On the Minimum Field Size of Network MDS Codes for Generalized Combination Networks

2026-08-04

AI Summary

This study investigates the minimum field size required to realize network MDS codes in generalized combination networks and Zosin–Khuller networks, with the goal of reducing computational complexity at network nodes. By establishing a unified framework for scalar and vector network MDS codes, the problem of network minimum distance is reformulated in terms of Hamming distance and covering Grassmannian code conditions from classical coding theory. Leveraging greedy constructions and MRD code techniques, the authors derive tight bounds on the required field size. Key contributions include the first demonstration of an equivalence between network MDS codes and classical MDS codes together with covering Grassmannian codes, a proof that vector codes offer no field-size advantage over scalar codes in several network classes, and a necessary and sufficient condition for the existence of vector MDS codes based on hypergraph homomorphisms. The derived bounds significantly improve upon existing general bounds and precisely quantify the MDS performance gap between scalar and vector approaches.

Abstract

This paper investigates the minimum field size required for network maximum distance separable (MDS) codes, a critical parameter affecting computational complexity at network nodes. Focusing on generalized combination networks and Zosin Khuller networks, we develop a systematic framework for both scalar and vector network MDS codes. For scalar codes on generalized combination networks, we establish an equivalence between the minimum distance of network codes and the minimum Hamming distance of classical linear codes, converting network-level MDS constraints into coding theory conditions. This yields necessary and sufficient existence conditions linked to classical MDS codes and covering Grassmannian codes. Using refined greedy constructions and MRD code designs, we obtain improved bounds on the minimum field size, outperforming the prior universal bound. For vector network codes, we develop an analogous distance equivalence and characterize existence via covering Grassmannian codes, yielding bounds on the minimum effective field size. Notably, the gap between optimal scalar and vector MDS codes vanishes for several parameter regimes; we explicitly identify a family of such networks where vector coding offers no field size advantage over scalar coding. For Zosin Khuller networks, we derive lower bounds on the minimal effective field size of vector MDS codes using hypergraph homomorphisms and subset intersection arguments, strictly improving prior scalar bounds. We further provide a necessary and sufficient condition built upon hypergraph homomorphisms for vector MDS construction, yielding an upper bound on the minimum field size. Finally, we bound the MDS gap between optimal scalar and vector solutions.

Problem

• network MDS codes
• minimum field size
• generalized combination networks
• Zosin Khuller networks
• vector network codes

Innovation

• network MDS codes
• minimum field size
• generalized combination networks
• vector network coding
• covering Grassmannian codes