Function Correcting Codes for Maximally-Unbalanced Boolean Functions

TL;DR

This paper investigates optimal single-error correcting function-correcting codes for maximally-unbalanced Boolean functions, analyzing their structure and performance over noisy channels with different decoding strategies.

Contribution

It characterizes the structure of optimal FCCs via their codeword distance matrices and evaluates how these structures affect error performance under various decoding methods.

Findings

FCC structure significantly influences error performance

Different FCC classes show distinct BER and function error behaviors

Decoding strategy impacts the effectiveness of FCC structures

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.