Multiple-Layer Integrated Interleaved Codes: A Class of Hierarchical Locally Recoverable Codes

TL;DR

This paper introduces a recursive, multi-layer construction of Extended Integrated Interleaved (EII) codes that offers hierarchical localities, improved erasure correction, and low-density parity-check matrices, enhancing locally recoverable code design.

Contribution

It presents a novel multi-layer EII code construction with hierarchical locality, broadening the class of codes beyond Reed-Solomon-based II codes.

Findings

Codes have hierarchical localities suitable for LRC applications.

The codes exhibit strong erasure-correcting capabilities.

Parity-check matrices are low density, facilitating efficient decoding.

Abstract

The traditional definition of Integrated Interleaved (II) codes generally assumes that the component nested codes are either Reed-Solomon (RS) or shortened Reed-Solomon codes. By taking general classes of codes, we present a recursive construction of Extended Integrated Interleaved (EII) codes into multiple layers, a problem that brought attention in literature for II codes. The multiple layer approach allows for a hierarchical scheme where each layer of the code provides for a different locality. In particular, we present the erasure-correcting capability of the new codes and we show that they are ideally suited as Locally Recoverable (LRC) codes due to their hierarchical locality and the small finite field required by the construction. Properties of the multiple layer EII codes, like their minimum distance and dimension, as well as their erasure decoding algorithms, parity-check matrices and performance analysis, are provided and illustrated with examples. Finally, we will observe that the parity-check matrices of high layer EII codes have low density.