New constructions of MSRD codes

TL;DR

The work tackles the construction of maximum sum-rank distance (MSRD) codes under the sum-rank metric, seeking explicit codes for parameter regimes that admit heterogeneous block sizes across positions.It proposes four construction paradigms: a Cartesian-product variant with faster decoding, a basis-combining method, lattices of MSRD codes, and a construction using systematic MSRD codes augmented by tail blocks via a structured isomorphism.These methods collectively yield MSRD codes for new parameter sets not reachable by prior constructions, while retaining compatibility with or generalization of known families such as linearized Reed--Solomon codes; the results also provide practical decoding advantages in certain regimes.Overall, the paper broadens the landscape of explicit MSRD constructions, enabling longer codes and flexible block-size configurations that are suitable for multishot network coding and related applications.

Abstract

In this work, we provide four methods for constructing new maximum sum-rank distance (MSRD) codes. The first method, a variant of cartesian products, allows faster decoding than known MSRD codes of the same parameters. The other three methods allow us to extend or modify existing MSRD codes in order to obtain new explicit MSRD codes for sets of matrix sizes (numbers of rows and columns in different blocks) that were not attainable by previous constructions. In this way, we show that MSRD codes exist (by giving explicit constructions) for new ranges of parameters, in particular with different numbers of rows and columns at different positions.

Theorems & Definitions (18)

• Lemma 1
• Theorem 1
• proof
• Lemma 2
• Theorem 2
• proof
• Theorem 3
• proof
• Proposition 3
• Theorem 4