(\mathbb{F}_{q^m}\)-rational point of \(\mathbb{P}^{k-1}\) is a point whose homogeneous coordinates are proportional to a vector in \(\mathbb{F}_{q^m}^k\). Then, the \(\mathbb{F}_{q^m}\)-rational points of \(X_{q^j}\) are \[X_{q^j}(\mathbb{F}_{q^m})=\left\{\left[N_0(a):N_1(a):\cdots:N_{k-1}(a)\right]:a\in \mathbb{F}_{q^m}\right\} \cup \{P_\infty\}.\] In particular, observe that \(\lvert X_{q^j}(\mathbb{F}_{q^m})\rvert=q^m+1\).

For all \(a\in\mathbb{F}_{q^m}\), we denote by \(\mathcal{V}_a(\mathbb{F}_{q^m})\) the set \[\mathcal{V}_a(\mathbb{F}_{q^m})\mathrel{\vcenter{:}}= \left\{\left[b:N_1(a)\sigma(b):N_2(a)\sigma^2(b):\cdots:N_{k-1}(a)\sigma^{k-1}(b)\right]:b\in\mathbb{F}_{q^m}^*\right\}.\]

For the sake of completeness, we restate and prove the following result, originally appearing as [26] and [25], in the notation of the present paper.

The \(\mathbb{F}_{q^m}\)-rational points of the curve \(X_{q^j}\) are therefore partitioned into linear sets associated with Gabidulin codes (together with two distinguished points). In the next proposition, we establish a stronger result: if the linear set arising from an MRD code is contained in the curve, then it must necessarily coincide with one of the components of this partition.

We conclude this section by showing that the associated set of projective points of a doubly-extended linearized Reed–Solomon code lies on a \(q^j\)-rational normal curve.

2.1 Characterization of Linearized Reed–Solomon Codes↩︎

In this subsection, we provide a characterization of linearized Reed–Solomon codes of dimension \(k\leq m\), building on the geometric interpretation of this class of codes. This result yields an effective method to determine whether a given generator matrix of a sum-rank metric code defines a linearized Reed–Solomon code, and hence a practical way to differentiate them from non-structured codes.

3 The Hilbert Sequence of a Linearized Reed–Solomon Code↩︎

The aim of this section is to advance further in the identification of invariants for linearized Reed–Solomon codes by studying the homogeneous forms that vanish on their associated geometric object. As observed in Remark 15, the homogeneous forms of degree \(q+1\) play a key role in the characterization of linearized Reed–Solomon codes. It is then natural to investigate whether forms of higher degrees could reveal other distinctive information about the code under analysis. In particular, in this section we study the \((q^m/q)\)-Hilbert sequence of doubly-extended linearized Reed–Solomon codes with parameters \(t=q+1\) and \(\mathbf{n}=(m,\dots,m,1,1)\). By Proposition 2, this is equivalent to studying the Hilbert function of the coordinate ring of their associated set of projective points. In the previous section, we showed that this set coincides with \(X_{q^j}(\mathbb{F}_{q^m})\), the set of \(\mathbb{F}_{q^m}\)-rational points of the \(q^j\)-rational normal curve. Now, our goal is to study its Hilbert function.

To simplify the computations, from now on we will only consider the Frobenius automorphism \(\sigma\) (\(j=1\)), but with due care our techniques also apply to any generator of the Galois group \(\mathop{\mathrm{Gal}}(\mathbb{F}_{q^m}/\mathbb{F}_q)\). Moreover, if \(k=2\), we simply have that \(X_q=\mathbb{P}^1(\overline{\mathbb{F}}_{q^m})\), while, for \(k=3\), \(X_q\) reduces to a plane curve. In particular, since it is generated by a single polynomial of degree \(q+1\), the Hilbert function is given by the formula \[\mathop{\mathrm{HF}}(d)=\binom{2+d}{2}-\binom{2+d-(q+1)}{2},\] where the second term is meant to be \(0\) if \(2+d-(q+1)<2\). Moreover, we immediately conclude that the Hilbert regularity is \(q\). Therefore, from now on, we assume \(k\in\{4,\dots,m\}\).

In the following, we denote by \[S\mathrel{\vcenter{:}}=\mathbb{F}_{q^m}[x_1,\dots,x_k]/I(X_q) \;\text{ and } \;S_{\mathbb{F}_{q^m}}\mathrel{\vcenter{:}}=\mathbb{F}_{q^m}[x_1,\dots,x_k]/I(X_q(\mathbb{F}_{q^m}))\] the homogeneous coordinate rings of \(X_q\) and \(X_q(\mathbb{F}_{q^m})\), respectively. Since \(X_q(\mathbb{F}_{q^m})\) is contained in \(X_q\), we immediately obtain \(\mathop{\mathrm{HF}}_S(d)\geq\mathop{\mathrm{HF}}_{S_{\mathbb{F}_{q^m}}}(d)\) for all \(d\geq0\). Even though the two Hilbert functions will eventually diverge from each other, we are interested in understanding until which point their equality holds.

In Proposition 9 we proved that \(X_q\) is a toric variety (see [40]). Therefore, it follows by [40] that \(I(X_q)\) is spanned as an \(\mathbb{F}_{q^m}\)-vector space by the set \(\{x^\mathbf{u}-x^\mathbf{v}:\mathbf{u},\mathbf{v}\in\mathbb{Z}_{\geq0}^k\text{ such that }A\mathbf{u}=A\mathbf{v}\}\), where \[\label{eq:32matrix32A} A\mathrel{\vcenter{:}}=\begin{pmatrix} \boldsymbol{\alpha}\\ \boldsymbol{\beta} \end{pmatrix} =\begin{pmatrix} 0 & 1 & \frac{q^2-1}{q-1} & \cdots &\frac{q^{k-1}-1}{q-1} \\ \frac{q^{k-1}-1}{q-1} & \frac{q^{k-1}-1}{q-1}-1 & \frac{q^{k-1}-1}{q-1}-\frac{q^2-1}{q-1} & \cdots & 0 \end{pmatrix}.\tag{9}\] In the next lemma, we provide a similar characterization for \(I(X_q(\mathbb{F}_{q^m}))\).

Although we do not have a closed formula for \(D\), we can easily establish a lower bound. The Hilbert functions of \(S\) and \(S_{\mathbb{F}_{q^m}}\) coincide up to the smallest degree \(D\) for which \(I(X_q(\mathbb{F}_{q^m}))_D\setminus I(X_q)_D\) becomes non-empty. If this is the case, i.e., there exists a polynomial \(p(x_1,\dots,x_k)\) of degree \(D\) in \(I(X_q(\mathbb{F}_{q^m}))\setminus I(X_q)\), then Bézout’s theorem [43] implies \[|\,X_q(\mathbb{F}_{q^m})\,|\leq |\,V(p)\cap X_q \,| \leq \deg(p)\deg(X_q),\] where \(V(p)\) is the variety of points in \(\mathbb{P}^{k-1}(\mathbb{F}_{q^m})\) that annihilate \(p\). Therefore, \[\deg(p)\geq \frac{|\,X_q(\mathbb{F}_{q^m})\,|}{\deg(X_q)}=\frac{(q^m+1)(q-1)}{q^{k-1}-1},\text{ that is } D\geq\left\lceil\frac{(q^m+1)(q-1)}{q^{k-1}-1}\right\rceil.\]

As a consequence, if \(k\ll m\), the two Hilbert functions \(\mathop{\mathrm{HF}}_S\) and \(\mathop{\mathrm{HF}}_{S_{\mathbb{F}_{q^m}}}\) coincide for many values. We can then simply focus on \(\mathop{\mathrm{HF}}_S(d)\).

We remark that obtaining a compact closed formula for \(\delta(X_q,P_\infty)\) is, in general, not straightforward. In practice, one must proceed case by case and choose the most suitable strategy for its computation.

3.1 Sumsets and Hilbert Regularity↩︎

The goal of this section is to compute the Hilbert regularity of the coordinate ring \(S\). To do so, we state the problem in terms of sumsets. We refer to [47] for a general introduction to the topic.

As in the previous subsection, we denote by \(\mathbf{S}(\boldsymbol{\alpha})\) the non-ordered set of entries \(\alpha_1,\dots,\alpha_k\) of the vector \(\boldsymbol{\alpha}=\left(0,1,\frac{q^2-1}{q-1},\cdots,\frac{q^{k-1}-1}{q-1}\right)\). For every non-negative integer \(s\), define the \(s\)-fold sumset of \(\Gamma(\boldsymbol{\alpha})\) as \(0\mathbf{S}(\boldsymbol{\alpha})\mathrel{\vcenter{:}}=\{0\}\) and \[s\mathbf{S}(\boldsymbol{\alpha})\mathrel{\vcenter{:}}=\{\alpha_{i_1}+\dots+\alpha_{i_s}:1\leq i_1\leq\dots\leq i_s\leq k\}\] for all \(s\geq1\). Then, there exists a direct relation among the \(s\)-fold sumsets of \(\Gamma(\boldsymbol{\alpha})\) and the Hilbert function of \(S\), that is \[\mathop{\mathrm{HF}}_S(s)=\lvert\, s\mathbf{S}(\boldsymbol{\alpha})\,\rvert,\] see [48] or [46]. In addition, the Castelnuovo–Mumford regularity of \(S\) can as-well be expressed in terms of sumsets.

In light of this result, our goal is then to estimate \(\mathrm{AP}(\Gamma(\boldsymbol{\alpha}))\) and \(\mathrm{E}(\Gamma(\boldsymbol{\alpha}))\).

Combining Theorem 20 with Lemma 4 and Lemma 5, we finally conclude that \[\mathop{\mathrm{CM_{reg}}}(S)=(q-1)(k-2)+1.\] In the next theorem we show that \(S\) is Cohen-Macaulay and, as a consequence, we determine the value of the Hilbert regularity of \(S\).

It remains to determine the Hilbert regularity of the homogeneous coordinate ring \(S_{\mathbb{F}_{q^m}}\). To this end, we develop an analogue of Theorem 20 in the context of finite abelian groups.
Denote by \(\mathbb{Z}/(q^m-1)\mathbb{Z}\) the cyclic group of order \(q^m-1\). We set \[T_1\mathrel{\vcenter{:}}=\left\{1, \frac{q^2-1}{q-1},\dots,\frac{q^{k-2}-1}{q-1}\right\}\subset\mathbb{Z}/(q^m-1)\mathbb{Z}.\] For all \(d\geq2\), we recursively define the set \(T_d\subseteq\mathbb{Z}/(q^m-1)\mathbb{Z}\) as \[T_d\mathrel{\vcenter{:}}= T_{d-1}\cup (T_{d-1}+1)\cup \left(T_{d-1}+\frac{q^2-1}{q-1}\right)\cup\dots\cup\left(T_{d-1}+\frac{q^{k-1}-1}{q-1}\right).\] Note that \(T_d\) does not equal \(d\mathbf{S}(\boldsymbol{\alpha})\) modulo \(q^m-1\) since \(T_1\) is obtained from \(\Gamma(\boldsymbol{\alpha})\) by removing \(0\) and \(\frac{q^{k-1}-1}{q-1}\).

3.2 Linearized Reed–Solomon Codes with an Arbitrary Number of Blocks↩︎

The analysis carried out in the previous subsection can be applied analogously to study the \((q^m/q)\)-Hilbert sequence of a (doubly-extended) linearized Reed–Solomon code \(\mathcal{C}\) with \(t<q+1\) blocks of maximal length. In this case, one considers the Hilbert sequence of the homogeneous coordinate ring associated with the set of projective points \(\bigsqcup_{i=1}^t \mathcal{L}_{\mathcal{U}_i}\subset X_q(\mathbb{F}_{q^m})\) and adapts the final results of the previous subsection to determine the corresponding Hilbert regularity. In particular, if \(\mathcal{C}\) is a standard linearized Reed–Solomon code, by Theorem 12, for every \(i \in \{1,\dots,t\}\) we have \(\mathcal{L}_{\mathcal{U}_i} = \mathcal{V}_{a_i}(\mathbb{F}_{q^m})\) for some \(a_i \in \mathbb{F}_{q^m}^*\). Moreover, by Proposition 2, we already know that the final stable value reached by the \((q^m/q)\)-Hilbert sequence of \(\mathcal{C}\) is \(\lvert \, \bigsqcup_{i=1}^t \mathcal{V}_{a_i}(\mathbb{F}_{q^m})\, \rvert = t\frac{q^m-1}{q-1}\). The doubly-extended case is analogous. The only difference is that \(\mathcal{L}_{\mathcal{U}_{t-1}} = P_0\) and \(\mathcal{L}_{\mathcal{U}_t} = P_\infty\), so that the final stable dimension becomes \((t-2)\frac{q^m-1}{q-1}+2\).

By further restricting to a single block, the same strategies presented here can be adapted to study the \((q^m/q)\)-Hilbert sequence of a \(k\)-dimensional (generalized) Gabidulin code \(\mathcal{C}\) over \(\mathbb{F}_{q^m}^m\). In particular, up to a change of basis, one can assume that \(\mathcal{C}\) is generated by a matrix \[G=\left(\begin{array}{cccc} 1 & & & \\ & N_1(a) & & \\ & & \ddots & \\ & & & N_{k-1}(a) \end{array}\right) \;\left(\begin{array}{ccc} b_1 & \cdots & b_m\\ \sigma(b_1) & \cdots & \sigma(b_m) \\ \vdots & & \vdots \\ \sigma^{k-1}(b_1) & \cdots & \sigma^{k-1}(b_m) \\ \end{array}\right)\] of the form \(\eqref{eq:32gen32matrix32LRS}\) above, for some \(a\in\mathbb{F}_{q^m}^*\) and some \(\mathbb{F}_q\)-linearly independent elements \(b_1,\dots,b_m\in\mathbb{F}_{q^m}\). In this case, it follows by Theorem 12 that the geometric object associated with \(\mathcal{C}\) is \(\mathcal{L}_G=\mathcal{V}_a(\mathbb{F}_{q^m})\). Hence, studying the \((q^m/q)\)-Hilbert sequence of \(\mathcal{C}\) reduces to studying the Hilbert sequence of \(\mathcal{V}_a(\mathbb{F}_{q^m})\). In this way, the results presented here extend the analysis of the \((q^m/q)\)-Hilbert sequence of Gabidulin codes carried out in [24]. Specifically, for the cases in which \(k-1\) divides \(m\), we can derive the following result.

As a direct consequence of the previous theorem and Lemma 2, we derive the following.

To illustrate our theoretical findings, we performed some computational experiments to study and visualize the Hilbert sequences of linearized Reed–Solomon codes using Magma [54] (see [55] for the detailed implementation). The results are presented in Figure 1 and Figure 2.

Example 5. Figure 1 displays the \((q^m/q)\)-Hilbert sequence of a \([(4,4,4,4), 4]_{5^4/5}\) linearized Reed–Solomon code \(\mathcal{C}\), together with some notable points. In particular, the value \(8=(q-1)(k-2)=\mathop{\mathrm{H_{reg}}}(S)\) marks the step at which the Hilbert sequence begins to coincide with the Hilbert polynomial \(31x=\frac{q^{k-1}-1}{q-1}x\). The sequence continues to agree with the Hilbert polynomial up to step \(20=\left\lfloor\frac{(q^m-1)(q-1)}{q^{k-1}-1}\right\rfloor=\mathop{\mathrm{H_{reg}}}(S)\), after which it slightly deviates before reaching its final stabilization. This behavior is a direct consequence of Corollary 1, since from this degree onward the vanishing ideal of \(\mathcal{L}_{\mathcal{U}_\mathcal{C}}\) is no longer generated solely by the defining polynomials of \(I(X_q)\). The sequence then proceeds until it attains its final stable dimension \(624=q^m-1\) at step \(28=\left\lfloor\frac{(q^m-1)(q-1)}{q^{k-1}-1}\right\rfloor+(q-1)(k-2)=\mathop{\mathrm{H_{reg}}}\left(S_{X_q(\mathbb{F}_{q^m})\setminus\{P_0,P_\infty\}}\right)\). Note that this value is coherent with Proposition 2 as this example tests a standard linearized Reed–Solomon code with \(q-1\) blocks whose associated geometric object is \(X_q\setminus\{P_0,P_\infty\}\) by Theorem 12.

Example 6. Figure 2 represents the \((q^m/q)\)-Hilbert sequences of two \([(4,4,4), 4]_{4^4/4}\) codes. In particular, the orange line is related to a linearized Reed–Solomon code, \(\mathcal{C}\), while the blue line is associated to a code \(\mathcal{G}\) obtained as the concatenation of three unrelated Gabidulin codes. This example extends Remark 15 and confirms the expected fact that the different geometric objects associated with these two codes are also reflected in their \((q^m/q)\)-Hilbert sequences. In particular, after initially coinciding, the two sequences diverge at step \(5=q+1\), when the \((q^m/q)\)-Hilbert sequence of \(\mathcal{G}\) begins to grow faster than that of \(\mathcal{C}\). As pointed out in Remark 15, this behavior is related with the fact that the geometric objects associated with each block of \(\mathcal{G}\) do not necessarily lie on the same \(q\)-rational normal curve. Hence, the \((q+1)\)-th part of \(I(\mathcal{L}_{\mathcal{U}_\mathcal{G}})\) will in general not contain all the polynomials determined in Proposition 9 to identify \(X_q\), while \(I(X_q)_{q+1}\subseteq I(\mathcal{L}_{\mathcal{U}_\mathcal{C}})_{q+1}\). The two sequences then reach their final stable dimension \(255=q^m-1\) at steps \(13\) and \(18=\left\lfloor\frac{(q^m-1)(q-1)}{q^{k-1}-1}\right\rfloor+(q-1)(k-2)=\mathop{\mathrm{H_{reg}}}\left(S_{X_q(\mathbb{F}_{q^m})\setminus\{P_0,P_\infty\}}\right)\), respectively.

Acknowledgments↩︎

The authors of this paper would like to thank Philippe Gimenez, Mario González-Sánchez, Christophe Levrat, Alessandro Neri, and Ferdinando Zullo for insightful discussions and suggestions.

References↩︎