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PIR Codes, Unequal-Data-Demand Codes, and the Griesmer Bound

Henk D. L. Hollmann, Martin Puškin, Ago-Erik Riet

TL;DR

The paper addresses generalizing the Griesmer bound to Unequal Data Demand (UDD) PIR codes by linking PIR codes to Unequal-Error-Protection (UEP) codes via a separation vector, and by formulating an Integer Linear Programming (ILP) bound mu(T) that bounds the minimum length. It provides two complementary proofs: a distance-based bound derived from the separation vector and a sharper ILP-based bound, showing that both yield Griesmer-type results for linear codes, linear UEP codes, and linear UDD-PIR codes. Moreover, the ILP framework unifies the proofs of these bounds, offering a uniform approach to deriving Griesmer bounds across different code families. The work advances the theory of private information retrieval in settings with unequal data demand and provides constructive tools (via the ILP) for assessing and achieving length-minimal code constructions.

Abstract

Unequal Error-Protecting (UEP) codes are error-correcting (EC) codes designed to protect some parts of the encoded data better than other parts. Here, we introduce a similar generalization of PIR codes that we call Unequal-Data-Demand (UDD) PIR codes. These codes are PIR-type codes designed for the scenario where some parts of the encoded data are in higher demand than other parts. We generalize various results for PIR codes to UDD codes. Our main contribution is a new approach to the Griesmer bound for linear EC codes involving an Integer Linear Programming (ILP) problem that generalizes to linear UEP codes and linear UDD PIR codes.

PIR Codes, Unequal-Data-Demand Codes, and the Griesmer Bound

TL;DR

The paper addresses generalizing the Griesmer bound to Unequal Data Demand (UDD) PIR codes by linking PIR codes to Unequal-Error-Protection (UEP) codes via a separation vector, and by formulating an Integer Linear Programming (ILP) bound mu(T) that bounds the minimum length. It provides two complementary proofs: a distance-based bound derived from the separation vector and a sharper ILP-based bound, showing that both yield Griesmer-type results for linear codes, linear UEP codes, and linear UDD-PIR codes. Moreover, the ILP framework unifies the proofs of these bounds, offering a uniform approach to deriving Griesmer bounds across different code families. The work advances the theory of private information retrieval in settings with unequal data demand and provides constructive tools (via the ILP) for assessing and achieving length-minimal code constructions.

Abstract

Unequal Error-Protecting (UEP) codes are error-correcting (EC) codes designed to protect some parts of the encoded data better than other parts. Here, we introduce a similar generalization of PIR codes that we call Unequal-Data-Demand (UDD) PIR codes. These codes are PIR-type codes designed for the scenario where some parts of the encoded data are in higher demand than other parts. We generalize various results for PIR codes to UDD codes. Our main contribution is a new approach to the Griesmer bound for linear EC codes involving an Integer Linear Programming (ILP) problem that generalizes to linear UEP codes and linear UDD PIR codes.
Paper Structure (11 sections, 5 theorems, 23 equations)

This paper contains 11 sections, 5 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Error-correcting codes
  3. PIR codes
  4. Unequal-Error-Protection codes
  5. Unequal-Data-Demand codes
  6. The Griesmer bound for UEP PIR codes
  7. An ILP problem related to PIR codes
  8. A sharper lower bound for the ILP problem
  9. The Griesmer bound for linear codes and for linear UEP-codes from the ILP problem
  10. Open problems
  11. Conclusions

Key Result

Theorem 6.1

Let $C$ be an $(n,q^k,d)_q$-code with encoder $\epsilon: \mathbb{F}_q^k\rightarrow \mathbb{F}_q^n$, and let $\epsilon$ have separation vector $\boldsymbol{s}(\epsilon)$. If $\epsilon$ is an UDD $T$-PIR code, where $T=(t_1, \ldots, t_{k})$ and $t_1\geq t_2\geq \cdots \geq t_{k}$, then $s_j(\epsilon)\

Theorems & Definitions (16)

  • Definition 3.1
  • Example 3.2
  • Example 4.1
  • Definition 5.1
  • Example 5.2
  • Theorem 6.1
  • proof
  • Theorem 6.2: Griesmer for UDD PIR codes
  • proof
  • Theorem 7.1
  • ...and 6 more