Sum-Rank Metric Codes: Theory & Applications

• Sum-Rank Metric Codes are block-structured codes where the distance is the sum of the ranks of differences, interpolating neatly between the rank and Hamming metrics.
• They employ algebraic and combinatorial constructions to achieve MSRD bounds, optimize decoding, and enable effective support theory and invariant classification.
• Applications include multishot network coding, distributed storage, and secure communication, while theoretical developments address asymptotic bounds, list decoding, and covering theory.

Sum-rank metric codes are codes on block-structured ambient spaces in which the distance between two codewords is the sum of the ranks of the blockwise differences. In matrix form, this places codewords in products of matrix spaces; in vector form, it places them in block decompositions of $\mathbb F_{q^m}^n$. The resulting metric interpolates exactly between the rank metric and the Hamming metric, and it is used in settings such as multishot network coding, distributed storage, and space-time coding. Recent work has developed a substantial theory around these codes, including support and anticode formalisms, MSRD extremality, asymptotic bounds, generic and list decoding, explicit algebraic and combinatorial constructions, and equivalence invariants (Martínez-Peñas, 2018, Abiad et al., 2023, Puchinger et al., 2020, Liu et al., 13 Mar 2025, Santonastaso et al., 5 Jul 2025).

1. Formal models and basic metric structure

A standard matrix model fixes blocks $X_i\in \mathbb F_q^{m\times \eta}$ and works in

$$M^\ell=\prod_{i=1}^{\ell}\mathbb F_q^{m\times \eta}, \qquad \boldsymbol X=(X_1,\dots,X_\ell).$$

The sum-rank weight is

$$\operatorname{wt}_{\mathrm{srk},\ell}(\boldsymbol X)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i),$$

and the induced distance is

$$d_{\mathrm{srk},\ell}(\boldsymbol X,\boldsymbol Y)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i-Y_i).$$

Under the common assumption $\eta\le m$, the total code length is $n=\eta\ell$, the maximum weight is $n$, and the metric is translation-invariant and invariant under multiplication by nonzero field scalars. In this setting, a general sum-rank code is any nonempty subset $C\subseteq M^\ell$, while an $\mathbb F_q$-linear sum-rank code is a subspace $X_i\in \mathbb F_q^{m\times \eta}$0 (Liu et al., 13 Mar 2025).

A complementary vector model writes $X_i\in \mathbb F_q^{m\times \eta}$1 in $X_i\in \mathbb F_q^{m\times \eta}$2 blocks of length $X_i\in \mathbb F_q^{m\times \eta}$3,

$X_i\in \mathbb F_q^{m\times \eta}$4

and expands each block over $X_i\in \mathbb F_q^{m\times \eta}$5. The block rank weight is the $X_i\in \mathbb F_q^{m\times \eta}$6-rank of the expansion matrix, and the sum-rank weight is the sum of those block ranks. In this notation, the weight decomposition $X_i\in \mathbb F_q^{m\times \eta}$7 records the rank of each block, with $X_i\in \mathbb F_q^{m\times \eta}$8. This model is especially convenient for decoding and asymptotic analysis over extension fields (Puchinger et al., 2020, Ott et al., 2021).

The metric contains the two classical extremes exactly. If $X_i\in \mathbb F_q^{m\times \eta}$9, there is a single block and the sum-rank metric is the ordinary rank metric. If every block is $$M^\ell=\prod_{i=1}^{\ell}\mathbb F_q^{m\times \eta}, \qquad \boldsymbol X=(X_1,\dots,X_\ell).$$0, or equivalently every vector block has length $$M^\ell=\prod_{i=1}^{\ell}\mathbb F_q^{m\times \eta}, \qquad \boldsymbol X=(X_1,\dots,X_\ell).$$1, then each nonzero block contributes rank $$M^\ell=\prod_{i=1}^{\ell}\mathbb F_q^{m\times \eta}, \qquad \boldsymbol X=(X_1,\dots,X_\ell).$$2, and the sum-rank metric becomes the Hamming metric. More general formulations also allow unequal block sizes

$$M^\ell=\prod_{i=1}^{\ell}\mathbb F_q^{m\times \eta}, \qquad \boldsymbol X=(X_1,\dots,X_\ell).$$3

with

$$M^\ell=\prod_{i=1}^{\ell}\mathbb F_q^{m\times \eta}, \qquad \boldsymbol X=(X_1,\dots,X_\ell).$$4

which makes the hybrid Hamming/rank nature explicit at the level of the ambient geometry (Abiad et al., 2023).

2. Supports, anticodes, generalized weights, and MSRD structure

A foundational development is the support theory of sum-rank codes. For a codeword $$M^\ell=\prod_{i=1}^{\ell}\mathbb F_q^{m\times \eta}, \qquad \boldsymbol X=(X_1,\dots,X_\ell).$$5, the sum-rank support is the tuple of row spaces of the block expansion matrices over the corresponding base fields. These supports form a lattice

$$M^\ell=\prod_{i=1}^{\ell}\mathbb F_q^{m\times \eta}, \qquad \boldsymbol X=(X_1,\dots,X_\ell).$$6

with componentwise sum, intersection, and orthogonal complement. The associated support spaces

$$M^\ell=\prod_{i=1}^{\ell}\mathbb F_q^{m\times \eta}, \qquad \boldsymbol X=(X_1,\dots,X_\ell).$$7

are in lattice isomorphism with supports, satisfy

$$M^\ell=\prod_{i=1}^{\ell}\mathbb F_q^{m\times \eta}, \qquad \boldsymbol X=(X_1,\dots,X_\ell).$$8

and provide the natural framework for restriction, shortening, duality, and effective length. In particular, the generalized sum-rank weights $$M^\ell=\prod_{i=1}^{\ell}\mathbb F_q^{m\times \eta}, \qquad \boldsymbol X=(X_1,\dots,X_\ell).$$9 admit Wei-type duality, and degeneracy is characterized by $$\operatorname{wt}_{\mathrm{srk},\ell}(\boldsymbol X)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i),$$0 (Martínez-Peñas, 2018).

A second structural axis is the anticode viewpoint. For a linear code $$\operatorname{wt}_{\mathrm{srk},\ell}(\boldsymbol X)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i),$$1, the sum-rank Anticode Bound states

$$\operatorname{wt}_{\mathrm{srk},\ell}(\boldsymbol X)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i),$$2

Codes attaining equality are optimal anticodes. These are classified: an optimal sum-rank anticode is precisely a product of optimal rank-metric anticodes on the blocks with $$\operatorname{wt}_{\mathrm{srk},\ell}(\boldsymbol X)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i),$$3, together with an optimal Hamming-metric anticode on the $$\operatorname{wt}_{\mathrm{srk},\ell}(\boldsymbol X)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i),$$4-row part. This classification is then used to define generalized sum-rank weights through intersections with optimal anticodes, extending both generalized Hamming weights and generalized rank weights (Moreno et al., 2021).

The same anticode framework clarifies MSRD codes. In the equal-row case, the Singleton-type bound becomes the usual

$$\operatorname{wt}_{\mathrm{srk},\ell}(\boldsymbol X)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i),$$5

in the vector model, while in the general matrix/block model it is expressed through the decomposition of $$\operatorname{wt}_{\mathrm{srk},\ell}(\boldsymbol X)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i),$$6 across block lengths. Codes meeting the corresponding bound are MSRD. Their generalized weights are parameter-determined, in direct analogy with MDS and MRD codes. The appendix of the anticode paper also shows that the support-space version of generalized weights measures information leakage in multishot network coding, linking the structural theory to secure communication (Moreno et al., 2021).

3. Upper bounds, asymptotics, and covering theory

The counting theory of sum-rank spheres and balls is central to asymptotic bounds. In the equal-block vector model, the number of words of weight $$\operatorname{wt}_{\mathrm{srk},\ell}(\boldsymbol X)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i),$$7 is a sum over bounded ordered partitions $$\operatorname{wt}_{\mathrm{srk},\ell}(\boldsymbol X)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i),$$8, and the dominant exponent of a sphere of total weight $$\operatorname{wt}_{\mathrm{srk},\ell}(\boldsymbol X)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i),$$9 is

$$d_{\mathrm{srk},\ell}(\boldsymbol X,\boldsymbol Y)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i-Y_i).$$0

This leads to simplified sphere-packing and Gilbert–Varshamov bounds for linear sum-rank codes, together with asymptotic tradeoffs in both the classical regime of fixed block size and growing number of blocks, and the newer regime in which block size itself grows with code length. In that latter regime, with $$d_{\mathrm{srk},\ell}(\boldsymbol X,\boldsymbol Y)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i-Y_i).$$1, the asymptotic GV-like rate behaves as

$$d_{\mathrm{srk},\ell}(\boldsymbol X,\boldsymbol Y)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i-Y_i).$$2

and random linear sum-rank codes are shown to attain near-GV behavior with high probability; for sufficiently large extension fields, random linear codes are also MSRD with high probability (Ott et al., 2021).

For finite-parameter upper bounds, graph methods have become important. The sum-rank-metric graph has vertex set $$d_{\mathrm{srk},\ell}(\boldsymbol X,\boldsymbol Y)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i-Y_i).$$3, adjacency given by sum-rank distance $$d_{\mathrm{srk},\ell}(\boldsymbol X,\boldsymbol Y)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i-Y_i).$$4, and decomposes as a Cartesian product of bilinear forms graphs. This graph is vertex-transitive and walk-regular, but it is distance-regular only in the Hamming-like case where all blocks are $$d_{\mathrm{srk},\ell}(\boldsymbol X,\boldsymbol Y)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i-Y_i).$$5 of the same width. Ratio-type spectral bounds applied to this graph yield new explicit upper bounds for minimum distances $$d_{\mathrm{srk},\ell}(\boldsymbol X,\boldsymbol Y)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i-Y_i).$$6, improve earlier bounds in many parameter sets, and produce new non-existence results for possibly nonlinear MSRD codes (Abiad et al., 2023).

A complementary development is a genuine Delsarte-style linear programming bound. Although the natural sum-rank distance relations do not generally form an association scheme directly, the Cartesian product decomposition embeds the problem into a direct product of bilinear forms schemes. The resulting LP imposes zero inner-distribution entries on all relation classes whose blockwise rank distances sum to less than the target minimum distance, and computational experiments show that the bound outperforms all previously known bounds on a range of relatively small instances (Abiad et al., 2024).

Covering theory adds a different extremal perspective. For prescribed sum-rank covering radius $$d_{\mathrm{srk},\ell}(\boldsymbol X,\boldsymbol Y)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i-Y_i).$$7, the minimum cardinality of a covering code is denoted $$d_{\mathrm{srk},\ell}(\boldsymbol X,\boldsymbol Y)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i-Y_i).$$8. The basic sphere-covering bound is

$$d_{\mathrm{srk},\ell}(\boldsymbol X,\boldsymbol Y)=\sum_{i=1}^{\ell}\operatorname{rank}(X_i-Y_i).$$9

and a nontrivial MSRD-based upper bound is

$\eta\le m$0

The same paper adapts elementary linear subspaces to the sum-rank setting and identifies ball-intersection formulas as the main missing ingredient for sharper recursive lower bounds (Ott et al., 2022). A later covering-code transfer from Hamming metric to sum-rank metric yields stronger Singleton-like bounds when block lengths are large, and also leads to explicit quasi-perfect and distance-optimal families, especially for minimum sum-rank distance $\eta\le m$1 (Liu et al., 9 Jan 2026).

4. Decoding, hardness, and list decoding

Generic decoding in the sum-rank metric was developed through support guessing. For an $\eta\le m$2-linear code with parity-check matrix $\eta\le m$3, the error model combines a blockwise weight decomposition with row and column support notions: each block error $\eta\le m$4 admits a factorization $\eta\le m$5, which yields a row support as the product of the row spaces of the $\eta\le m$6, and a column support as the product of the column spaces of the expansions of the $\eta\le m$7. Knowing a row super-support or column super-support reduces decoding to a linear system. The paper gives erasure-decoding complexities $\eta\le m$8 and $\eta\le m$9 operations over $n=\eta\ell$0, respectively, and then builds a Las Vegas generic decoder that repeatedly samples random supports. Its expected work factor is controlled by a quantity $n=\eta\ell$1, and a simple upper bound has exponential term

$n=\eta\ell$2

The same work also proves a probabilistic hardness reduction from decisional Hamming syndrome decoding to decisional sum-rank syndrome decoding; under the stated extension-degree condition, $n=\eta\ell$3-SRSD in $n=\eta\ell$4 would imply $n=\eta\ell$5 (Puchinger et al., 2020).

For highly structured small-block codes, faster decoding is possible. Binary linear $n=\eta\ell$6 sum-rank-metric codes built from quaternary BCH, Goppa, and additive quaternary codes admit a reduction from sum-rank decoding to Hamming decoding over $n=\eta\ell$7. In the main decoding theorem, one decodes once in one constituent code and three times in twists of the other constituent, and this yields quadratic-time decoders with cost $n=\eta\ell$8 operations in $n=\eta\ell$9. The paper also constructs asymptotically good sequences of quadratically encodable and decodable binary $n$0 sum-rank codes satisfying

$n$1

from Goppa codes (Chen et al., 2023).

List decoding has recently reached a capacity theorem for the sum-rank metric. In the matrix model $n$2, with $n$3, the asymptotic log-volume exponent of a radius-$n$4 ball is

$n$5

so the capacity expression is

$n$6

Random general sum-rank codes of rate $n$7 are $n$8-list-decodable with high probability, and the same optimal $n$9 list size is proved for random $C\subseteq M^\ell$0-linear sum-rank codes after establishing a limited correlation property between sum-rank balls and $C\subseteq M^\ell$1-subspaces. A notable nuance is that the specialization $C\subseteq M^\ell$2 yields $C\subseteq M^\ell$3, which does not recover the classical Hamming list-decoding capacity $C\subseteq M^\ell$4 (Liu et al., 13 Mar 2025).

5. Explicit constructions and geometric families

A geometric construction of sum-rank codes is given by linearized Algebraic Geometry codes. Starting from a cyclic Galois cover of curves $C\subseteq M^\ell$5, the construction uses Ore polynomial quotients

$C\subseteq M^\ell$6

noncommutative Riemann–Roch spaces $C\subseteq M^\ell$7, and evaluations into products of endomorphism algebras

$C\subseteq M^\ell$8

The resulting codes satisfy a Goppa-like distance bound

$C\subseteq M^\ell$9

together with a lower bound on the dimension derived from the noncommutative Riemann–Roch theorem. In genus $\mathbb F_q$0, the construction recovers linearized Reed–Solomon codes exactly, while over asymptotically optimal towers of curves it yields families asymptotically better than the sum-rank Gilbert–Varshamov bound (Berardini et al., 2023).

A more explicit $\mathbb F_q$1 algebraic-geometric route uses quadratic Kummer extensions. Two constructions, one built from a totally ramified place and one from a completely split place, produce $\mathbb F_q$2 sum-rank codes with parameters

$\mathbb F_q$3

Their determinant analysis relies on local integral bases and valuation control, and the paper emphasizes that these constructions can have superior code lengths compared with linearized Reed–Solomon codes under equivalent parameter constraints. An elliptic-function-field example gives concrete codes of parameters $\mathbb F_q$4 (Yunlong et al., 23 Jun 2025).

Several combinatorial constructions target small fields and special weights. Binary $\mathbb F_q$5 codes obtained from BCH, Goppa, and additive quaternary codes can exceed the dimensions of sum-rank BCH codes with the same guaranteed minimum distance, and the exact weight formula

$\mathbb F_q$6

explains the refinement over earlier lower bounds (Chen et al., 2023). A geometric study of one-weight codes identifies $\mathbb F_q$7-dimensional $\mathbb F_q$8-linear sum-rank codes with tuples of $\mathbb F_q$9-subspaces of $X_i\in \mathbb F_q^{m\times \eta}$00, proves that doubly-extended linearized Reed–Solomon codes remain MSRD, and shows that when $X_i\in \mathbb F_q^{m\times \eta}$01 they are one-weight. The same work introduces $X_i\in \mathbb F_q^{m\times \eta}$02-simplex codes in the sum-rank metric via Singer subgroup orbits and constructs a new family of one-weight MSRD codes for $X_i\in \mathbb F_q^{m\times \eta}$03, called $X_i\in \mathbb F_q^{m\times \eta}$04-fold linearized Reed–Solomon codes (Neri et al., 2021).

Distance-optimal and quasi-perfect families have also been constructed from covering-code ideas. The covering lift from Hamming metric produces infinite families of quasi-perfect $X_i\in \mathbb F_q^{m\times \eta}$05-ary sum-rank codes with matrix sizes $X_i\in \mathbb F_q^{m\times \eta}$06, explicit binary quasi-perfect codes of matrix size $X_i\in \mathbb F_q^{m\times \eta}$07, distance-optimal sum-rank codes of matrix sizes $X_i\in \mathbb F_q^{m\times \eta}$08 and $X_i\in \mathbb F_q^{m\times \eta}$09 with minimum distance $X_i\in \mathbb F_q^{m\times \eta}$10, and almost-MSRD $X_i\in \mathbb F_q^{m\times \eta}$11 codes of block length up to $X_i\in \mathbb F_q^{m\times \eta}$12 and Singleton defect $X_i\in \mathbb F_q^{m\times \eta}$13 (Liu et al., 9 Jan 2026).

6. Geometry of equivalence, invariants, and current directions

The code equivalence problem in the sum-rank metric has motivated a new invariant theory. In the matrix model $X_i\in \mathbb F_q^{m\times \eta}$14, additive isometries are classified as block permutations followed by blockwise rank-metric isometries. Building on this, the invariant paper introduces $X_i\in \mathbb F_q^{m\times \eta}$15 generalized idealisers

$X_i\in \mathbb F_q^{m\times \eta}$16

the centraliser $X_i\in \mathbb F_q^{m\times \eta}$17, the center $X_i\in \mathbb F_q^{m\times \eta}$18, and a refined notion of $X_i\in \mathbb F_q^{m\times \eta}$19-linearity defined via the left idealiser rather than the ambient coordinate model. The associated nuclear parameters

$X_i\in \mathbb F_q^{m\times \eta}$20

are equivalence invariants. A skew-polynomial realization $X_i\in \mathbb F_q^{m\times \eta}$21 makes these quantities explicitly computable, and the paper uses them to separate the major known MSRD families, including linearized Reed–Solomon, additive twisted linearized Reed–Solomon, and TZ-type twisted linearized Reed–Solomon codes (Santonastaso et al., 5 Jul 2025).

Several open directions are stated explicitly across the literature. The list-decoding capacity work is existential and does not provide explicit constructions or efficient list-decoding algorithms at the capacity radius; it also identifies random $X_i\in \mathbb F_q^{m\times \eta}$22-linear vector sum-rank codes as an unresolved case (Liu et al., 13 Mar 2025). The covering paper identifies explicit formulas for ball intersections $X_i\in \mathbb F_q^{m\times \eta}$23 as the main missing piece for stronger lower bounds (Ott et al., 2022). The algebraic-geometric papers point to the construction of MSRD algebraic-geometric sum-rank codes as a natural next step (Berardini et al., 2023, Yunlong et al., 23 Jun 2025). The covering-code approach to strong Singleton-like bounds leaves open the construction of almost-MSRD families with longer block lengths and larger minimum distance (Liu et al., 9 Jan 2026). Taken together, these directions show that the field has moved beyond the initial definition of the metric toward a mature research program in asymptotic theory, extremal combinatorics, algorithmics, and equivalence classification.