Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Length of Functional Batch and PIR Codes

Altan B. Kilic, Alberto Ravagnani, Flavio Salizzoni

Abstract

We consider the problem of computing the minimum length of functional batch and PIR codes of fixed dimension and for a fixed list size, over an arbitrary finite field. We recover, generalize, and refine several results that were previously obtained for binary codes. We present new upper and lower bounds for the minimum length, and discuss the asymptotic behaviour of this parameter. We also compute its value for several parameter sets. The paper also offers insights into the "correct" list size to consider for the Functional Batch Conjecture over non-binary finite fields, and establishes various supporting results.

The Length of Functional Batch and PIR Codes

Abstract

We consider the problem of computing the minimum length of functional batch and PIR codes of fixed dimension and for a fixed list size, over an arbitrary finite field. We recover, generalize, and refine several results that were previously obtained for binary codes. We present new upper and lower bounds for the minimum length, and discuss the asymptotic behaviour of this parameter. We also compute its value for several parameter sets. The paper also offers insights into the "correct" list size to consider for the Functional Batch Conjecture over non-binary finite fields, and establishes various supporting results.

Paper Structure

This paper contains 7 sections, 31 theorems, 76 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Problem statement and preliminaries
  3. Exact values
  4. Length of 2-dimensional FB codes
  5. Explicit computations for other special parameters
  6. Bounds and field size
  7. Asymptotic behaviour

Key Result

Proposition 2.5

Let $G \in \mathbb F_q^{k\times k}$ be any invertible matrix. We have that $M$ achieves $\textnormal{FB}(k,t,q)$, or $\textnormal{FP}(k,t,q)$, if and only if $GM$ does.

Theorems & Definitions (61)

  • Conjecture 1.1: Functional Batch Conjecture
  • Conjecture 1.2: see balister2011coloring
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • ...and 51 more