Extended Integrated Interleaved Codes over any Field with Applications to Locally Recoverable Codes

TL;DR

This paper introduces a new, more general definition of Extended Integrated Interleaved (EII) codes that can be constructed over any field, including binary fields, enhancing their applicability in locally recoverable codes with improved properties.

Contribution

A comprehensive new definition of EII codes over arbitrary fields, expanding their applicability and generalizing previous definitions, along with improved constructions and decoding algorithms.

Findings

EII codes can be constructed over any field, including GF(2).

New constructions meet upper bounds on minimum distance.

An iterative decoding algorithm for EII codes is developed.

Abstract

Integrated Interleaved (II) and Extended Integrated Interleaved (EII) codes are a versatile alternative for Locally Recoverable (LRC) codes, since they require fields of relatively small size. II and EII codes are generally defined over Reed-Solomon type of codes. A new comprehensive definition of EII codes is presented, allowing for EII codes over any field, and in particular, over the binary field $GF(2)$. The traditional definition of II and EII codes is shown to be a special case of the new definition. Improvements over previous constructions of LRC codes, in particular, for binary codes, are given, as well as cases meeting an upper bound on the minimum distance. Properties of the codes are presented as well, in particular, an iterative decoding algorithm on rows and columns generalizing the iterative decoding algorithm of product codes. Two applications are also discussed: one is finding a systematic encoding of EII codes such that the parity symbols have a balanced distribution on rows, and the other is the problem of ordering the symbols of an EII code such that the maximum length of a correctable burst is achieved.