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Optimal Functional $2^{s-1}$-Batch Codes: Exploring New Sufficient Conditions

Lev Yohananov, Isaac Barouch Essayag

TL;DR

The paper addresses the problem of optimal functional $2^{s-1}$-batch codes and the conjecture that the minimum number of servers is $2^s-1$. It introduces a polynomial-algebraic framework that uses all vectors in $GF(2)^s$ (plus a virtual zero-server) as servers and reduces recovery to finding two-server pairs via the multivariate polynomial $f_v$, studying solvability through a ring modulo $x_i^q+x_i$ and Alon's Nullstellensatz-style results. The authors derive equivalences between solvability and algebraic conditions on $f_v$, establish several sufficient conditions, and explicitly solve the $s=2$ and $s=3$ cases, with computer-assisted avenues suggested for larger $s$. These results advance understanding of when $FB(s,2^{s-1})=2^s-1$ and provide a concrete algebraic toolkit for constructing optimal functional batch codes with two-server recovery.

Abstract

A functional $k$-batch code of dimension $s$ consists of $n$ servers storing linear combinations of $s$ linearly independent information bits. These codes are designed to recover any multiset of $k$ requests, each being a linear combination of the information bits, by $k$ disjoint subsets of servers. A recent conjecture suggests that for any set of $k = 2^{s-1}$ requests, the optimal solution requires $2^s-1$ servers. This paper shows that the problem of functional $k$-batch codes is equivalent to several other problems. Using these equivalences, we derive sufficient conditions that improve understanding of the problem and enhance the ability to find the optimal solution.

Optimal Functional $2^{s-1}$-Batch Codes: Exploring New Sufficient Conditions

TL;DR

The paper addresses the problem of optimal functional -batch codes and the conjecture that the minimum number of servers is . It introduces a polynomial-algebraic framework that uses all vectors in (plus a virtual zero-server) as servers and reduces recovery to finding two-server pairs via the multivariate polynomial , studying solvability through a ring modulo and Alon's Nullstellensatz-style results. The authors derive equivalences between solvability and algebraic conditions on , establish several sufficient conditions, and explicitly solve the and cases, with computer-assisted avenues suggested for larger . These results advance understanding of when and provide a concrete algebraic toolkit for constructing optimal functional batch codes with two-server recovery.

Abstract

A functional -batch code of dimension consists of servers storing linear combinations of linearly independent information bits. These codes are designed to recover any multiset of requests, each being a linear combination of the information bits, by disjoint subsets of servers. A recent conjecture suggests that for any set of requests, the optimal solution requires servers. This paper shows that the problem of functional -batch codes is equivalent to several other problems. Using these equivalences, we derive sufficient conditions that improve understanding of the problem and enhance the ability to find the optimal solution.
Paper Structure (5 sections, 11 theorems, 50 equations)

This paper contains 5 sections, 11 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Analysis of $f_v$
  4. The $s=2$ and $s=3$ Cases
  5. Conclusion

Key Result

Theorem 3

Let $\mathbb{F}$ be an arbitrary field, and let $f = f(x_1, \dots , x_n)$ be a polynomial in $\mathbb{F}[x_1, \dots, x_n]$. Let $S_1, \dots , S_n$ be nonempty subsets of $\mathbb{F}$ and define $g_i(x_i) = \prod_{s \in S_i} (x_i - s)$. If $f$ vanishes over all the common zeros of $g_1, \dots , g_n$ Moreover, if $f, g_1, \dots, g_n$ lie in $R[x_1, \dots , x_n]$ for some subring $R$ of $\mathbb{F}$

Theorems & Definitions (31)

  • Conjecture 1
  • Conjecture 2
  • Theorem 3
  • Theorem 4
  • proof
  • Example 1
  • proof
  • Remark 1
  • Theorem 5
  • Theorem 6
  • ...and 21 more