Hamming and simplex codes for the sum-rank metric

TL;DR

This paper introduces sum-rank Hamming and simplex codes, establishing their properties, bounds, and decoding algorithms, with applications in error correction and locally repairable codes, especially over small fields.

Contribution

It defines sum-rank Hamming and simplex codes, explores their properties, bounds, and isometry classes, and proposes applications in error correction and locally repairable codes.

Findings

Sum-rank Hamming codes are characterized as longest codes with minimum sum-rank distance 3.

Sum-rank Hamming codes are perfect codes when the field-extension degree is 1.

Efficient syndrome decoding algorithms are developed for sum-rank Hamming codes.

Abstract

Sum-rank Hamming codes are introduced in this work. They are essentially defined as the longest codes (thus of highest information rate) with minimum sum-rank distance at least $ 3 $ (thus one-error-correcting) for a fixed redundancy $ r $, base-field size $ q $ and field-extension degree $ m $ (i.e., number of matrix rows). General upper bounds on their code length, number of shots or sublengths and average sublength are obtained based on such parameters. When the field-extension degree is $ 1 $, it is shown that sum-rank isometry classes of sum-rank Hamming codes are in bijective correspondence with maximal-size partial spreads. In that case, it is also shown that sum-rank Hamming codes are perfect codes for the sum-rank metric. Also in that case, estimates on the parameters (lengths and number of shots) of sum-rank Hamming codes are given, together with an efficient syndrome decoding algorithm. Duals of sum-rank Hamming codes, called sum-rank simplex codes, are then introduced. Bounds on the minimum sum-rank distance of sum-rank simplex codes are given based on known bounds on the size of partial spreads. As applications, sum-rank Hamming codes are proposed for error correction in multishot matrix-multiplicative channels and to construct locally repairable codes over small fields, including binary.

Tables

Table 1. Table 1: Some parameters for the linear locally repairable codes described in Theorem 9 for q = 2 𝑞 2 q=2 , which may correct any 2 2 2 extra erasures on top of 1 1 1 erasure per local group.

Locality $N$	2	2	3	3	4	4	5	5
No. local groups $\ell$	4	21	9	73	17	273	33	1057
Global parities $r$	4	6	6	9	8	12	10	15
Dimension $k$	6	36	21	210	60	1080	155	5270
Length $M$	12	63	36	292	85	1365	198	6342

Equations

$$\operatorname{diag}(A_{1},A_{2},\ldots,A_{\ell})=\left(\begin{array}[]{cccc}A_{1}&0&\ldots&0\\ 0&A_{2}&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&A_{\ell}\\ \end{array}\right)\in\mathbb{F}_{q}^{n\times n}.$$

$$M_{\mathcal{A}}\left(\mathbf{c}\right)=\left(\begin{array}[]{cccc}c_{11}&c_{12}&\ldots&c_{1n}\\ c_{21}&c_{22}&\ldots&c_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ c_{m1}&c_{m2}&\ldots&c_{mn}\\ \end{array}\right),$$

$${\mathrm{wt}}_{SR}(\mathbf{c})=\sum _{i=1}^{\ell }\mathrm{Rk}(M_{\mathcal{A}}({\mathbf{c}}^{(i)})),$$

$$d_{SR}(\mathbf{c},\mathbf{d})={\mathrm{wt}}_{SR}(\mathbf{c}-\mathbf{d}),$$

$${\mathrm{d}}_{SR}(\mathcal{C})=\min \{{\mathrm{wt}}_{SR}(\mathbf{c})\mid \mathbf{c}\in \mathcal{C}\setminus \{0\}\}.$$

$$\begin{array}{rl}{\mathrm{d}}_{SR}(\mathcal{C})=\min \{{\mathrm{d}}_H(\mathcal{C}A)& \mid A=\mathrm{diag}(A_1,A_2,\dots ,A_{\ell }),\\ & A_i\in {\mathbb{F}}_q^{n_i\times n_i}\text{ invertible},1\le i\le \ell \}.\end{array}$$

$$\mathrm{d}(\varphi (\mathbf{c}),\varphi (\mathbf{d}))=\mathrm{d}(\mathbf{c},\mathbf{d}),$$

$$\varphi ({\mathbf{c}}^{(1)},{\mathbf{c}}^{(2)},\dots ,{\mathbf{c}}^{(\ell )})=({\beta }_1{\mathbf{c}}^{(\sigma (1))}{A}_1,{\beta }_2{\mathbf{c}}^{(\sigma (2))}{A}_2,\dots ,{\beta }_{\ell }{\mathbf{c}}^{(\sigma (\ell ))}{A}_{\ell }),$$

$$\varphi (\mathbf{c})=({\varphi }_1({\mathbf{c}}_1),{\varphi }_2({\mathbf{c}}_2),\dots ,{\varphi }_v({\mathbf{c}}_v)),$$

$${\psi }_i({\mathbf{c}}^{(i)})=\varphi (0,\dots ,0,{\mathbf{c}}^{(i)},0,\dots ,0),$$

$$1={\mathrm{wt}}_R({\mathbf{e}}_{j_1}^{(i)}+{\mathbf{e}}_{j_2}^{(i)})={\mathrm{wt}}_{SR}({\psi }_i({\mathbf{e}}_{j_1}^{(i)})+{\psi }_i({\mathbf{e}}_{j_2}^{(i)}))\ge 2,$$

$$n\le \min \left\{\frac{q^{mr}-1}{q^m-1},\lfloor \frac{\ell mr}{2}\rfloor \right\},$$

$$\frac{{\sum }_{i=1}^{\ell }{n}_i}{\ell }\le \frac{mr}{2}.$$

$$\ell \le \frac{q^{mr}-1}{N(q^n-1)}.$$

$$|\mathcal{C}|\le {\left(q^{n/\ell }\right)}^{\ell m-{\mathrm{d}}_{SR}(\mathcal{C})+1}.$$

$$3\le {\mathrm{d}}_{SR}(\mathcal{C})\le \frac{\ell mr}{n}+1,$$

$$2n\le rm.$$

$$H=(H_1,H_2,\dots ,H_{\ell })\in {\mathbb{F}}_q^{r\times n},$$

$$H=(H_1,H_2,\dots ,H_{\ell })\in {\mathbb{F}}_q^{r\times n}\text{and}L=(L_1,L_2,\dots ,L_{\ell })\in {\mathbb{F}}_q^{r\times n},$$

$$\{{\mathcal{H}}_1,{\mathcal{H}}_2,\dots ,{\mathcal{H}}_{\ell }\}=\{{\mathcal{L}}_1,{\mathcal{L}}_2,\dots ,{\mathcal{L}}_{\ell }\}.$$

$$\frac{q^r-q^s}{q^N-1}-q^s+1\le |I|\le \frac{q^r-q^s}{q^N-1}.$$

$$|I|=\frac{q^r-1}{q^N-1}.$$

$$N\left(\frac{q^r-q^s}{q^N-1}-q^s+1\right)\le n\le N\frac{q^r-q^s}{q^N-1},$$

$$|\mathcal{C}|\cdot |{\mathcal{B}}_1|=|{\mathbb{F}}_q^n|,$$

$$|{\mathcal{B}}_1|=1+\ell (q^N-1).$$

$$\ell =\frac{q^r-1}{q^N-1}.$$

$$|\mathcal{C}|\cdot |{\mathcal{B}}_1|=q^k(1+\ell (q^N-1))=q^k\left(1+\frac{q^r-1}{q^N-1}(q^N-1)\right)=q^k{q}^r=q^n=|{\mathbb{F}}_q^n|.$$

$$H=(H_1,H_2,\dots ,H_{\ell })\in {\mathbb{F}}_q^{r\times n},$$

$$H=(H_1,H_2,\dots ,H_{\ell })\in {\mathbb{F}}_q^{r\times n}\text{and}L=(L_1,L_2,\dots ,L_{\ell })\in {\mathbb{F}}_q^{r\times n},$$

$$\{{\mathcal{H}}_1,{\mathcal{H}}_2,\dots ,{\mathcal{H}}_{\ell }\}=\{{\mathcal{L}}_1,{\mathcal{L}}_2,\dots ,{\mathcal{L}}_{\ell }\}.$$

Full text

Keywords: Hamming codes, Hamming metric, locally repairable codes, multishot network coding, rank metric, simplex codes, sum-rank metric.

MSC: 94B05, 94B35, 94B65.

1 Introduction

The Hamming metric and Hamming codes were both introduced by Hamming in the seminal work [11]. In simple terms, Hamming codes can be understood as the longest linear codes that can correct a single symbol-wise error (thus having minimum Hamming distance at least $3$) for a fixed redundancy (i.e., number of redundant symbols). Their binary version has been extensively applied in practice (usually interleaved or concatenated) for channels with rare bit-wise errors, since their decoding algorithms are very efficient and easy to implement even for large code lengths.

Since then, other metrics have attracted the attention of coding theorists. Among them, the rank metric [5, 8] is a remarkable example due to its applications mainly in universal error-correction in linearly coded networks [14] and to construct locally repairable codes [22]. Traditionally, linear codes endowed with the rank metric are also called rank-metric codes.

However, in most applications, the length of a rank-metric code does not play the same role as if the Hamming metric were considered instead. The Hamming metric measures the number of symbol-wise errors in discrete memoryless channels (see [13, Sec. 1.11.2]). Here, the code length represents the number of shots, corresponding to symbols, that the code covers in such a channel (a block of symbols, thus also the name block code for such codes). When one considers (in a discrete memoryless way) multiple uses or shots of the channels for which the rank metric measures the number of errors, one naturally obtains the so-called sum-rank metric. This metric was formally defined in [21] and implicitly considered earlier for space-time coding in [16]. It is the appropriate metric to measure the number of errors and erasures in multishot network coding universally [19], obtain maximally recoverable locally repairable codes with arbitrary local codes [20] and codes with optimal rate-diversity in space-time coding with multiple fading blocks [16, Sec. III].

In this work, we introduce and briefly study Hamming codes tailored for the sum-rank metric, together with their duals, which we will call sum-rank simplex codes. We will define proper sum-rank Hamming codes (Definition 3), which are defined as linear codes capable of correcting a single sum-rank error for the largest number of shots (terms in the sum-rank sum) for a fixed base field, redundancy and matrix sizes. We will also define improper sum-rank Hamming codes (Definition 4) where the numbers of matrix columns are also allowed to grow. We will briefly comment on the longest rank-metric codes that may correct a single rank error, for fixed base field, redundancy and field-extension degree (or number of matrix rows). We will then turn to the case of field-extension degree $1$ or $1$ row per matrix. In this case, we will show that sum-rank isometry classes of sum-rank Hamming codes correspond bijectively with partial spreads for corresponding parameters. We will estimate the lengths, sublengths and number of shots for these sum-rank Hamming codes based on known bounds on the size of partial spreads. We will show that sum-rank Hamming codes are perfect codes for the sum-rank metric in this case, in contrast with the rank metric, for which there exist no non-trivial perfect codes. We will provide an efficient syndrome decoding algorithm for such codes, and we will lower bound the minimum sum-rank distance of their duals, called sum-rank simplex codes.

The codes introduced in this work have applications in single error correction in multishot matrix-multiplicative channels (Subsection 5.1), which model for instance multishot network coding [19]. They also provide locally repairable codes over small fields, which can correct $2$ extra erasures on top of the admissible local erasures and are backwards-compatible with arbitrary local linear codes, see Subsection 5.2.

The remainder of this work is organized as follows. In Section 2, we define the sum-rank metric and characterize their linear isometries. In Section 3, we give the main definitions and properties of sum-rank Hamming codes. In Section 4, we introduce and study their duals, that is, sum-rank simplex codes. Finally, in Section 5, we provide the applications mentioned in the previous paragraph.

2 The sum-rank metric and its linear isometries

We start by fixing the notation that we will use throughout this paper. Fix a prime power $q$ and positive integers $m$, $\ell$ and $n=n_{1}+n_{2}+\cdots+n_{\ell}$. We will denote by $\mathbb{F}_{q}$ the finite field with $q$ elements. We will also denote by $\mathbb{F}_{q}^{m\times n}$ the set of $m\times n$ matrices with entries in $\mathbb{F}_{q}$, and we denote $\mathbb{F}_{q}^{n}=\mathbb{F}_{q}^{1\times n}$.

For matrices $A_{i}\in\mathbb{F}_{q}^{n_{i}\times n_{i}}$, for $i=1,2,\ldots,\ell$, we define the block-diagonal matrix

[TABLE]

Fix an ordered basis $\mathcal{A}=\{\alpha_{1},\alpha_{2},\ldots,\alpha_{m}\}$ of $\mathbb{F}_{q^{m}}$ over $\mathbb{F}_{q}$. We define $M_{\mathcal{A}}:\mathbb{F}_{q^{m}}^{n}\longrightarrow\mathbb{F}_{q}^{m\times n}$ as the matrix representation map, given by

[TABLE]

for $\mathbf{c}=(c_{1},c_{2},\ldots,c_{n})\in\mathbb{F}_{q^{m}}^{n}$, where $c_{1,j},c_{2,j},\ldots,c_{m,j}\in\mathbb{F}_{q}$ are the unique scalars such that $c_{j}=\sum_{i=1}^{m}\alpha_{i}c_{i,j}\in\mathbb{F}_{q^{m}}$, for $j=1,2,\ldots,n$.

Given $X\in\mathbb{F}_{q}^{m\times n}$, we denote by $\operatorname{Row}(X)\subseteq\mathbb{F}_{q}^{n}$ and $\operatorname{Col}(X)\subseteq\mathbb{F}_{q}^{m}$ the vector spaces generated by the rows and the columns of $X$, respectively. For $\mathbf{c}\in\mathbb{F}_{q^{m}}^{n}$, we denote $\operatorname{Row}(\mathbf{c})=\operatorname{Row}(M_{\mathcal{A}}(\mathbf{c}))\subseteq\mathbb{F}_{q}^{n}$ and $\operatorname{Col}(\mathbf{c})=\operatorname{Col}(M_{\mathcal{A}}(\mathbf{c}))\subseteq\mathbb{F}_{q}^{m}$. The latter depends on $\mathcal{A}$, but we omit this for simplicity.

We now define the sum-rank metric in $\mathbb{F}_{q^{m}}^{n}$, which was introduced in [21]. It was implicitly considered earlier in the space-time coding literature (see [16, Sec. III]). The definition that we will consider was first given in [17, Def. 25] and is more general, as the definition in [21] considers equals sublengths $n_{1}=n_{2}=\ldots=n_{\ell}$. The case of unequal sublengths is of interest, for instance, for locally repairable codes (see [20]).

Definition 1** (Sum-rank metric [21]).**

Let $\mathbf{c}=(\mathbf{c}^{(1)},$ $\mathbf{c}^{(2)},$ $\ldots,$ $\mathbf{c}^{(\ell)})\in\mathbb{F}_{q^{m}}^{n}$, where $\mathbf{c}^{(i)}\in\mathbb{F}_{q^{m}}^{n_{i}}$, for $i=1,2,\ldots,\ell$. We define the sum-rank weight of $\mathbf{c}$ as

[TABLE]

where $M_{\mathcal{A}}$ is as in (1). Finally, we define the sum-rank metric $\operatorname{d}_{SR}:(\mathbb{F}_{q^{m}}^{n})^{2}\longrightarrow\mathbb{N}$ as

[TABLE]

for all $\mathbf{c},\mathbf{d}\in\mathbb{F}_{q^{m}}^{n}$. In this work, a linear code is an $\mathbb{F}_{q^{m}}$-linear vector subspace $\mathcal{C}\subseteq\mathbb{F}_{q^{m}}^{n}$. We define its minimum sum-rank distance as

[TABLE]

Observe that indeed the sum-rank metric is a metric. Observe also that sum-rank weights and metrics depend on the sum-rank length partition $n=n_{1}+n_{2}+\cdots+n_{\ell}$ and the subfield $\mathbb{F}_{q}\subseteq\mathbb{F}_{q^{m}}$. However, we do not write this dependency for brevity. Note, on the other hand, that they do not depend on the choice of ordered basis $\mathcal{A}$.

Finally, we observe that the Hamming metric [11] and the rank metric [5, 8] are recovered from the sum-rank metric by setting $n_{1}=n_{2}=\ldots=n_{\ell}=1$ and $\ell=1$, respectively. We will also use the notation ${\rm wt}_{R}={\rm wt}_{SR}$ and ${\rm d}_{R}={\rm d}_{SR}$ if $\ell=1$, and ${\rm wt}_{H}={\rm wt}_{SR}$ and ${\rm d}_{H}={\rm d}_{SR}$ if $n_{1}=n_{2}=\ldots=n_{\ell}=1$.

The next theorem was proven in [20, Th. 1]. It will be a central tool in the study of sum-rank Hamming codes.

Theorem 1** ([20]).**

Given a linear code $\mathcal{C}\subseteq\mathbb{F}_{q^{m}}^{n}$, it holds that

[TABLE]

In particular, we see that multiplying on the right by invertible block-diagonal matrices over $\mathbb{F}_{q}$ constitutes a linear sum-rank isometry, for the corresponding sum-rank length partition. In [18, Subsec. 3.3], we anticipated a characterization of all linear sum-rank isometries. We now write such a characterization formally.

In this work, a map $\phi:\mathbb{F}_{q^{m}}^{n}\longrightarrow\mathbb{F}_{q^{m}}^{n}$ is called linear if it is $\mathbb{F}_{q^{m}}$-linear. As usual, we say that it is an isometry for a metric ${\rm d}$ if

[TABLE]

for all $\mathbf{c},\mathbf{d}\in\mathbb{F}_{q^{m}}^{n}$. Observe that a linear isometry is always injective (hence a vector space isomorphism if the domain and codomain have the same dimension) since ${\rm d}(\mathbf{c},\mathbf{d})>0$ if, and only if, $\mathbf{c}\neq\mathbf{d}$, therefore $\phi(\mathbf{c})\neq\phi(\mathbf{d})$ if, and only if, $\mathbf{c}\neq\mathbf{d}$.

We may now prove the result mentioned above.

Theorem 2**.**

Assume first that $N=n_{1}=n_{2}=\ldots=n_{\ell}$ and $n=\ell N$. A map $\phi:\mathbb{F}_{q^{m}}^{n}\longrightarrow\mathbb{F}_{q^{m}}^{n}$ is a linear sum-rank isometry if, and only if, there exist elements $\beta_{1},\beta_{2},\ldots,\beta_{\ell}\in\mathbb{F}_{q^{m}}^{*}$, invertible matrices $A_{1},A_{2},\ldots,A_{\ell}\in\mathbb{F}_{q}^{N\times N}$ and a permutation $\sigma:[\ell]\longrightarrow[\ell]$ such that

[TABLE]

for all $\mathbf{c}=(\mathbf{c}^{(1)},\mathbf{c}^{(2)},\ldots,\mathbf{c}^{(\ell)})\in\mathbb{F}_{q^{m}}^{n}$, where $\mathbf{c}^{(i)}\in\mathbb{F}_{q^{m}}^{N}$, for $i=1,2,\ldots,\ell$.

More generally, assume that the sum-rank partition of $n$ into sublengths is $n=\ell_{1}N_{1}+\ell_{2}N_{2}+\cdots+\ell_{v}N_{v}$, where $0<N_{1}<N_{2}<\ldots<N_{v}$ and $\ell=\ell_{1}+\ell_{2}+\cdots+\ell_{v}$. That is, the first $\ell_{1}$ sublengths are equal, the next $\ell_{2}$ sublengths are equal and strictly larger than the previous ones, etcetera. In this case, a map $\phi:\mathbb{F}_{q^{m}}^{n}\longrightarrow\mathbb{F}_{q^{m}}^{n}$ is a linear sum-rank isometry if, and only if, there exist linear sum-rank isometries $\phi_{j}:\mathbb{F}_{q^{m}}^{\ell_{j}N_{j}}\longrightarrow\mathbb{F}_{q^{m}}^{\ell_{j}N_{j}}$ that are given as in (2), for $j=1,2,\ldots,v$, such that

[TABLE]

for all $\mathbf{c}=(\mathbf{c}_{1},\mathbf{c}_{2},\ldots,\mathbf{c}_{v})$, where $\mathbf{c}_{j}\in\mathbb{F}_{q^{m}}^{\ell_{j}N_{j}}$, for $j=1,2,\ldots,v$.

Proof.

In both the particular and general cases, the reversed implication is trivial, so we only prove the direct one.

We first prove the particular case where $N=n_{1}=n_{2}=\ldots=n_{\ell}$. Reindex the canonical basis of $\mathbb{F}_{q^{m}}^{n}$ as $\mathbf{e}^{(i)}_{j}=\mathbf{e}_{(i-1)N+j}$, for $i=1,2,\ldots,\ell$ and $j=1,2,\ldots N$, and fix one such index $i$. For simplicity, we identify $\mathbf{e}^{(i)}_{j}\in\mathbb{F}_{q^{m}}^{n}$ with the $j$th vector of the canonical basis in $\mathbb{F}_{q^{m}}^{N}$, keeping in mind that we have fixed the index $i$. Define now the linear map $\psi_{i}:\mathbb{F}_{q^{m}}^{N}\longrightarrow\mathbb{F}_{q^{m}}^{n}$ such that

[TABLE]

where $\mathbf{c}^{(i)}\in\mathbb{F}_{q^{m}}^{N}$ is placed in the $i$th block of $N$ coordinates in $\mathbb{F}_{q^{m}}^{n}$. Obviously, we have that ${\rm wt}_{SR}(\psi_{i}(\mathbf{c}^{(i)}))={\rm wt}_{R}(\mathbf{c}^{(i)})$.

First, we see that $\psi_{i}(\mathbf{e}_{j}^{(i)})^{(k)}=\mathbf{0}$ for all $k\in[\ell]$ except for exactly one, which may depend on $j$, for $j=1,2,\ldots,N$. Indeed if $\psi_{i}(\mathbf{e}_{j}^{(i)})^{(k)}\neq\mathbf{0}$ for two $k$, then ${\rm wt}_{SR}(\psi_{i}(\mathbf{e}_{j}^{(i)}))\geq 2$, which contradicts that ${\rm wt}_{R}(\mathbf{e}_{j}^{(i)})=1$.

Similarly we see that if $\psi_{i}(\mathbf{e}_{j_{1}}^{(i)})^{(i_{1})}\neq\mathbf{0}$, $\psi_{i}(\mathbf{e}_{j_{2}}^{(i)})^{(i_{2})}\neq\mathbf{0}$, and $j_{1}\neq j_{2}$, then $i_{1}=i_{2}$. If this does not hold, then by the previous paragraph, we have that

[TABLE]

which is absurd. Therefore for all $i\in[\ell]$, there exists $\sigma(i)\in[\ell]$ such that $\psi_{i}(\mathbf{e}_{j}^{(i)})^{(\sigma(i))}\neq\mathbf{0}$ and $\psi_{i}(\mathbf{e}_{j}^{(i)})^{(k)}=\mathbf{0}$ for $k\neq\sigma(i)$. In other words, $\pi_{i}\circ\psi_{i}:\mathbb{F}_{q^{m}}^{N}\longrightarrow\mathbb{F}_{q^{m}}^{N}$ is a linear rank isometry, where $\pi_{i}:\mathbb{F}_{q^{m}}^{n}\longrightarrow\mathbb{F}_{q^{m}}^{N}$ is the projection onto the $i$th block of $N$ coordinates. By [2, Th. 1], there exists an element $\beta_{i}\in\mathbb{F}_{q^{m}}^{*}$ and an invertible matrix $A_{i}\in\mathbb{F}_{q}^{N\times N}$ such that $\pi_{i}(\psi_{i}(\mathbf{c}^{(i)}))=\beta_{i}\mathbf{c}^{(i)}A_{i}$, for all $\mathbf{c}^{(i)}\in\mathbb{F}_{q^{m}}^{N}$.

Finally, $\sigma:[\ell]\longrightarrow[\ell]$ must be a bijection, i.e., a permutation, because otherwise $\phi$ would not be a vector space isomorphism. Thus the result follows.

To extend this result to the general case (where the sublengths $n_{i}$ may not be equal), we only need to prove that a block of $n_{i}$ coordinates cannot be mapped to a block of $n_{k}$ coordinates if $n_{i}\neq n_{k}$. The proof of this claim is similar to the rest of this proof and is left to the reader. ∎

Finally, since MacWilliams’ extension theorem does not hold in the case $\ell=1$ (see [1, Ex. 2.9(c)]), we choose to define sum-rank isometric codes (of the same length) as follows.

Definition 2**.**

We say that two linear codes $\mathcal{C},\mathcal{D}\subseteq\mathbb{F}_{q^{m}}^{n}$ are sum-rank isometric if there exists a linear sum-rank isometry $\phi:\mathbb{F}_{q^{m}}^{n}\longrightarrow\mathbb{F}_{q^{m}}^{n}$ such that $\mathcal{D}=\phi(\mathcal{C})$.

3 Hamming codes for the sum-rank metric

In this section, we introduce Hamming codes for the sum-rank metric, which we will call sum-rank Hamming codes for brevity.

3.1 Definition and basic properties

We start by giving the main definitions and basic properties. We will distinguish between codes with equal sublengths, which we will call proper, and those with unequal sublengths, which we will call improper.

Definition 3** (Proper sum-rank Hamming codes).**

Fix the base field size $q$, the extension degree $m$ and a redundancy $r$. Let $n=\ell N$ be a sum-rank length partition with equal sublengths $N=n_{1}=n_{2}=\ldots=n_{\ell}$. We say that a linear code $\mathcal{C}\subseteq\mathbb{F}_{q^{m}}^{n}$ is a proper sum-rank Hamming code if ${\rm d}_{SR}(\mathcal{C})\geq 3$, $r=n-\dim(\mathcal{C})$ and there is no other linear code $\mathcal{C}^{\prime}\subseteq\mathbb{F}_{q^{m}}^{n^{\prime}}$ with ${\rm d}_{SR}(\mathcal{C}^{\prime})\geq 3$ and $r=n^{\prime}-\dim(\mathcal{C}^{\prime})$ for a sum-rank length partition $n^{\prime}=N\ell^{\prime}$ where $\ell^{\prime}>\ell$.

Definition 4** (Improper sum-rank Hamming codes).**

Fix the base field size $q$, the extension degree $m$ and a redundancy $r$. Let $n=n_{1}+n_{2}+\cdots+n_{\ell}$ be a sum-rank length partition. We say that a linear code $\mathcal{C}\subseteq\mathbb{F}_{q^{m}}^{n}$ is an improper sum-rank Hamming code if ${\rm d}_{SR}(\mathcal{C})\geq 3$, $r=n-\dim(\mathcal{C})$ and there is no other linear code $\mathcal{C}^{\prime}\subseteq\mathbb{F}_{q^{m}}^{n^{\prime}}$ with ${\rm d}_{SR}(\mathcal{C}^{\prime})\geq 3$ and $r=n^{\prime}-\dim(\mathcal{C}^{\prime})$ for a sum-rank length partition $n^{\prime}=n_{1}^{\prime}+n_{2}^{\prime}+\cdots+n^{\prime}_{\ell^{\prime}}$ where $n^{\prime}>n$, $\ell^{\prime}\geq\ell$ and $n^{\prime}_{i}\geq n_{i}$, for $i=1,2,\ldots,\ell$.

Remark 5**.**

Classical Hamming codes [11] (see also [13, Sec. 1.8]) are therefore proper sum-rank Hamming codes where $m=N=1$. To the best of our knowledge, the case $\ell=1$, or Hamming codes for the rank metric, has not yet been considered. We will briefly treat it in Subsection 3.2.

In other words, sum-rank Hamming codes are, by definition, the longest linear codes with minimum sum-rank distance at least $3$ and redundancy $r$, for a given base field and field-extension degree. The whole point of this definition is to obtain the codes with highest information rate for codes that can correct one sum-rank error, for a given base field and field-extension degree. The base field and field-extension degrees are fixed, since in general better codes (longer with equal or larger information rates) can be obtained over larger fields, establishing a natural trade-off, since larger fields imply higher computational complexity and less fine symbol partitions (meaning that errors on smaller symbols, such as bit errors, are treated as errors on larger symbols, such as bytes).

In practice, one usually wishes to fix a code length and find the single error-correcting code with the largest information rate (i.e., largest dimension). However, as in the classical case, we will see that it is easier to fix the redundancy and try to find the longest single error-correcting code for that redundancy.

Due to Theorem 1, it will be useful to use a basic characterization of codes with minimum Hamming distance at least $3$. The following lemma is [13, Cor. 1.4.14].

Lemma 6**.**

Let $\mathcal{C}\subseteq\mathbb{F}_{q^{m}}^{n}$ be a linear code with parity-check matrix $H\in\mathbb{F}_{q^{m}}^{r\times n}$. Then ${\rm d}_{H}(\mathcal{C})\geq 3$ if, and only if, any two columns of $H$ are linearly independent over $\mathbb{F}_{q^{m}}$.

Equipped with Theorem 1 and Lemma 6, we may show that Definitions 3 and 4 are consistent, meaning that both the length $n$ and number of sublengths $\ell$ of a sum-rank Hamming code (proper or improper) cannot be infinite. However, the next proposition does not imply that sum-rank Hamming codes exist for all the considered parameters. That is, if a code with minimum sum-rank distance at least $3$ exists, for given base and extension fields, and given redundancy, then one such code with maximum possible length exists, which would then be a sum-rank Hamming code. We will actually give two bounds, using also [20, Cor. 3], and give an upper bound on the average sublength. However, these bounds are not sharp in general, as shown later by Corollary 13.

Proposition 7**.**

Let $\mathcal{C}\subseteq\mathbb{F}_{q^{m}}^{n}$ be a linear code such that ${\rm d}_{SR}(\mathcal{C})\geq 3$ and $n=n_{1}+n_{2}+\cdots+n_{\ell}$. Then it holds that

[TABLE]

where $r=n-\dim(\mathcal{C})$. In particular, we have the following upper bound on the average sublength:

[TABLE]

In particular, for proper sum-rank Hamming codes with equal sublengths $N=n_{1}=n_{2}=\ldots=n_{\ell}$, we deduce that $2N\leq mr$ and

[TABLE]

Proof.

From Theorem 1, we deduce that ${\rm d}_{H}(\mathcal{C})\geq 3$. Thus $\mathcal{C}$ satisfies the conditions in Lemma 6. Hence its code length $n$ is not larger than the size of the projective space $\mathbb{P}_{\mathbb{F}_{q^{m}}}(\mathbb{F}_{q^{m}}^{r})$, which is the first term in the minimum in (3).

On the other hand, the result [20, Cor. 3] says that

[TABLE]

Using that $|\mathcal{C}|=q^{m(n-r)}$, solving for ${\rm d}_{SR}(\mathcal{C})$ and using that ${\rm d}_{SR}(\mathcal{C})\geq 3$, we deduce that

[TABLE]

from which the second term in the minimum in (3) follows, and we are done. ∎

Remark 8**.**

Note that, for classical Hamming codes, we have that $N=n_{1}=n_{2}=\ldots=n_{\ell}=1$ and $m=1$, thus $n=\ell$ and (3) simply says that $r\geq 2$ (and actually any $r\geq 2$ can be chosen), which is obvious from the Singleton bound, and $n\leq(q^{r}-1)/(q-1)$, which is the actual length of classical Hamming codes (see [13, Sec. 1.8]).

We next give a basic characterization of having minimum sum-rank distance at least $3$. The proof is immediate from Theorem 1 and Lemma 6 after unfolding the definitions. We will make use of this result mainly for the case $m=1$ in Subsection 3.3.

Proposition 9**.**

Let $n=n_{1}+n_{2}+\cdots+n_{\ell}$ be a sum-rank length partition. Let $\mathcal{C}\subseteq\mathbb{F}_{q^{m}}^{n}$ be a linear code with a parity check matrix $H=(H_{1},H_{2},\ldots,H_{\ell})\in\mathbb{F}_{q^{m}}^{r\times n}$, where $H_{i}\in\mathbb{F}_{q^{m}}^{r\times n_{i}}$, for $i=1,2,\ldots,\ell$. Then it holds that ${\rm d}_{SR}(\mathcal{C})\geq 3$ if, and only if, the following two conditions hold:

$H_{i}\mathbf{a}_{i}^{T}\in\mathbb{F}_{q^{m}}^{r}$* and $H_{j}\mathbf{a}_{j}^{T}\in\mathbb{F}_{q^{m}}^{r}$ are linearly independent over $\mathbb{F}_{q^{m}}$, for all non-zero $\mathbf{a}_{i}\in\mathbb{F}_{q}^{n_{i}}$ and $\mathbf{a}_{j}\in\mathbb{F}_{q}^{n_{j}}$, and for all $1\leq i<j\leq\ell$, and* 2. 2.

$H_{i}\mathbf{a}^{T}\in\mathbb{F}_{q^{m}}^{r}$* and $H_{i}\mathbf{b}^{T}\in\mathbb{F}_{q^{m}}^{r}$ are linearly independent over $\mathbb{F}_{q^{m}}$, for all $\mathbf{a}\in\mathbb{F}_{q}^{n_{i}}$ and $\mathbf{b}\in\mathbb{F}_{q}^{n_{i}}$ that are linearly independent over $\mathbb{F}_{q}$, for all $i=1,2,\ldots,\ell$.*

3.2 On the case $\ell=1$: Rank-metric Hamming codes

In this subsection, we briefly discuss the rank-metric case, that is, $\ell=1$. Observe that, in principle, there may not exist sum-rank Hamming codes with $\ell=1$, whether proper or improper. This is because, both in Definitions 3 and 4, we do not fix $\ell$ and we allow it to grow. In other words, it may be the case that, for a given redundancy, base field and extension field, there always exists a code with minimum sum-rank distance at least $3$ and $\ell>1$. Thus, for such redundancy, base and extension fields, no rank-metric Hamming code would exist. We believe that looking for the largest number of shots $\ell$ is the right way of considering Hamming codes, as $\ell$ is the dominant part of the code length and has analogous interpretations to the classical code length in the applications. See the Introduction or Section 5 for details.

However, we briefly discuss what happens if we decide to fix $\ell=1$ and find the longest linear rank-metric code $\mathcal{C}\subseteq\mathbb{F}_{q^{m}}^{n}$ with ${\rm d}_{R}(\mathcal{C})\geq 3$ for a fixed redundancy $r=n-\dim(\mathcal{C})$, base field $q$ and field-extension degree $m$. Whether such codes are sum-rank Hamming codes according to Definitions 3 and 4 is left open.

First, it is not difficult to see that the minimum in (3) is attained by the second term. Hence we have the bound

[TABLE]

The bound (6) can be attained for many choices of parameters, although we leave it as open problem to see if it can always be achieved as long as $2$ divides $rm$.

The following constructions from the literature based on Gabidulin codes [8] achieve the bound (6). Assume that $2n=rm$. First note that the case $r=1$ may not happen, because then it would hold that $2n=m$ by (6) and $3\leq{\rm d}_{R}(\mathcal{C})\leq r+1=2$ by the Singleton bound. Hence $r\geq 2$ and therefore, $n\geq m$.

If $r=2$, then $n=m$. In this case, the Gabidulin code [8] of length $n$ and dimension $n-2$ has minimum rank distance exactly ${\rm d}_{R}(\mathcal{C})=n-\dim(\mathcal{C})+1=r+1=3$. 2. 2.

In general for $r$ even, let $r=2h$ for a positive integer $h$. Then a Cartesian product of $h$ Gabidulin codes, each of length $m$ and dimension $m-2$, gives a linear code $\mathcal{C}\subseteq\mathbb{F}_{q^{m}}^{n}$ of length $n=hm$, dimension $k=h(m-2)=n-r$ and minimum rank distance ${\rm d}_{R}(\mathcal{C})=3$. It is easy to check that $2n=mr$, hence (6) is attained.

Decoding such codes can be done using decoding algorithms for Gabidulin codes. Unfortunately, as is well-known, decoding one Gabidulin code of length $n$ is not efficient in practice, as the known decoding algorithms require in general a super-linear (in $n$) number of operations over $\mathbb{F}_{q^{m}}$, whose size is exponential in the code length $n$. The sum-rank Hamming codes with $m=1$ from Subsection 3.3, in contrast, admit very efficient decoding algorithms (see Subsection 3.4).

Finally, we observe that there exists a generalization of Gabidulin codes for the sum-rank metric in general. Such codes are called linearized Reed-Solomon codes and were introduced in general in [17]. Such codes recover classical Reed-Solomon codes when $n_{1}=n_{2}=\ldots=n_{\ell}=1$, that is, when the sum-rank metric becomes the Hamming metric. For this reason, and since Reed-Solomon codes and Hamming codes are not equal (for the Hamming metric), we do not expect linearized Reed-Solomon codes to be sum-rank Hamming codes in general.

3.3 The case $m=1$: Maximal-size partial spreads

In this subsection, we study the case $m=1$. Although this case may seem mathematically simpler, it is arguably the most interesting case from a practical point of view, as the size of the field $q^{m}$ becomes simply $q$. As we shall see, sum-rank isometry classes of sum-rank Hamming codes are in correspondence with maximal-size partial spreads [3].

We observe that, when $m=1$, the sum-rank metric becomes the Hamming metric over the alphabets $\mathbb{F}_{q}^{n_{1}},\mathbb{F}_{q}^{n_{2}},\ldots,\mathbb{F}_{q}^{n_{\ell}}$. If not all sublengths $n_{1},n_{2},\ldots,n_{\ell}$ are equal, the corresponding codes are called mixed codes or polyalphabetic codes in the literature [7, 12, 23]. Constructions of Hamming codes for a single alphabet of the form $\mathbb{F}_{q}^{N}$ (that is, the case $N=n_{1}=n_{2}=\ldots=n_{\ell}$), with $N>1$, were also given in [6]. However, the results in this subsection were not obtained in previous works, e.g., [6, 7, 12, 23], to the best of our knowledge.

We start by revisiting the definition of partial spreads.

Definition 10** (Partial spreads).**

Let $1\leq N\leq r$. A partial $N$-spread in $\mathbb{F}_{q}^{r}$ is a family $\mathcal{P}=\{\mathcal{H}_{i}\}_{i\in I}$ of $N$-dimensional vector subspaces $\mathcal{H}_{i}\subseteq\mathbb{F}_{q}^{r}$, for $i\in I$, such that $\mathcal{H}_{i}\cap\mathcal{H}_{j}=\{\mathbf{0}\}$, whenever $i\neq j$. A maximal-size partial $N$-spread is a partial $N$-spread $\mathcal{P}=\{\mathcal{H}_{i}\}_{i\in I}$ such that $I$ has maximal size among partial $N$-spreads in $\mathbb{F}_{q}^{r}$.

We next characterize proper sum-rank Hamming codes when $m=1$ in terms of partial spreads.

Theorem 3**.**

Let $n=\ell N$ be a sum-rank length partition with equal sublengths $N=n_{1}=n_{2}=\ldots=n_{\ell}$. A linear code $\mathcal{C}\subseteq\mathbb{F}_{q}^{n}$ is a proper sum-rank Hamming code if, and only if, $N\leq r=n-\dim(\mathcal{C})$ and it has a parity-check matrix of the form

[TABLE]

where $H_{i}\in\mathbb{F}_{q}^{r\times N}$ and $\mathcal{H}_{i}={\rm Col}(H_{i})\subseteq\mathbb{F}_{q}^{r}$, for $i=1,2,\ldots,\ell$, and $\mathcal{P}=\{\mathcal{H}_{1},\mathcal{H}_{2},\ldots,\mathcal{H}_{\ell}\}$ is a maximal-size partial $N$-spread in $\mathbb{F}_{q}^{r}$.

Proof.

First, Item 2 in Proposition 9 shows that all $\mathcal{H}_{i}\subseteq\mathbb{F}_{q}^{r}$ must have dimension $N$ and, in particular, $N\leq r$. Second, Item 1 in Proposition 9 shows that it must hold that $\mathcal{H}_{i}\cap\mathcal{H}_{j}=\{\mathbf{0}\}$, whenever $i\neq j$. Finally, $\ell$ is maximum as in Definition 3 if, and only if, $\mathcal{P}$ is a maximal-size partial $N$-spread, since $\ell=|\mathcal{P}|$. ∎

Now we show that sum-rank isometry classes of sum-rank Hamming codes are in bijective correspondence with maximal-size partial spreads.

Theorem 4**.**

Let $n=\ell N$ be a sum-rank length partition with equal sublengths $N=n_{1}=n_{2}=\ldots=n_{\ell}$. Let $\mathcal{C},\mathcal{D}\subseteq\mathbb{F}_{q}^{n}$ be two proper sum-rank Hamming codes with $r=n-\dim(\mathcal{C})=n-\dim(\mathcal{D})$ and with parity-check matrices given by

[TABLE]

respectively. Denote $\mathcal{H}_{i}={\rm Col}(H_{i})\subseteq\mathbb{F}_{q}^{r}$ and $\mathcal{L}_{i}={\rm Col}(L_{i})\subseteq\mathbb{F}_{q}^{r}$, for $i=1,2,\ldots,\ell$. Then $\mathcal{C}$ and $\mathcal{D}$ are sum-rank isometric if, and only if,

[TABLE]

In particular, sum-rank isometry classes of proper sum-rank Hamming codes with the sum-rank length partition $n=\ell N$ and redundancy $r$ correspond bijectively with maximal-size partial $N$-spreads in $\mathbb{F}_{q}^{r}$.

Proof.

It follows by combining Definition 2, Theorem 2 and Theorem 3. ∎

Remark 11**.**

Observe that, by choosing $N=1$, we recover the correspondence between classical Hamming codes [11] and projective spaces, which obviously are maximal-size partial $1$-spreads. See [13, Sec. 1.8].

Next we recall some exact formulas and explicit bounds for the size of maximal-size proper partial spreads. We will use these results to estimate the length of sum-rank Hamming codes and the minimum sum-rank distance of sum-rank Simplex codes in Section 4. There are numerous works in that respect. However, we will use the following two bounds from the literature due to their simplicity.

Proposition 12**.**

Let $1\leq N\leq r$, and let $s\geq 0$ be the remainder of $r$ divided by $N$. If $\mathcal{P}=\{\mathcal{H}_{i}\}_{i\in I}$ is a maximal-size partial $N$-spread in $\mathbb{F}_{q}^{r}$, then

[TABLE]

In particular, if $s=0$, that is, if $N$ divides $r$, then $\mathcal{P}$ is a maximal-size partial $N$-spread if, and only if,

[TABLE]

Proof.

The lower bound in (8) is proven in [4]. A simple proof of the upper bound in (8) is given in [10], although sharper bounds were known earlier (see, e.g., [3]). ∎

Therefore, we may estimate the length of proper sum-rank Hamming codes when $m=1$ as follows.

Corollary 13**.**

Let $n=\ell N$ be a sum-rank length partition with equal sublengths $N=n_{1}=n_{2}=\ldots=n_{\ell}$. Let $r\geq N$, and let $s\geq 0$ be the remainder of $r$ divided by $N$. If $\mathcal{C}\subseteq\mathbb{F}_{q}^{n}$ is a proper sum-rank Hamming code with $r=n-\dim(\mathcal{C})$, then

[TABLE]

and equality holds in (9) if $s=0$.

Remark 14**.**

Observe that the length $n$ of a proper sum-rank Hamming code in (9) coincides with that of classical Hamming codes if $N=m=1$, which implies that $s=0$. See [13, Sec. 1.8].

Finally, we show that proper sum-rank Hamming codes for $m=1$ are perfect codes for such a sum-rank metric when $N$ divides $r$. In particular, we conclude that their sum-rank error-correction capability is exactly $1$.

Corollary 15**.**

Let $n=\ell N$ be a sum-rank length partition with equal sublengths $N=n_{1}=n_{2}=\ldots=n_{\ell}$. Let $r\geq N$ be such that $N$ divides $r$. If $\mathcal{C}\subseteq\mathbb{F}_{q}^{n}$ is a proper sum-rank Hamming code with $r=n-\dim(\mathcal{C})$, then $\mathcal{C}$ is a perfect code for such a sum-rank metric. In other words, if $\mathcal{B}_{t}=\{\mathbf{c}\in\mathbb{F}_{q}^{n}\mid{\rm wt}_{SR}(\mathbf{c})\leq t\}$ denotes the ball centered in the origin with radius $t>0$, then

[TABLE]

where $1=\left\lfloor({\rm d}_{SR}(\mathcal{C})-1)/2\right\rfloor$ is the sum-rank error-correcting capability of $\mathcal{C}$.

Proof.

It is straightforward to see that

[TABLE]

Next, from Corollary 13 it holds that

[TABLE]

Therefore, if we denote $k=\dim(\mathcal{C})$, we conclude that

[TABLE]

Finally, it must hold that $t=\left\lfloor({\rm d}_{SR}(\mathcal{C})-1)/2\right\rfloor=1$, since $t\geq 1$ and $|\mathcal{C}|\cdot|\mathcal{B}_{t}|\leq|\mathbb{F}_{q}^{n}|=|\mathcal{C}|\cdot|\mathcal{B}_{1}|$. ∎

Remark 16**.**

In [15], it was proven that there exist no non-trivial perfect code for the rank-metric, that is, for the sum-rank metric where $\ell=1$. Note however that Corollary 15 holds only for $\ell\geq{\rm d}_{SR}(\mathcal{C})=3$ since $m=1$, hence it has no intersection with the case $\ell=1$. However, the case of classical Hamming codes also falls into the case $m=1$. Therefore, it is still an open problem to see if there exist non-trivial perfect codes for the sum-rank metric when $m>1$.

Partial spreads allowed us to deal with proper Hamming codes. For improper Hamming codes, we need to define partial spreads whose subspaces may have distinct dimensions.

Definition 17** (Improper partial spreads).**

Let $\ell$ and $n_{1},n_{2},\ldots,n_{\ell}$ be positive integers such that $n_{i}\leq r$, for $i=1,2,\ldots,\ell$, and denote $\mathbf{n}=(n_{1},n_{2},\ldots,n_{\ell})$. An improper partial $\mathbf{n}$-spread in $\mathbb{F}_{q}^{r}$ is a family $\mathcal{P}=\{\mathcal{H}_{i}\}_{i=1}^{\ell}$ of vector subspaces $\mathcal{H}_{i}\subseteq\mathbb{F}_{q}^{r}$, for $i=1,2,\ldots,\ell$, such that $\dim(\mathcal{H}_{i})=n_{i}$, for $i=1,2,\ldots,\ell$, and $\mathcal{H}_{i}\cap\mathcal{H}_{j}=\{\mathbf{0}\}$, whenever $i\neq j$. A maximal-size improper partial $\mathbf{n}$-spread in $\mathbb{F}_{q}^{r}$ is an improper partial $\mathbf{n}$-spread in $\mathbb{F}_{q}^{r}$ such that there is no improper partial $\mathbf{n}^{\prime}$-spread in $\mathbb{F}_{q}^{r}$ for $\mathbf{n}^{\prime}=(n_{1}^{\prime},n_{2}^{\prime},\ldots,n_{\ell^{\prime}}^{\prime})$, where $\ell^{\prime}\geq\ell$, $n_{i}^{\prime}\geq n_{i}$, for $i=1,2,\ldots,\ell$, and $n_{1}^{\prime}+n_{2}^{\prime}+\cdots+n_{\ell^{\prime}}^{\prime}>n_{1}+n_{2}+\cdots+n_{\ell}$.

All the previous results on proper sum-rank Hamming codes still hold for improper sum-rank Hamming codes with $m=1$, except for being perfect codes, which we leave as open problem. We also leave as an open problem estimating their code length $n$.

Next we collect the analogous results to Theorems 3 and 4. Proofs are left to the reader.

Theorem 5**.**

Let $\ell$ and $n_{1},n_{2},\ldots,n_{\ell}$ be positive integers such that $n_{i}\leq r$, for $i=1,2,\ldots,\ell$, and denote $\mathbf{n}=(n_{1},n_{2},\ldots,n_{\ell})$ and $n=n_{1}+n_{2}+\cdots+n_{\ell}$. A linear code $\mathcal{C}\subseteq\mathbb{F}_{q}^{n}$ is an improper sum-rank Hamming code for such a sum-rank length partition if, and only if, $\max\{n_{1},n_{2},\ldots,n_{\ell}\}\leq r=n-\dim(\mathcal{C})$ and it has a parity-check matrix of the form

[TABLE]

where $H_{i}\in\mathbb{F}_{q}^{r\times n_{i}}$ and $\mathcal{H}_{i}={\rm Col}(H_{i})\subseteq\mathbb{F}_{q}^{r}$, for $i=1,2,\ldots,\ell$, and $\mathcal{P}=\{\mathcal{H}_{1},\mathcal{H}_{2},\ldots,\mathcal{H}_{\ell}\}$ is a maximal-size improper partial $\mathbf{n}$-spread in $\mathbb{F}_{q}^{r}$.

Theorem 6**.**

Let $\ell$ and $n_{1},n_{2},\ldots,n_{\ell}$ be positive integers such that $n_{i}\leq r$, for $i=1,2,\ldots,\ell$, and denote $\mathbf{n}=(n_{1},n_{2},\ldots,n_{\ell})$ and $n=n_{1}+n_{2}+\cdots+n_{\ell}$. Let $\mathcal{C},\mathcal{D}\subseteq\mathbb{F}_{q}^{n}$ be two improper sum-rank Hamming codes for such a sum-rank length partition and with $r=n-\dim(\mathcal{C})=n-\dim(\mathcal{D})$ and parity-check matrices given by

[TABLE]

respectively. Denote $\mathcal{H}_{i}={\rm Col}(H_{i})\subseteq\mathbb{F}_{q}^{r}$ and $\mathcal{L}_{i}={\rm Col}(L_{i})\subseteq\mathbb{F}_{q}^{r}$, for $i=1,2,\ldots,\ell$. Then $\mathcal{C}$ and $\mathcal{D}$ are sum-rank isometric if, and only if,

[TABLE]

In particular, sum-rank isometry classes of improper sum-rank Hamming codes with the sum-rank length partition $n=n_{1}+n_{2}+\cdots+n_{\ell}$ and redundancy $r$ correspond bijectively with maximal-size improper partial $\mathbf{n}$-spreads in $\mathbb{F}_{q}^{r}$.

3.4 Syndrome decoding when $m=1$

In this subsection, we show how to adapt the single-error syndrome decoding algorithm for classical Hamming codes (see [13, Sec. 1.11]) to sum-rank Hamming codes when $m=1$, both proper and improper. Actually, the algorithm works exactly in the same way for any linear code of sum-rank distance at least $3$ (thus being able to correct at least a single sum-rank error) whenever $m=1$.

Set $m=1$ and let $n=n_{1}+n_{2}+\cdots+n_{\ell}$ be a sum-rank length partition. Consider in general a linear code $\mathcal{C}\subseteq\mathbb{F}_{q}^{n}$ such that ${\rm d}_{SR}(\mathcal{C})\geq 3$. As in the proof of Theorem 3, any parity-check matrix of $\mathcal{C}$ is of the form

[TABLE]

where $r=n-\dim(\mathcal{C})$, and if $\mathcal{H}_{i}={\rm Col}(H_{i})\subseteq\mathbb{F}_{q}^{r}$, for $i=1,2,\ldots,\ell$, then $\mathcal{P}=\{\mathcal{H}_{1},\mathcal{H}_{2},\ldots,\mathcal{H}_{\ell}\}$ forms a (not necessarily maximal-size) improper partial $\mathbf{n}$-spread, where $\mathbf{n}=(n_{1},n_{2},\ldots,n_{\ell})$. In particular, the rank of $H_{i}$ is $n_{i}$, for $i=1,2,\ldots,\ell$, and thus $r\geq\max\{n_{1},n_{2},\ldots,n_{\ell}\}$.

We briefly recall how syndrome decoding works in general (for any metric). We refer to [13, Sec. 1.11] for more details. Let $t$ be a positive integer such that $2t$ is less than the minimum distance of $\mathcal{C}$. Then there exists a unique vector (called coset leader) of weight at most $t$ in every coset $\mathbf{x}+\mathcal{C}$, where $\mathbf{x}\in\mathbb{F}_{q}^{n}$, that contains a vector of weight at most $t$. Recall that, for any $\mathbf{x}\in\mathbb{F}_{q}^{n}$, it holds that $H\mathbf{y}^{T}=H\mathbf{x}^{T}$, for all $\mathbf{y}\in\mathbf{x}+\mathcal{C}$. Thus, we may identify coset leaders $\mathbf{e}$ of weight at most $t$ with syndromes $\mathbf{s}=H\mathbf{x}^{T}$ of cosets $\mathbf{x}+\mathcal{C}$ of weight at most $t$.

With this in mind, the general algorithm works as follows: Let $\mathbf{y}=\mathbf{c}+\mathbf{e}\in\mathbb{F}_{q}^{n}$ be the received vector, where $\mathbf{c}\in\mathcal{C}$ is the correct codeword and $\mathbf{e}\in\mathbb{F}_{q}^{n}$ is an error vector of weight at most $t$. Compute the syndrome $\mathbf{s}=H\mathbf{y}^{T}$. Then $\mathbf{e}$ is the unique coset leader of the coset corresponding to $\mathbf{s}$. If we have a method (e.g., by a precomputed lookup table) to obtain $\mathbf{e}$ from $\mathbf{s}$, then we obtain $\mathbf{e}$ and compute $\mathbf{c}=\mathbf{y}-\mathbf{e}$.

Thus the difficult part is to obtain the coset leader $\mathbf{e}$ from the syndrome $\mathbf{s}$. This is simple for the code $\mathcal{C}$ described in the beginning of this subsection. Since we target $t=1$, we only need to consider $\mathbf{e}\in\mathbb{F}_{q}^{n}$ such that ${\rm wt}_{SR}(\mathbf{e})\leq 1$. These are either $\mathbf{e}=\mathbf{0}$ or

[TABLE]

where $\mathbf{a}\in\mathbb{F}_{q}^{n_{i}}$ is a non-zero vector in the $i$th block of $n_{i}$ coordinates over $\mathbb{F}_{q}$. Therefore, we only need to find $i$ and $\mathbf{a}$ from the syndrome $H\mathbf{e}_{i,\mathbf{a}}$. To that end, we only need to try to solve the systems of linear equations

[TABLE]

given the syndrome $\mathbf{s}\in\mathbb{F}_{q}^{r}$, iteratively in $j=1,2,\ldots,\ell$. Such a solution $\mathbf{a}\in\mathbb{F}_{q}^{n_{j}}$ only exists for a unique value of $j$, say $i$ (at which point the iteration stops), and it is unique, since $\mathcal{P}=\{\mathcal{H}_{1},\mathcal{H}_{2},\ldots,\mathcal{H}_{\ell}\}$ is an improper partial $\mathbf{n}$-spread.

In conclusion, given a received vector $\mathbf{y}=\mathbf{c}+\mathbf{e}\in\mathbb{F}_{q}^{n}$, where $\mathbf{c}\in\mathcal{C}$ is the correct codeword and ${\rm wt}_{SR}(\mathbf{e})\leq 1$, we proceed as follows:

Compute $\mathbf{s}=H\mathbf{y}^{T}\in\mathbb{F}_{q}^{r}$. If $\mathbf{s}=\mathbf{0}$, then output $\mathbf{c}=\mathbf{y}$ and exit the algorithm. 2. 2.

Until a solution is found, try to solve the linear system $H_{j}\mathbf{a}^{T}=\mathbf{s}$ in the variables $\mathbf{a}\in\mathbb{F}_{q}^{n_{j}}$, iteratively in $j=1,2,\ldots,\ell$. 3. 3.

Given the unique solution $(i,\mathbf{a})$ from Step 2, output $\mathbf{c}=\mathbf{y}-\mathbf{e}_{i,\mathbf{a}}$, where $\mathbf{e}_{i,\mathbf{a}}$ is as in (11).

Step 1 has a complexity of $\mathcal{O}(nr)$, whereas Step 2 has a complexity of $\mathcal{O}(\ell r^{3})$, both in number of operations in the field $\mathbb{F}_{q}$. Hence, the total number of operations over $\mathbb{F}_{q}$ is $\mathcal{O}(nr+\ell r^{3})$. In the case of proper sum-rank Hamming codes where $N=n_{1}=n_{2}=\ldots=n_{\ell}$ divides $r$, this complexity is

[TABLE]

It is worth noting that the term $(q^{r}-1)(q^{N}-1)$ is dominant over $r^{3}+Nr$ and thus the complexity is close to linear in the code length $n$.

4 Sum-rank simplex codes

In this section, we define sum-rank simplex codes as duals of sum-rank Hamming code, in analogy with the classical case (see [13, Sec. 1.8]). Since we distinguished between proper and improper sum-rank Hamming codes, we do the same for sum-rank simplex codes. We consider dual codes with respect to the standard inner product in $\mathbb{F}_{q^{m}}^{n}$.

Definition 18** (Sum-rank simplex codes).**

A linear code $\mathcal{C}\subseteq\mathbb{F}_{q^{m}}^{n}$ is called a proper (resp. improper) sum-rank simplex code if its dual $\mathcal{C}^{\perp}\subseteq\mathbb{F}_{q^{m}}^{n}$ is a proper (resp. improper) sum-rank Hamming code.

One may consider the case $\ell=1$. The duals of the codes in Subsection 3.2 are again of the same form (Cartesian products of Gabidulin codes), thus we do not investigate this case further.

In the rest of the section, we only consider again the case $m=1$. We will also make use of partial spreads as in Subsection 3.3. However, now our objective is to lower bound the minimum sum-rank distance of the corresponding proper sum-rank simplex code.

The main result of this subsection is the following.

Theorem 7**.**

Let $n=\ell N$ be a sum-rank length partition with equal sublengths $N=n_{1}=n_{2}=\ldots=n_{\ell}$. Let $r\geq N$ and let $s\geq 0$ be the remainder of $r$ divided by $N$. If $\mathcal{C}\subseteq\mathbb{F}_{q}^{n}$ is a proper sum-rank simplex code of dimension $r$, then it holds that

[TABLE]

Proof.

By Definition 18 and Theorem 3, $\mathcal{C}$ has a generator matrix of the form

[TABLE]

where $H_{i}\in\mathbb{F}_{q}^{r\times N}$ and $\mathcal{H}_{i}={\rm Col}(H_{i})\subseteq\mathbb{F}_{q}^{r}$, for $i=1,2,\ldots,\ell$, and $\mathcal{P}=\{\mathcal{H}_{1},\mathcal{H}_{2},\ldots,\mathcal{H}_{\ell}\}$ is a maximal-size partial $N$-spread in $\mathbb{F}_{q}^{r}$.

Let $\mathbf{x}\in\mathbb{F}_{q}^{r}$ be distinct from zero. Then it holds that

[TABLE]

where $\mathbf{x}^{\perp}\subseteq\mathbb{F}_{q}^{r}$ is the dual of the linear code generated by the vector $\mathbf{x}$. In other words,

[TABLE]

where $\mathcal{Q}=\{\mathcal{H}_{i}\mid\mathcal{H}_{i}\subseteq\mathbf{x}^{\perp},1\leq i\leq\ell\}$ is the partial $N$-spread $\mathcal{P}$ restricted to the hyperplane $\mathbf{x}^{\perp}$.

Let $s\geq 0$ and $t\geq 0$ be the remainders of $r$ and $r-1$ divided by $N$, respectively. By Proposition 12, we deduce that

[TABLE]

We will consider now the cases $s=0$ and $s>0$ separately.

Assume first that $s=0$. Then $t=N-1$ and

[TABLE]

[TABLE]

and (12) is proven in this case.

Next assume that $s>0$. Then $t=s-1\geq 0$ and

[TABLE]

Let $h\geq 0$ be the unique integer such that $r=hN+s$. We have that

[TABLE]

where the term $(q^{s}-q^{s-1})(q^{hN}-1)/(q^{N}-1)$ is an integer, and moreover

[TABLE]

since $s<N$. Thus we deduce that

[TABLE]

Hence (12) is proven also in the case $s>0$, and we are done. ∎

Remark 19**.**

Observe that (12) is an equality in the case of classical simplex codes, which is recovered from Theorem 7 by further setting $N=1$, hence $s=0$. Furthermore, a classical simplex code is a constant-weight code, meaning that any of its non-zero codewords has weight equal to $q^{r-1}$. See [13, Th. 1.8.3]. We leave the sharpness of (12) as an open problem, together with whether sum-rank simplex codes are in general constant-weight codes. Obviously, the use of sharper bounds on the sizes of maximal-size partial spreads than those in Proposition 12 would help in this regard.

5 Some applications

In this section, we provide two possible applications of sum-rank Hamming and simplex codes.

5.1 Multishot matrix-multiplicative channels

Multishot matrix-multiplicative channels model, among others, multiple uses of linearly coded networks where the transmitter has no knowledge of the network or linear network code, and the linear combinations in intermediate nodes are not seen by the receiver. We refer to [21] and [19] for details on multishot network coding. We note that a similar channel may be used for space-time coding, see [16, Sec. III].

We follow [19, Sec. II] to define multishot matrix-multiplicative channels with errors and erasures. Fix positive integers $\ell,n_{1},n_{2},\ldots,n_{\ell}$, $m_{1},m_{2},\ldots,m_{\ell}$ and denote $\mathbf{n}=(n_{1},n_{2},\ldots,n_{\ell})$ and $\mathbf{m}=(m_{1},m_{2},\ldots,m_{\ell})$. An $(\mathbf{m}\times\mathbf{n})$-multishot matrix-multiplicative channel with noise over the base field $\mathbb{F}_{q}$ takes as input a sequence of matrices

[TABLE]

and outputs a sequence of matrices

[TABLE]

for some matrix $A_{i}\in\mathbb{F}_{q}^{n_{i}\times N_{i}}$, called the $i$th transfer matrix, and some matrix $E_{i}\in\mathbb{F}_{q}^{m_{i}\times N_{i}}$, called the $i$th error matrix, for $i=1,2,\ldots,\ell$, and for some positive integers $N_{1},N_{2},\ldots,N_{\ell}$.

The number of sum-rank errors and erasures are given, under this model, by

[TABLE]

respectively. As it was the case for the sum-rank metric recovering the Hamming and rank metrics, it holds that multishot matrix-multiplicative channels with errors and erasures as in (14) recover discrete memoryless channels with symbol-wise errors and erasures by setting $m_{1}=m_{2}=\ldots=m_{\ell}=n_{1}=n_{2}=\ldots=n_{\ell}=N_{1}=N_{2}=\ldots=N_{\ell}=1$, and they recover singleshot matrix-multiplicative channels simply by setting $\ell=1$.

We will only consider the case $m=m_{1}=m_{2}=\ldots=m_{\ell}$. In such a case, we may consider a sequence of matrices in $\mathbb{F}_{q}^{m\times n_{1}}\times\mathbb{F}_{q}^{m\times n_{2}}\times\cdots\times\mathbb{F}_{q}^{m\times n_{\ell}}$ as a vector in $\mathbb{F}_{q^{m}}^{n}$, where $n=n_{1}+n_{2}+\cdots+n_{\ell}$, via the matrix representation map in (1).

With such considerations, we say that a $k$-dimensional linear code $\mathcal{C}\subseteq\mathbb{F}_{q^{m}}^{n}$ can coherently correct $t$ sum-rank errors and $\rho$ sum-rank erasures if, for all transfer matrices $A_{i}\in\mathbb{F}_{q}^{n_{i}\times N_{i}}$, for $i=1,2,\ldots,\ell$, such that $\sum_{i=1}^{\ell}{\rm Rk}(A_{i})\geq n-\rho$, there exists a decoder $D:\mathbb{F}_{q^{m}}^{N}\longrightarrow\mathbb{F}_{q^{m}}^{k}$, possibly depending on $A_{1},A_{2},\ldots,A_{\ell}$, where $N=N_{1}+N_{2}+\cdots+N_{\ell}$, such that

[TABLE]

where $\mathbf{c}=(\mathbf{c}^{(1)},\mathbf{c}^{(2)},\ldots,\mathbf{c}^{(\ell)})\in\mathbb{F}_{q^{m}}^{n}$ is the encoding of $\mathbf{x}\in\mathbb{F}_{q^{m}}^{n}$ by some fixed generator matrix of $\mathcal{C}$, for all $\mathbf{x}\in\mathbb{F}_{q^{m}}^{k}$ and all $\mathbf{e}=(\mathbf{e}^{(1)},\mathbf{e}^{(2)},\ldots,\mathbf{e}^{(\ell)})\in\mathbb{F}_{q^{m}}^{N}$ such that

[TABLE]

where $E_{i}=M_{\mathcal{A}}(\mathbf{e}^{(i)})\in\mathbb{F}_{q}^{m\times N_{i}}$, for $i=1,2,\ldots,\ell$. The word coherently refers to the fact that the transfer matrices $A_{1},A_{2},\ldots,A_{\ell}$ are known to the decoder (that is why it may depend on such matrices). An adaptation to the non-coherent case may be done by lifting, see [19, Subsec. IV-C].

With these definitions, the following result is a particular case of [19, Th. 1].

Proposition 20** ([19]).**

Let $n=n_{1}+n_{2}+\cdots+n_{\ell}$ be a sum-rank length partition, and let $t\geq 0$ and $\rho\geq 0$ be integers. A linear code $\mathcal{C}\subseteq\mathbb{F}_{q^{m}}^{n}$ can coherently correct up to $t$ sum-rank errors and $\rho$ sum-rank erasures in any $(\mathbf{m}\times\mathbf{n})$-multishot matrix-multiplicative channel if, and only if, it satisfies that ${\rm d}_{SR}(\mathcal{C})>2t+\rho$.

Hence, sum-rank Hamming codes are the longest linear codes that can correct a single sum-rank error ($t=1$ and $\rho=0$) over such channels.

Theorem 8**.**

Let $n=\ell N$ be a sum-rank length partition with equal sublengths $N=n_{1}=n_{2}=\ldots=n_{\ell}$. A proper sum-rank Hamming code $\mathcal{C}\subseteq\mathbb{F}_{q^{m}}^{n}$ with redundancy $r=n-\dim(\mathcal{C})$ is the linear code of redundancy $r$ capable of correcting a single sum-rank error in an $(\mathbf{m}\times\mathbf{n})$-multishot matrix-multiplicative channel with $m_{1}=m_{2}=\ldots=m_{\ell}$ and $n_{1}=n_{2}=\ldots=n_{\ell}$ for the largest number of shots $\ell$.

Improper sum-rank Hamming code admit a similar interpretation for multishot matrix-multiplicative channels. However, their main feature is not necessarily admitting the largest number of shots $\ell$, but having the largest overall sum-rank length $n$ attainable by increasing $\ell$ and/or the numbers of matrix columns $n_{1},n_{2},\ldots,n_{\ell}$.

We conclude by noting that, if $m=1$, then no code can correct a single sum-rank error if $\ell\leq 2$, since we have that ${\rm d}_{SR}(\mathcal{C})\leq\ell$ when $m=1$. In particular, no rank error-correcting code exists if $m=1$, as in that case $\ell=1$. Therefore, in this work we have provided the first known error-correcting codes for matrix-multiplicative channels with $m=1$ but $N=n_{1}=n_{2}=\ldots=n_{\ell}>1$, by making use of several shots of the channel, i.e. $\ell\geq 3$. In particular, choosing $q=2$ and $m=1$, we have provided the first codes that can correct bit-wise errors in linearly coded networks where the transmitter has no knowledge of the network or linear network code and the linear combinations of intermediate nodes are not known by the receiver. The decoding algorithm from Subsection 3.4 can be trivially implemented in this scenario.

5.2 Locally repairable codes

Locally repairable codes [9] are an attractive alternative to MDS codes for large distributed storage systems, since they allow to repair a single erasure (most common erasure pattern) by contacting a small number, called locality, of other nodes, while being able to correct more erasures in catastrophic cases.

In [20], we established a connection between the sum-rank metric and locally repairable codes. We start by recalling the following result, which is [20, Cor. 4].

Lemma 21** ([20]).**

Let $n=n_{1}+n_{2}+\cdots+n_{\ell}$ and let $\mathcal{C}_{out}\subseteq\mathbb{F}_{q^{m}}^{n}$ be a linear code. Choose an $n_{i}$-dimensional local linear code $\mathcal{C}_{i}\subseteq\mathbb{F}_{q}^{N_{i}}$ with generator matrix $A_{i}\in\mathbb{F}_{q}^{n_{i}\times N_{i}}$ (thus $n_{i}\leq N_{i}$), for $i=1,2,\ldots,\ell$. Define the global code $\mathcal{C}_{glob}\subseteq\mathbb{F}_{q^{m}}^{M}$ with total length $M=N_{1}+N_{2}+\cdots+N_{\ell}$ as

[TABLE]

Denote by $\Gamma_{i}\subseteq[M]$ the set of coordinates ranging from $\sum_{j=1}^{i-1}N_{j}+1$ to $\sum_{j=1}^{i}N_{j}$, for $i=1,2,\ldots,\ell$. Then $\dim(\mathcal{C}_{glob})=\dim(\mathcal{C}_{out})$ and any erasure pattern $\mathcal{E}_{i}\subseteq\Gamma_{i}$ with $|\mathcal{E}_{i}|<{\rm d}_{H}(\mathcal{C}_{i})$ can be corrected by $\mathcal{C}_{glob}$ with the same complexity over the same field as with $\mathcal{C}_{i}$ and only using the $N_{i}-|\mathcal{E}_{i}|$ symbols in $\Gamma_{i}\setminus\mathcal{E}_{i}$, for $i=1,2,\ldots,\ell$.

In other words, the global code $\mathcal{C}_{glob}$ keeps the same dimension as the outer code $\mathcal{C}_{out}$, but it also has the local erasure-correction capability of the local codes $\mathcal{C}_{1},\mathcal{C}_{2},\ldots,\mathcal{C}_{\ell}$. Next, we recall the global erasure-correction capability of the global code in terms of the sum-rank erasure-correction capability of the outer code. The following result follows from Theorem 1 and it gives a simple sufficient condition for a global erasure pattern to be correctable by the global code.

Proposition 22**.**

Let notation be as in Lemma 21. Let $\mathcal{E}\subseteq[M]$ be an erasure pattern and define $\mathcal{E}_{i}=\mathcal{E}\cap\Gamma_{i}$ and $\mathcal{R}_{i}=\Gamma_{i}\setminus\mathcal{E}_{i}$, for $i=1,2,\ldots,\ell$. Then the erasure pattern $\mathcal{E}$ can be corrected by $\mathcal{C}_{glob}$ if

[TABLE]

where ${\rm d}_{SR}$ is considered with respect to the sum-rank length partition $n=n_{1}+n_{2}+\cdots+n_{\ell}$.

Observe that the local codes could in principle help in the global erasure correction by repairing erasures locally (inside each $\Gamma_{i}$) whenever they can, as in Lemma 21. However, they would only be adding redundant symbols, hence they would not increase the ranks of the matrices $A_{i}|_{\mathcal{R}_{i}}$, thus such a local erasure correction does not affect the global erasure-correction capability, as shown in Proposition 22.

Usually, the local codes are be chosen to be MDS, as they must be short codes. Moreover, they are usually considered such that

[TABLE]

preferably with $q=2$. In that case, all local codes may correct up to one erasure very efficiently with only one XOR operation. The global code $\mathcal{C}_{glob}$ would possibly have unequal localities $n_{1},n_{2},\ldots,n_{\ell}$ (locality $n_{i}$ for the $i$th local group $\Gamma_{i}$). However, the construction in (15) allows any choice of local linear codes over $\mathbb{F}_{q}$.

In the case of equal localities $N=n_{1}=n_{2}=\ldots=n_{\ell}$, we may provide locally repairable codes over any field $\mathbb{F}_{q}$, including $q=2$, by making use of proper sum-rank Hamming codes with $m=1$. In that case, we deduce the following construction from Corollary 13.

Theorem 9**.**

Let $q$ be an arbitrary prime power (e.g. $q=2$) and let $n=\ell N$ be a sum-rank length partition with equal sublengths $N=n_{1}=n_{2}=\ldots=n_{\ell}$. Let $r\geq N$ be such that $N$ divides $r$. Let $\mathcal{C}_{out}\subseteq\mathbb{F}_{q}^{n}$ be a proper sum-rank Hamming code with redundancy $r=n-\dim(\mathcal{C}_{out})$ and let the $i$th local code $\mathcal{C}_{i}\subseteq\mathbb{F}_{q}^{N_{i}}$ have a generator matrix as in (16). Then the global code given as in (15) has locality $N$ and number of local groups, total length and dimension given by

[TABLE]

respectively, where $r=hN$. Therefore, according to [20, Def. 6], the number of local parities is $1$ per local group (i.e., the redundancy of each local code) and the number of global parities is $r$ (i.e., the redundancy of the outer code). Finally, $\mathcal{C}_{glob}$ may correct one erasure locally by a simple XOR of the remaining nodes in that local group, and it may correct any erasure pattern with one erasure per local group plus any other $2$ extra erasures anywhere.

Table 1 gives some choices of parameters for linear codes as in Theorem 9 for $q=2$.

We have chosen to provide a construction based on sum-rank Hamming codes with $m=1$, since it allows to work on any finite field $\mathbb{F}_{q}$ (in particular $q=2$) and since simplex codes have low information rate. Having high information rate even though only up to $2$ extra erasures may be corrected is a desirable parameter regime for distributed storage applications.

Finally, note that we could have chosen any family of local linear codes in Theorem 9. This would make sense, for instance, to obtain hierachical locally repairable codes by choosing local codes that are in turn locally repairable codes (see [20, Subsec. V-C]).

Acknowledgement

The author wishes to thank Frank R. Kschischang for valuable discussions on this manuscript. The author also gratefully acknowledges the support from The Independent Research Fund Denmark (Grant No. DFF-7027-00053B).