Codes with Unequal Locality

TL;DR

This paper introduces codes with unequal locality, providing bounds on their minimum distance, adapting existing constructions to achieve these bounds, and proposing algorithms for locality assignment to optimize code recoverability.

Contribution

It generalizes the concept of locality in codes by allowing different symbols to have different localities, deriving bounds, and adapting constructions for optimal minimum distance.

Findings

Derived tight upper bounds on minimum distance for unequal locality codes.

Adapted Pyramid and rank-metric code constructions to achieve bounds.

Proposed a greedy algorithm for locality assignment to maximize minimum distance.

Abstract

For a code $\code$, its $i$-th symbol is said to have locality $r$ if its value can be recovered by accessing some other $r$ symbols of $\code$. Locally repairable codes (LRCs) are the family of codes such that every symbol has locality $r$. In this paper, we focus on (linear) codes whose individual symbols can be partitioned into disjoint subsets such that the symbols in one subset have different locality than the symbols in other. We call such codes as "codes with unequal locality". For codes with "unequal information locality", we compute a tight upper bound on the minimum distance as a function of number of information symbols of each locality. We demonstrate that the construction of Pyramid codes can be adapted to design codes with unequal information locality that achieve the minimum distance bound. This result generalizes the classical result of Gopalan et al. for codes with unequal locality. Next, we consider codes with "unequal all symbol locality", and establish an upper bound on the minimum distance as a function of number of symbols of each locality. We show that the construction based on rank-metric codes by Silberstein et al. can be adapted to obtain codes with unequal all symbol locality that achieve the minimum distance bound. Finally, we introduce the concept of "locality requirement" on a code, which can be viewed as a recoverability requirement on symbols. Information locality requirement on a code essentially specifies the minimum number of information symbols of different localities that must be present in the code. We present a greedy algorithm that assigns localities to information symbols so as to maximize the minimum distance among all codes that satisfy a given locality requirement.