Almost Optimal Construction of Functional Batch Codes Using Hadamard Codes

TL;DR

This paper presents new constructions of functional batch codes using Hadamard codes, reducing the gap towards the conjectured optimal number of servers for given request sizes and providing near-optimal solutions for various parameters.

Contribution

The paper introduces improved constructions of functional batch codes based on Hadamard codes, narrowing the gap towards the conjectured optimal server count for specific request sizes.

Findings

Existence of codes supporting approximately 5/6 of the maximum requests with minimal servers.

Construction of codes with 2^{s+1}-2 servers supporting 2^{s} requests, achieving optimality.

New bounds and constructions that approach the conjectured optimal server count for various parameters.

Abstract

A \textit{functional $k$-batch} code of dimension $s$ consists of $n$ servers storing linear combinations of $s$ linearly independent information bits. Any multiset request of size $k$ of linear combinations (or requests) of the information bits can be recovered by $k$ disjoint subsets of the servers. The goal under this paradigm is to find the minimum number of servers for given values of $s$ and $k$. A recent conjecture states that for any $k=2^{s-1}$ requests the optimal solution requires $2^s-1$ servers. This conjecture is verified for $s\leq 5$ but previous work could only show that codes with $n=2^s-1$ servers can support a solution for $k=2^{s-2} + 2^{s-4} + \left\lfloor \frac{ 2^{s/2}}{\sqrt{24}} \right\rfloor$ requests. This paper reduces this gap and shows the existence of codes for $k=\lfloor \frac{5}{6}2^{s-1} \rfloor - s$ requests with the same number of servers. Another construction in the paper provides a code with $n=2^{s+1}-2$ servers and $k=2^{s}$ requests, which is an optimal result.These constructions are mainly based on Hadamard codes and equivalently provide constructions for \textit{parallel Random I/O (RIO)} codes.