Theory of supports for linear codes endowed with the sum-rank metric

Abstract

The sum-rank metric naturally extends both the Hamming and rank metrics in coding theory over fields. It measures the error-correcting capability of codes in multishot matrix-multiplicative channels (e.g. linear network coding or the discrete memoryless channel on fields). Although this metric has already shown to be of interest in several applications, not much is known about it. In this work, sum-rank supports for codewords and linear codes are introduced and studied, with emphasis on duality. The lattice structure of sum-rank supports is given; characterizations of the ambient spaces (support spaces) they define are obtained; the classical operations of restriction and shortening are extended to the sum-rank metric; and estimates (bounds and equalities) on the parameters of such restricted and shortened codes are found. Three main applications are given: (1) Generalized sum-rank weights are introduced, together with their basic properties and bounds; (2) It is shown that duals, shortened and restricted codes of maximum sum-rank distance (MSRD) codes are in turn MSRD; (3) Degenerateness and effective lengths of sum-rank codes are introduced and characterized. In an Appendix, skew supports are introduced, defined by skew polynomials, and their connection to sum-rank supports is given.

A The skew metric and skew supports

In this Appendix, we briefly revisit the relation between the sum-rank metric (Definition 1) and the skew metric introduced in [37, Def. 9]. We extend such a relation to sum-rank supports and skew supports (which we introduce in this Appendix), and the corresponding support spaces. The exposition in this appendix follows the lines in [37].

Let $\sigma : \mathbb {F} \longrightarrow \mathbb {F}$ be a field endomorphism and let $\delta : \mathbb {F} \longrightarrow \mathbb {F}$ be a $\sigma$-derivation, that is, $\delta$ is additive and $\delta (ab) = \sigma (a) \delta (b) + \delta (a) b$, for all $a,b \in \mathbb {F}$. Define the skew polynomial ring $\mathbb {F}[x; \sigma , \delta ]$ as the vector space over $\mathbb {F}$ with basis $\{ x^i \mid i \in \mathbb {N} \}$ and with product given by the rules $x^i x^j = x^{i + j}$, for $i,j \in \mathbb {N}$, and

$$\begin{aligned} xa = \sigma (a) x + \delta (a), \end{aligned}$$

(5)

for $a \in \mathbb {F}$. Define the degree of a non-zero skew polynomial $F = \sum _{i \in \mathbb {N}} F_i x^i \in \mathbb {F}[x; \sigma , \delta ]$, denoted by $\deg (F)$, as the maximum $i \in \mathbb {N}$ such that $F_i \ne 0$. We also define $\deg (0) = \infty$. Skew polynomial rings were introduced by Ore in [49] and the products given by (5) are the only products in $\mathbb {F}[x; \sigma , \delta ]$ such that $\deg (FG) = \deg (F) + \deg (G)$, for $F,G \in \mathbb {F}[x; \sigma , \delta ]$. The extension to several variables was recently given in [41]. Conventional polynomial rings are recovered by setting $\sigma = \mathrm{Id}$ and $\delta = 0$.

Since $\mathbb {F}[x ; \sigma , \delta ]$ is a right Euclidean domain, we may define the evaluation of $F \in \mathbb {F}[x; \sigma , \delta ]$ in $a \in \mathbb {F}$ as the unique $F(a) \in \mathbb {F}$ such that there exists $G \in \mathbb {F}[x; \sigma , \delta ]$ with

$$\begin{aligned} F = G \cdot (x-a) + F(a). \end{aligned}$$

This concept of evaluation was introduced in [29, 30].

Given a subset $\varOmega \subseteq \mathbb {F}$, we may define its associated ideal as $I(\varOmega ) = \{ F \in \mathbb {F}[x;\sigma , \delta ] \mid F(a) = 0, \forall a \in \varOmega \}$. Observe that $I(\varOmega )$ is a left ideal in $\mathbb {F}[x;\sigma , \delta ]$. Since $\mathbb {F}[x;\sigma , \delta ]$ is a right Euclidean domain, there exists a unique monic skew polynomial $F_\varOmega \in I(\varOmega )$ of minimal degree among those in $I(\varOmega )$, which in turn generates $I(\varOmega )$ as left ideal. Such a skew polynomial is called the minimal skew polynomial of $\varOmega$ [30].

Next, given a subset $\varOmega \subseteq \mathbb {F}$, we define its P-closure as $\overline{\varOmega } = Z(F_\varOmega ) \subseteq \mathbb {F}$ (the set of zeros of $F_\varOmega$), and we say that $\varOmega$ is P-closed if $\overline{\varOmega } = \varOmega$. A set $\varOmega \subseteq \mathbb {F}$ is called P-independent if $a \notin \overline{\varOmega \setminus \{ a \}}$, for all $a \in \varOmega$. We say that $\mathcal {B} \subseteq \varOmega$ is a P-basis of a P-closed set $\varOmega$ if $\mathcal {B}$ is P-independent and $\varOmega = \overline{\mathcal {B}}$. We also say that $\varOmega$ is a finitely generated P-closed set if it admits a finite P-basis, which is the case as long as $\varOmega \ne \mathbb {F}$, or $\varOmega = \mathbb {F}$ and $\mathbb {F}$ is finite.

Given a finitely generated P-closed set $\varOmega \subseteq \mathbb {F}$, any two of its P-bases are finite and have the same number of elements, which moreover coincides with $\deg (F_\varOmega )$. This motivates the definition rank of $\varOmega$ as

$$\begin{aligned} \mathrm{Rk}(\varOmega ) = \deg (F_\varOmega ) < \infty . \end{aligned}$$

Fix a finitely generated P-closed set $\varOmega \subseteq \mathbb {F}$ of rank n and fix one of its P-bases $\mathcal {B}$. An important tool to define skew metrics is skew polynomial Lagrange interpolation. Let $\mathbb {F}[x; \sigma , \delta ]_n$ be the n-dimensional vector space of skew polynomials of degree less than n. It follows from [29, Th. 8] that the evaluation map over the points in $\mathcal {B}$,

$$\begin{aligned} E_\mathcal {B} : \mathbb {F}[x; \sigma , \delta ]_n \longrightarrow \mathbb {F}^\mathcal {B}, \end{aligned}$$

is a vector space isomorphism. Hence we may define skew weights [37, Def. 9] as follows.

Definition 15

(Skew weights [37]) Given $F \in \mathbb {F}[x; \sigma , \delta ]_n$ and $f = E_\mathcal {B}(F) \in \mathbb {F}^\mathcal {B}$, we define their skew weight over $\varOmega$ as

$$\begin{aligned} \mathrm{wt}_\mathcal {B}(f) = \mathrm{wt}_\varOmega (F) = n - \mathrm{Rk}(Z_\varOmega (F)), \end{aligned}$$

where $Z_\varOmega (F) = Z(F) \cap \varOmega = Z(\{ F, F_\varOmega \})$ is the P-closed set of zeros of F in $\varOmega$.

Skew weights are indeed weights [37, Prop. 10] and define a metric in $\mathbb {F}^\mathcal {B}$, called the skew metric [37, Def. 11], by the usual formula: $\mathrm{d}_\mathcal {B}(f, g) = \mathrm{wt}_\mathcal {B}(f - g)$, for $f,g \in \mathbb {F}^\mathcal {B}$. To relate this metric with the sum-rank metric, we need the concept of conjugacy from [30]: We say that $a, c \in \mathbb {F}$ are conjugates if there exists $\beta \in \mathbb {F}^*$ such that

$$\begin{aligned} c = a^\beta {\mathop {=}\limits ^{def}} \sigma (\beta )\beta ^{-1} a + \delta (\beta )\beta ^{-1}. \end{aligned}$$

Putting together the results [29, Th. 23] and [30, Th. 4.5], we obtain the following characterization: A finite subset $\mathcal {B} \subseteq \mathbb {F}$ with n elements is a P-basis of $\varOmega = \overline{\mathcal {B}}$ if, and only if, $n = n_1 + n_2 + \cdots + n_\ell$, for some $\ell$, and there exists pair-wise non-conjugate elements $a^{(1)}, a^{(2)}, \ldots , a^{(\ell )} \in \mathbb {F}$ and a set of linearly independent elements $\{ \beta _1^{(i)}, \beta _2^{(i)}, \ldots , \beta _{n_i}^{(i)} \} \subseteq \mathbb {F}$, over the subfield $K_i = K_{a^{(i)}} = \{ \beta \in \mathbb {F}^* \mid \left( a^{(i)} \right) ^{\beta } = a^{(i)} \} \cup \{ 0 \} \subseteq \mathbb {F}$, for each $i = 1,2, \ldots , \ell$, such that

$$\begin{aligned} \mathcal {B} = \bigcup _{i = 1}^\ell \left\{ \left( a^{(i)} \right) ^{\beta _j^{(i)}} \mid j = 1,2, \ldots , n_i \right\} , \end{aligned}$$

(6)

where the union is disjoint. With this characterization at hand, we may give a vector space isomorphism connecting both metrics. The result follows from [37, Th. 2 & 3].

Theorem 6

[37] With notation as above, define the vector space isomorphism $\phi _\mathcal {B} : \mathbb {F}^n \longrightarrow \mathbb {F}^\mathcal {B}$ by $\phi _\mathcal {B} (\mathbf {c}^{(1)},$$\mathbf {c}^{(2)},$$\ldots ,$$\mathbf {c}^{(\ell )}) = f$, where $\mathbf {c}^{(i)} = (c_1^{(i)}, c_2^{(i)}, \ldots , c_{n_i}^{(i)}) \in \mathbb {F}^{n_i}$ and

$$\begin{aligned} f \left( \left( a^{(i)} \right) ^{\beta _j^{(i)}} \right) = c_j^{(i)} (\beta _j^{(i)})^{-1}, \end{aligned}$$

(7)

for $j = 1,2, \ldots , n_i$ and $i = 1,2,\ldots , \ell$. Then $\phi _\mathcal {B}$ is an isometry: For $\mathbf {c} \in \mathbb {F}^n$, it holds that

$$\begin{aligned} \mathrm{wt}_\mathcal {B}(\phi _\mathcal {B}(\mathbf {c})) = \mathrm{wt}_{SR}(\mathbf {c}), \end{aligned}$$

where $\mathrm{wt}_{SR}$ is the sum-rank weight from Definition 1 with $K_i = K_{a^{(i)}}$, for $i = 1,2, \ldots , \ell$.

The representation (6) and the map given by (7) establish a dictionary between the sum-rank metric and the skew metric. This dictionary, however, depends on the conjugacy representatives $a^{(1)}, a^{(2)}, \ldots , a^{(\ell )}$ and the P-basis $\mathcal {B}$ of $\varOmega$. The elements $\beta _1^{(i)}, \beta _2^{(i)},$$\ldots ,$$\beta _{n_i}^{(i)}$ are determined up to scalar factor in $K_i^*$ (thus uniquely as projective points in $\mathbb {P}_{K_i}(\mathbb {F})$) by the conjugacy representatives and $\mathcal {B}$, for $i = 1,2, \ldots , \ell$. It is important to notice that in the case $\sigma = \mathrm{Id}$ and $\delta = 0$, which corresponds to conventional polynomials and the Hamming metric, the dependency disappears since conjugacy classes only have one element and the only P-basis of $\varOmega$ is $\mathcal {B} = \varOmega$.

In particular, the concept of sum-rank support can be readily translated into the concept of skew support. First, define the lattice of skew supports in $\varOmega$ as

$$\begin{aligned} \mathcal {P}_{Sk}(\varOmega ) = \{ \varPsi \subseteq \varOmega \mid \varPsi \text { is P-closed} \}. \end{aligned}$$

Thus skew supports will simply be P-closed subsets of $\varOmega$, which form a lattice with intersections $\varPsi _1 \cap \varPsi _2$ and sums defined as $\varPsi _1 + \varPsi _2 = \overline{\varPsi _1 \cup \varPsi _2} = Z(F_{\varPsi _1 \cup \varPsi _2})$. The results [37, Prop. 43] and [37, Prop. 47] state that $\mathcal {P}_{Sk}(\varOmega )$ is a lattice isomorphic to $\mathcal {P}(\mathbf {K}^\mathbf {n})$, by mapping P-bases into lists of bases via (6), where $K_i = K_{a^{(i)}}$, for $i = 1,2, \ldots , \ell$. This mapping will now be used to define skew supports. As for vector and projective spaces, we implicitly associate the zero vector space with the empty P-closed set.

Definition 16

(Skew supports) With notation as above, let $f \in \mathbb {F}^\mathcal {B}$ and define $\mathbf {c} = (\mathbf {c}^{(1)}, \mathbf {c}^{(2)}, \ldots , \mathbf {c}^{(\ell )}) = \phi _\mathcal {B}^{-1}(f) \in \mathbb {F}^n$, where $\mathbf {c}^{(i)} \in \mathbb {F}^{n_i}$, for $i = 1,2, \ldots , \ell$. Next, let $\gamma _h^{(i)} = \sum _{j=1}^{n_i} c_{h,j}^{(i)} \beta _j^{(i)} \in \mathbb {F}$, where $(c_{h,1}^{(i)}, c_{h,2}^{(i)}, \ldots , c_{h,n_i}^{(i)}) \in K_i^{n_i}$ form the rows of $M_{\mathcal {A}_i}(\mathbf {c}^{(i)}) \in K_i^{m_i \times n_i}$, for $h = 1,2, \ldots , m_i$, and let $\mathcal {G}_i \subseteq \mathbb {F}^*$ be a basis of the vector space generated by $\gamma _1^{(i)}, \gamma _2^{(i)},$$\ldots ,$$\gamma _{m_i}^{(i)} \subseteq \mathbb {F}$ over $K_i$, for $i = 1,2, \ldots , \ell$. Define the P-independent set

$$\begin{aligned} \mathcal {B}_f = \bigcup _{i = 1}^\ell \left\{ \left( a^{(i)} \right) ^{\gamma } \mid \gamma \in \mathcal {G}_i \right\} . \end{aligned}$$

We define the skew support of $f \in \mathbb {F}^\mathcal {B}$ as

$$\begin{aligned} \mathrm{Supp}_{Sk}(f) = \varOmega _f = \overline{\mathcal {B}_f} \in \mathcal {P}_{Sk}(\varOmega ). \end{aligned}$$

Finally, for a vector subspace $\mathcal {F} \subseteq \mathbb {F}^\mathcal {B}$, we define its skew support as

$$\begin{aligned} \mathrm{Supp}_{Sk}(\mathcal {F}) = \sum _{f \in \mathcal {F}} \mathrm{Supp}_{Sk}(f) \in \mathcal {P}_{Sk}(\varOmega ), \end{aligned}$$

which allows to define the skew weight of $\mathcal {F}$ as $\mathrm{wt}_\mathcal {B}(\mathcal {F}) = \mathrm{Rk}(\mathrm{Supp}_{Sk}(\mathcal {F}))$.

As it was the case with the map in (7), the skew support $\mathrm{Supp}_{Sk}(f) \in \mathcal {P}_{Sk}(\varOmega )$ depends only on the conjugacy representatives and the choice of P-basis $\mathcal {B}$ of $\varOmega$. To see this, note that the vector space generated by the rows of $M_{\mathcal {A}_i}(\mathbf {c}^{(i)})$ does not depend on $\mathcal {A}_i$, and secondly, the P-basis corresponding to different bases of the subspace generated by $\gamma _1^{(i)}, \gamma _2^{(i)}, \ldots , \gamma _{m_i}^{(i)} \in \mathbb {F}$ over $K_i$ generate the same P-closed set $\varOmega _f$ by [37, Cor. 27].

Using the same arguments, we may prove the following properties:

Proposition 11

The following properties hold.

• (1) For $f \in \mathbb {F}^\mathcal {B}$ and $a \in \mathbb {F}^*$, it holds that $\mathrm{Supp}_{Sk}(af) = \mathrm{Supp}_{Sk}(\langle f \rangle ) = \mathrm{Supp}_{Sk}(f)$ and

  $$\begin{aligned} \mathrm{Rk}(\mathrm{Supp}_{Sk}(f)) = \mathrm{wt}_\mathcal {B}(f). \end{aligned}$$

• (2) $\phi _\mathcal {B}^{-1}(f)$ and $\phi _\mathcal {B}^{-1}(g)$ have the same sum-rank support if, and only if, f and g have the same skew support, for $f,g \in \mathbb {F}^\mathcal {B}$. The same holds for subspaces of $\mathbb {F}^\mathcal {B}$.

• (3) If $\mathcal {F} \subseteq \mathbb {F}^\mathcal {B}$ and $\mathcal {D} = \phi _\mathcal {B}^{-1}(\mathcal {F}) \subseteq \mathbb {F}^n$ are subspaces, then

  $$\begin{aligned} \mathrm{wt}_\mathcal {B}(\mathcal {F}) = \mathrm{Rk}(\mathrm{Supp}_{Sk}(\mathcal {F})) = \mathrm{Rk}(\mathrm{Supp}(\mathcal {D})) = \mathrm{wt}_{SR}(\mathcal {D}) . \end{aligned}$$

The concept of skew support space may also be considered. It may be introduced as a lattice of subspaces of $\mathbb {F}^\mathcal {B}$.

Definition 17

(Skew support spaces) Given a P-closed subset $\varPsi \subseteq \varOmega$ (i.e. $\varPsi \in \mathcal {P}_{Sk}(\varOmega )$), we define the skew support space associated to $\varPsi$ over $\mathcal {B}$ as

$$\begin{aligned} \mathcal {W}_\varPsi = \{ f \in \mathbb {F}^\mathcal {B} \mid \mathrm{Supp}_{Sk}(f) \subseteq \varPsi \} \subseteq \mathbb {F}^\mathcal {B} . \end{aligned}$$

We may add to Theorem 1 the following characterizations. They follow from the results in this Appendix, except for the arithmetic characterizations in Items 3 and 4. These follow by combining Item 6 in Theorem 1 and the recent result [38, Th. 2], which gives the connection between coordinate-wise matrix products as in Theorem 1 and products of skew polynomials given by (5).

Proposition 12

The following are equivalent:

• (1) $\mathcal {W}$ is a skew support space, that is, there exists $\varPsi \in \mathcal {P}_{Sk}(\varOmega )$ such that $\mathcal {W} = \mathcal {W}_\varPsi$.

• (2) $\mathcal {V} = \phi _\mathcal {B}^{-1}(\mathcal {W}) \subseteq \mathbb {F}^n$ is a sum-rank support space.

• (3) $\mathcal {W}$ is a left ideal of $\mathbb {F}^\mathcal {B}$ for the product in $\mathbb {F}^\mathcal {B}$ given by $f g \in \mathbb {F}^\mathcal {B}$, where

  $$\begin{aligned} (fg)(a) = (FG)(a), \end{aligned}$$

  (8)

  for $a \in \mathcal {B}$, $f,g \in \mathbb {F}^\mathcal {B}$ and $F,G \in \mathbb {F}[x; \sigma , \delta ]_n$ such that $f = E_\mathcal {B}(F)$ and $g = E_\mathcal {B}(G)$.

• (4) There exists a P-closed subset $\varPhi \subseteq \varOmega$ such that $\mathcal {W} = E_\mathcal {B}(I(\varPhi ))$.

In particular, by Item 2, skew support spaces are also vector subspaces of $\mathbb {F}^\mathcal {B}$. Notice also that, in general, $(fg)(a) \ne f(a) g(a)$ in Item 3 (see [30, Th. 2.7]).

In conclusion, in this Appendix we have introduced skew supports and support spaces, and we have given the precise connections with sum-rank supports and support spaces. Except for the Hamming-metric case, the dictionary between both types of concepts depends on the choice of conjugacy representatives and P-basis of the ambient P-closed set via (6). With this dictionary, all of the remaining results and definitions in this paper can be translated to skew supports and support spaces. We leave however as open problem defining skew supports and support spaces independently of a set of conjugacy representatives and a P-basis.