Function-Correcting Codes for Sum-Rank Metric

Abstract: Function-Correcting Codes (FCCs) are a class of codes designed to protect the evaluation of a specific function of a message against channel errors at a higher level than the level of protection for the message, while requiring significantly less redundancy than conventional error-correcting codes. In this paper, we study function-correcting codes under the sum-rank metric, which is a natural generalization of both the Hamming metric and the rank-metric and also we derive general upper and lower bounds on the optimal redundancy of FCCs in the sum-rank metric. In particular, we establish a Plotkin-like bound for irregular-distance codes in sum-rank metric. Furthermore, we present explicit construction of function-correcting sum-rank metric codes (FCSRCs) for locally binary functions with optimal redundancy.

• The paper introduces a unified framework for function-correcting codes in the sum-rank metric, deriving new upper and lower redundancy bounds including a Plotkin-like limit.
• It presents explicit constructions for locally binary FCCs and sum-rank weight functions that achieve optimal redundancy by leveraging MSRD codes.
• The work significantly impacts network coding and distributed storage by reducing redundancy while ensuring reliable partial function recovery under channel errors.

Function-Correcting Codes for the Sum-Rank Metric: A Technical Analysis

Introduction

The paper "Function-Correcting Codes for Sum-Rank Metric" (2607.03857) extends the concept of function-correcting codes (FCCs) to the sum-rank metric, yielding a unified theoretical and constructive framework of relevance for multi-shot network coding and distributed storage systems. The sum-rank metric generalizes the Hamming and rank metrics, allowing error analysis and code design applicable to both scenarios in a single algebraic formalism. This work contributes new upper and lower bounds on redundancy for FCCs in this metric, derives a Plotkin-like bound for irregular-distance codes, and provides explicit constructions of FCCs for sum-rank locally binary functions with optimal redundancy.

Sum-Rank Metric Codes and Function-Correcting Codes

The sum-rank metric, as formalized in recent works (Gorla et al., 2023), amalgamates the Hamming and rank metrics. It defines the distance on collections of block matrices as the sum of ranks across blocks. This is especially applicable for network coding in multi-shot or subpacketized storage regimes, where errors may manifest in either coordinate or matrix rank.

Function-correcting codes, originally formulated for the Hamming metric, are designed to guarantee reliable recovery of a specified function of the message, not just the message itself, under channel errors, while requiring less redundancy than traditional ECCs. If the function is bijective, an FCC coincides with a classical ECC; for non-bijective functions, significant reductions in redundancy can be achieved.

Extending FCCs to the sum-rank metric generalizes prior results for classical and rank-metric cases, including the study of maximum sum-rank distance codes (MSRD) [(Gorla et al., 2023), 67(10):6456–6475], maximizing minimum distance under the sum-rank metric, and relevant for network-coded storage.

Core Technical Contributions

Two principal directions are pursued:

1. Bounds on Redundancy: The work provides general upper and lower bounds for the minimum redundancy required by an FCC in the sum-rank metric, specifically adapting and extending Plotkin-type combinatorial bounds to irregular-distance codes. The bounds subsume the Hamming and rank-metric cases as special cases.
2. Explicit Constructions: The paper gives explicit, optimal constructions of function-correcting codes for locally binary functions in the sum-rank setting. These constructions are shown to attain the lower bound on redundancy, thus demonstrating tightness for these function classes.

Plotkin-Like Bound for Irregular Sum-Rank Distance Codes

A key theoretical result is a Plotkin-like bound on the minimum required block length (and thus redundancy) of an irregular-distance code under the sum-rank metric. For a given distance requirement matrix $\mathbf{D}$ and code cardinality $M$, the bound is:

$$N_{srk}(\mathbf{D}) \geq \frac{2q^m}{m(M^2(q^m-1) - a(q^m - a))} \sum_{1 \le i < j \le M} [\mathbf{D}]_{i, j}$$

where $a = M \bmod q^m$. This bound reduces to classical Plotkin bounds in the appropriate metric limits (Hamming for $m=1$, rank-metric for $k=1$).

FCCs for Locally Binary and Sum-Rank Weight Functions

For $2t$-sum-rank locally binary functions (i.e., functions whose value does not change for more than two "nearby" messages within sum-rank distance $2t$), the authors construct FCCs with redundancy exactly $\lceil 2t/m \rceil$. This is achieved by encoding the distinguishing function outcome using repeated blocks of identity or zero matrices, ensuring that any error pattern of up to $t$ sum-rank can be resolved with the minimal number of redundant blocks.

For sum-rank weight functions ($M$0), an explicit construction leveraging MSRD codes assigns parity in a way that uniquely identifies the function value, again with redundancy $M$1 under mild field size constraints.

Notable Numerical and Structural Results

• Optimality for Locally Binary Functions: For any sum-rank locally binary function, the minimal redundancy to correct $M$2 errors is $M$3, matching the corollary lower bound $M$4.
• Plotkin Bound Generalization: The sum-rank Plotkin-like bound collapses to the known variants for Hamming and rank-metric cases, and is applicable for arbitrary irregular distance requirement matrices.
• Explicit Code Construction: The paper provides code constructions for both locally binary and weight functions that not only meet the bound but also admit a simple, explicit encoding and decoding structure.

Broader Implications and Future Directions

Extending FCCs to the sum-rank metric has several practical and theoretical implications. In network coding or distributed storage, the function of interest may be a partial attribute (e.g., a checksum, parity function, or other invariant) rather than the full message, making highly efficient FCCs attractive for bandwidth- and redundancy-constrained environments. The ability to minimize redundancy with proofs of optimality directly impacts storage overhead and network throughput.

The techniques adapted for the sum-rank metric can inform the analysis and design of codes for other intermediate or composite metrics pertinent to multi-shot and subpacketized regimes. The general irregular-distance code analysis paves the way for more refined bounds and potentially universally optimal constructions for classes of functions beyond locally binary or weight.

Theoretically, these results enrich understanding of the tradeoffs between code length, function complexity, channel error parameters, and achievable redundancy in non-classical coding scenarios. They highlight the value of matrix-analytic and algebraic techniques in approaching composite metrics.

Future research might consider:

• Generalizing explicit constructions to wider classes of functions (e.g., polynomial evaluations, modular reductions, or multilinear maps)
• Further tightening of upper and lower redundancy bounds for functions not admitting simple locality properties
• Efficient decoding algorithms and complexity analysis tailored to sum-rank FCCs
• Connections to secure computation or private information retrieval in network-coded settings

Conclusion

This paper provides a rigorous extension of function-correcting code theory to the sum-rank metric, encompassing both foundational combinatorial bounds and explicit constructions for function classes relevant in modern coding applications. The generalization of Plotkin bounds and demonstration of optimal redundancy for certain function types constitute significant technical contributions. This work sets the stage for further exploration of FCCs under generalized metrics, with evident applications in network coding, distributed storage, and beyond.