TL;DR
This paper introduces Johnson graph codes, a new class of codes based on Johnson graphs, and explores their properties and applications in distributed storage, including a comparison with Hamming graph codes and Reed-Muller codes.
Contribution
The paper defines Johnson graph codes, provides their construction and properties, and demonstrates their use in layered code concatenation for distributed storage systems.
Findings
Johnson graph codes are uniquely determined by local neighborhood restrictions.
These codes can be constructed with specific properties suitable for distributed storage.
They relate to and differ from Reed-Muller codes, especially on the hypercube.
Abstract
We define a Johnson graph code as a subspace of labelings of the vertices in a Johnson graph with the property that labelings are uniquely determined by their restriction to vertex neighborhoods specified by the parameters of the code. We give a construction and main properties for the codes and show their role in the concatenation of layered codes that are used in distributed storage systems. A similar class of codes for the Hamming graph is discussed in an appendix. Codes of the latter type are in general different from affine Reed-Muller codes, but for the special case of the hypercube they agree with binary Reed-Muller codes.
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