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Function Correcting Codes for Maximally-Unbalanced Boolean Functions

Rajlaxmi Pandey, Shiven Bajpai, Anjana A Mahesh, B. Sundar Rajan

TL;DR

The results show that FCCs with different distance-matrix structures can exhibit markedly different Data BER and function error behavior, and that the influence of code structure depends strongly on the decoding strategy.

Abstract

Function-Correcting Codes (FCCs) enable reliable computation of a function of a $k$-bit message over noisy channels without requiring full message recovery. In this work, we study optimal single-error correcting FCCs (SEFCCs) for maximally-unbalanced Boolean functions, where $k$ denotes the message length and $t$ denotes the error-correction capability. We analyze the structure of optimal SEFCC constructions through their associated codeword distance matrices and identify distinct FCC classes based on this structure. We then examine the impact of these structural differences on error performance by evaluating representative FCCs over the additive white Gaussian noise (AWGN) channel using both soft-decision and hard-decision decoding. The results show that FCCs with different distance-matrix structures can exhibit markedly different Data BER and function error behavior, and that the influence of code structure depends strongly on the decoding strategy.

Function Correcting Codes for Maximally-Unbalanced Boolean Functions

TL;DR

The results show that FCCs with different distance-matrix structures can exhibit markedly different Data BER and function error behavior, and that the influence of code structure depends strongly on the decoding strategy.

Abstract

Function-Correcting Codes (FCCs) enable reliable computation of a function of a -bit message over noisy channels without requiring full message recovery. In this work, we study optimal single-error correcting FCCs (SEFCCs) for maximally-unbalanced Boolean functions, where denotes the message length and denotes the error-correction capability. We analyze the structure of optimal SEFCC constructions through their associated codeword distance matrices and identify distinct FCC classes based on this structure. We then examine the impact of these structural differences on error performance by evaluating representative FCCs over the additive white Gaussian noise (AWGN) channel using both soft-decision and hard-decision decoding. The results show that FCCs with different distance-matrix structures can exhibit markedly different Data BER and function error behavior, and that the influence of code structure depends strongly on the decoding strategy.
Paper Structure (7 sections, 5 theorems, 18 equations, 7 figures, 3 tables)

This paper contains 7 sections, 5 theorems, 18 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notations
  3. Preliminaries
  4. Main Results
  5. Illustrative Example: $k=2$
  6. Grouping of FCCs by DM
  7. Conclusion

Key Result

Theorem 1

For a maximally-unbalanced Boolean function $f: \mathbb{F}_2^k \rightarrow \mathbb{F}_2$, the number of optimal FCCs with single-error correction capability is given by

Figures (7)

  • Figure 1: System model for the function-correcting code under noisy channel.
  • Figure 2: Error performance of FCCs for OR function with $k=2$ under soft-decision decoding.
  • Figure 3: Error performance of FCCs for OR function with $k=2$ under hard-decision decoding.
  • Figure 4: Error performance of FCCs for OR function with $k=3$ for soft decision decoding.
  • Figure 5: Error performance of FCCs for OR function with $k=3$ for hard decision decoding.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Definition 3: Maximally-Unbalanced Boolean Function
  • Definition 4: Codeword Distance Matrix (Codeword DM)
  • Definition 5: Comparable Codeword Distance Matrices
  • Definition 6: Comparable Chain
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • ...and 2 more