Improved batch code lower bounds

TL;DR

This paper establishes a new lower bound on the redundancy of linear batch codes, showing they require at least on the order of the square root of the product of code length and request size, advancing understanding of their limitations.

Contribution

It introduces a novel lower bound of a(\u221a{Nk}) on the redundancy of linear batch codes, improving previous bounds and aiding in determining optimal code parameters.

Findings

Proves a lower bound of a(a(a{Nk})) on batch code redundancy.

Improves previous bounds of a(a(a{N}+k)).

Analyzes the tensor dimension of the code's dual to derive bounds.

Abstract

Batch codes are a useful notion of locality for error correcting codes, originally introduced in the context of distributed storage and cryptography. Many constructions of batch codes have been given, but few lower bound (limitation) results are known, leaving gaps between the best known constructions and best known lower bounds. Towards determining the optimal redundancy of batch codes, we prove a new lower bound on the redundancy of batch codes. Specifically, we study (primitive, multiset) linear batch codes that systematically encode $n$ information symbols into $N$ codeword symbols, with the requirement that any multiset of $k$ symbol requests can be obtained in disjoint ways. We show that such batch codes need $\Omega(\sqrt{Nk})$ symbols of redundancy, improving on the previous best lower bounds of $\Omega(\sqrt{N}+k)$ at all $k=n^\varepsilon$ with $\varepsilon\in(0,1)$. Our proof follows from analyzing the dimension of the order-$O(k)$ tensor of the batch code's dual code.