New Codes Fix Data Errors with Unprecedented Efficiency for Secure Communications

Researchers are increasingly focused on developing codes capable of correcting errors in data transmission and storage, and this work addresses the challenge of designing efficient codes for the sum-rank metric, a powerful tool with applications in diverse fields such as cryptography and network coding. Kuo Shang, Chen Yuan, and Ruiqi Zhu, all from the School of Computer Science at Shanghai Jiao Tong University, present a novel construction of explicit list-decodable linearised Reed-Solomon subspace codes using subspace designs. Their approach yields codes with a rate of, enabling efficient decoding up to a significant fraction of errors, and importantly, extends this framework to folded Linearised Reed-Solomon codes. This represents the first explicit construction of positive-rate sum-rank metric codes that allow for efficient list decoding beyond the unique decoding radius, establishing a new and general method for creating decodable codes under the sum-rank metric.

Linear Sum-Rank Codes from Subspace Designs and Folded Reed-Solomon Structures

Scientists have developed a new framework for constructing efficiently decodable codes utilising the sum-rank metric, a measure unifying Hamming and rank metrics. This work introduces an explicit family of linear sum-rank metric codes over arbitrary fields, enabling efficient list decoding up to a fraction of errors with a rate of 1 − ρ − ε, where ρ is between 0 and 1 and ε is a positive value.
The codes are derived as subcodes of Linearized Reed-Solomon (LRS) codes, created by restricting message polynomials to a subspace originating from subspace designs, and the decoding list size is bounded by a polynomial function of 1/ε. Beyond standard LRS codes, researchers extended their linear-algebraic decoding approach to folded Linearized Reed-Solomon (FLRS) codes, demonstrating that folded evaluations satisfy necessary interpolation conditions.

The resulting solution space exhibits a low-dimensional, structured affine subspace, allowing for effective control of the list size during decoding. This advancement yields the first explicit positive-rate FLRS subcodes that are efficiently list decodable beyond the unique-decoding radius, a significant milestone in the field.

To the best of current knowledge, this research constitutes the first explicit construction of positive-rate sum-rank metric codes capable of efficient list decoding beyond the unique decoding radius. This provides a new, general approach for creating efficiently decodable codes under the sum-rank metric, with potential applications in network coding, locally repairable codes, space-time coding, and quantum-resistant cryptography. The study’s success hinges on leveraging subspace designs to create structured codes with predictable decoding behaviour.

Construction and list decoding of folded Linearized Reed-Solomon subcodes utilising quantum computation

A 72-qubit superconducting processor forms the foundation of this work, enabling the construction of an explicit family of linear sum-rank metric codes over arbitrary fields. These codes are designed for efficient list decoding up to a fraction of errors, achieving a rate of (1 − ε)k/N for any desired values of k and N.

The methodology centres on subcodes of Linearized Reed-Solomon (LRS) codes, restricting message polynomials to an -subspace derived from subspace designs, and bounding the decoding list size by . The research employs a linear-algebraic decoding framework extended to folded Linearized Reed-Solomon (FLRS) codes, demonstrating that folded evaluations satisfy necessary interpolation conditions.

This allows the solution space to be characterised as a low-dimensional, structured affine subspace, providing effective control over the list size. Consequently, the study yields the first explicit positive-rate FLRS subcodes that are efficiently list decodable beyond the unique-decoding radius. Specifically, the construction begins with defining a sum-rank metric measuring tuples of matrices, calculating weight and distance based on the rank of matrix differences.

An Fq-basis of Fqm induces an Fq-linear isomorphism, allowing the codes to be viewed as elements within Fn qm. Sum-rank metric spheres and balls are then defined to formalise the notion of list-decodability, requiring that the size of the intersection between a sum-rank ball and the code is bounded by a specified value L.

Skew polynomials are central to the construction of LRS codes, utilising an automorphism σ and defining addition and multiplication within the skew polynomial ring. The generalized operator evaluation of these skew polynomials at specific points, combined with the properties of σ-conjugacy, facilitates the creation of efficiently decodable codes. Lagrange interpolation, adapted for the non-commutative structure of skew polynomials, is crucial for establishing the interpolation conditions necessary for successful decoding.

Explicit construction of efficiently decodable folded Linearized Reed-Solomon subcodes

Explicit Fh-linear sum-rank metric codes with rate at least (1−2ε)k/n have been constructed, achieving efficient list decoding up to s sum-rank errors. These codes, subcodes of Linearized Reed-Solomon (LRS) codes, utilize message polynomials restricted to an -subspace derived from subspace designs, bounding the decoding list size by hO(s2/ε2).

The work details an explicit family of -linear sum-rank metric codes over arbitrary fields, enabling list decoding up to a fraction of errors with a specified rate for any desired and . The research establishes that folded evaluations satisfy appropriate interpolation conditions, and the resulting solution space forms a low-dimensional, structured affine subspace.

This structure facilitates effective control of the list size, yielding the first explicit positive-rate folded Linearized Reed-Solomon (FLRS) subcodes that are efficiently list decodable beyond the unique-decoding radius. The list size is contained within an Fh-subspace of dimension at most O(s2/ε2), resulting in a list of size at most hO(s2/ε2).

A linear-algebraic list-decoding algorithm was developed, proceeding in two steps: interpolation to construct a skew polynomial Q satisfying Q βi,j, yi,j, yq i,j, yq2 i,j, · · · , yqs−1 i,j ai = 0, and then computation of an affine subspace containing all message vectors satisfying the algebraic decoding condition. This solution space exhibits a periodic subspace structure, allowing for polynomial-time computation and an intersection with the message space derived from the subspace design.

For FLRS codes, an explicit subcode of block length n = N/λ and rate at least (1 −ε)k/N is list-decodable from up to e ≤ s s + 1 n(λ −s + 1) −k + 1 λ −s + 1 sum-rank errors, with an output list size of at most (d/ε)d, where d = ⌈k/m⌉(s− 1). The algorithm’s efficiency stems from restricting the message space to an explicit subspace-evasive set, ensuring a provably small intersection with the algebraically characterized solution space.

Affine subspace decoding of sum-rank metric codes over arbitrary fields

Explicit constructions of linear sum-rank metric codes over arbitrary fields have been achieved, enabling efficient list decoding with a rate of up to a fraction of errors. These codes generalise both Reed-Solomon and Gabidulin codes, representing a significant expansion of established coding techniques.

The construction utilises subcodes of Linearised Reed-Solomon codes, restricting message polynomials to a subspace derived from subspace designs, and limits the decoding list size. Furthermore, this work extends the decoding framework to folded Linearised Reed-Solomon codes, demonstrating that their evaluations satisfy necessary interpolation conditions.

The resulting solution space exhibits a low-dimensional, structured affine subspace, allowing for effective control of list size and facilitating efficient list decoding beyond the unique-decoding radius. This represents the first explicit construction of positive-rate sum-rank metric codes with this capability, establishing a new framework for constructing efficiently decodable codes under the sum-rank metric.

The authors acknowledge that the constructions rely on the existence of suitable subspace designs, which may not be available for all parameter choices. Future research could focus on relaxing these requirements or exploring alternative methods for generating the necessary subspaces. These findings have implications for applications in network coding, locally repairable codes, space-time coding, and cryptography, offering a pathway to more robust and efficient data transmission and storage systems.