Decoding Frequency Permutation Arrays under Infinite norm

TL;DR

This paper studies frequency permutation arrays under the infinity norm, providing bounds, a construction with efficient encoding/decoding, and a locally decodable design useful for private information retrieval.

Contribution

It introduces new bounds and a novel construction for frequency permutation arrays under the infinity norm, including local decoding capabilities.

Findings

Established bounds on FPA size under $\, ext{l}_ ext{infinity}$-norm

Developed an efficient encoding and decoding construction

Demonstrated local decodability for private information retrieval

Abstract

A frequency permutation array (FPA) of length $n=m\lambda$ and distance $d$ is a set of permutations on a multiset over $m$ symbols, where each symbol appears exactly $\lambda$ times and the distance between any two elements in the array is at least $d$. FPA generalizes the notion of permutation array. In this paper, under the distance metric $\ell_\infty$-norm, we first prove lower and upper bounds on the size of FPA. Then we give a construction of FPA with efficient encoding and decoding capabilities. Moreover, we show our design is locally decodable, i.e., we can decode a message bit by reading at most $\lambda+1$ symbols, which has an interesting application for private information retrieval.