Linearized Reed-Solomon Codes with Support-Constrained Generator Matrix and Applications in Multi-Source Network Coding

TL;DR

The paper investigates linearized Reed-Solomon (LRS) MSRD codes under the sum-rank metric with support-constrained generator matrices. It proves that the GM-MDS-type condition $|igcap_{i\in\Omega} Z_i|+| abla\Omega|\le k$ is necessary and sufficient for the existence of MSRD codes with a given zero pattern, and it provides explicit field-size bounds $q \ge \ell+1$ and $m \ge \max_{l}\{ k-1+\log_q k, n_l \}$. When the constraint is not satisfied, the paper shows how the largest possible sum-rank distance can be achieved via subcodes of LRS codes with sufficiently large field sizes. As an application, it demonstrates how these results enable the design of distributed LRS codes for distributed multi-source network coding through an integer-programming framework that enforces the support constraints while guaranteeing decodability.

Abstract

Linearized Reed-Solomon (LRS) codes are evaluation codes based on skew polynomials. They achieve the Singleton bound in the sum-rank metric and therefore are known as maximum sum-rank distance (MSRD) codes. In this work, we give necessary and sufficient conditions for the existence of MSRD codes with a support-constrained generator matrix. The conditions on the support constraints are identical to those for MDS codes and MRD codes. The required field size for an $[n,k]_{q^m}$ LRS codes with support-constrained generator matrix is $q\geq \ell+1$ and $m\geq \max_{l\in[\ell]}\{k-1+\log_qk, n_l\}$, where $\ell$ is the number of blocks and $n_l$ is the size of the $l$-th block. The special cases of the result coincide with the known results for Reed-Solomon codes and Gabidulin codes. For the support constraints that do not satisfy the necessary conditions, we derive the maximum sum-rank distance of a code whose generator matrix fulfills the constraints. Such a code can be constructed from a subcode of an LRS code with a sufficiently large field size. Moreover, as an application in network coding, the conditions can be used as constraints in an integer programming problem to design distributed LRS codes for a distributed multi-source network.

Figures (4)

• Figure 1: Illustration of the distributed multi-source network model.
• Figure 2: Illustration of the support-constrained generator matrix of the $[23,9,15]_{4^9}$ LRS code for the distributed multi-source network.
• Figure 3: Proof logic for $\ref{['item:equiv2']}\implies\ref{['item:equiv1']}$ with initial hypothesis \ref{['H:fromPre']} and \ref{['H:fromHere']}.
• Figure 4: Illustration of the induction for \ref{['item:equiv2']}$\implies$\ref{['item:equiv1']} under difference cases.

Theorems & Definitions (38)

• Theorem 1: GM--MDS Condition yildiz2018optimumlovett2018mds
• Theorem 2: GM--MRD Condition yildiz2019gabidulin
• Proposition 1
• proof
• Proposition 2
• Conjecture 1
• Lemma 1
• proof
• Definition 1
• Lemma 2: ott2022covering