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A Note on Function Correcting Codes for b-Symbol Read Channels

Sachin Sampath, B. Sundar Rajan

TL;DR

This work extends Function-Correcting Codes to $b$-symbol read channels for linear functions, proposing a Plotkin-like lower bound on the optimal redundancy $r_b^f(k,t)$ that depends on the function's rank $l$ and the kernel weight sum $s$. The bound specializes to the symbol-pair case when $b=2$ and recovers the classical ECC Plotkin bounds in the bijective (invertible) case, thereby unifying function-protection with traditional error-control limits. By leveraging coset structure and $b$-symbol distances, the results provide precise redundancy guarantees and guidance for designing FCCs in multi-symbol read channels. The findings have practical impact on efficient protection of function evaluations in channels with $b$-symbol reads, bridging FCC theory with established ECC bounds. In particular, the work offers concrete lower bounds for linear functions that inform code construction and performance trade-offs in such channels.

Abstract

Function-Correcting Codes (FCCs) is a novel paradigm in Error Control Coding introduced by Lenz et. al. 2023 for the binary substitution channel \cite{FCC}. FCCs aim to protect the function evaluation of data against errors instead of the data itself, thereby relaxing the redundancy requirements of the code. Later R. Premlal et. al. \cite{LFCC} gave new bounds on the optimal redundancy of FCCs and also extensively studied FCCs for linear functions. The notion of FCCs has also been extended to different channels such as symbol-pair read channel over the binary field by Xia et. al. \cite{FCSPC} and b-symbol read channel over finite fields by A.Singh et. al. \cite{FCBSC} In this work, we study FCCs for linear functions for the b-symbol read channel. We provide the Plotkin-like bound on FCCs for b-symbol read channel which reduces to a Plotkin-like bound for FCCs for the symbol-pair read channel when $b$=2. FCCs reduce to classical Error Correcting Codes (ECCs) when the function is bijective. Analogous to this our bound reduces to the Plotkin-bound for classical ECCS for both the b-symbol and symbol-pair read channels \cite{Plotkin-b-symbol, Plotkin-symbol-pair} when we consider linear bijective functions.

A Note on Function Correcting Codes for b-Symbol Read Channels

TL;DR

This work extends Function-Correcting Codes to -symbol read channels for linear functions, proposing a Plotkin-like lower bound on the optimal redundancy that depends on the function's rank and the kernel weight sum . The bound specializes to the symbol-pair case when and recovers the classical ECC Plotkin bounds in the bijective (invertible) case, thereby unifying function-protection with traditional error-control limits. By leveraging coset structure and -symbol distances, the results provide precise redundancy guarantees and guidance for designing FCCs in multi-symbol read channels. The findings have practical impact on efficient protection of function evaluations in channels with -symbol reads, bridging FCC theory with established ECC bounds. In particular, the work offers concrete lower bounds for linear functions that inform code construction and performance trade-offs in such channels.

Abstract

Function-Correcting Codes (FCCs) is a novel paradigm in Error Control Coding introduced by Lenz et. al. 2023 for the binary substitution channel \cite{FCC}. FCCs aim to protect the function evaluation of data against errors instead of the data itself, thereby relaxing the redundancy requirements of the code. Later R. Premlal et. al. \cite{LFCC} gave new bounds on the optimal redundancy of FCCs and also extensively studied FCCs for linear functions. The notion of FCCs has also been extended to different channels such as symbol-pair read channel over the binary field by Xia et. al. \cite{FCSPC} and b-symbol read channel over finite fields by A.Singh et. al. \cite{FCBSC} In this work, we study FCCs for linear functions for the b-symbol read channel. We provide the Plotkin-like bound on FCCs for b-symbol read channel which reduces to a Plotkin-like bound for FCCs for the symbol-pair read channel when =2. FCCs reduce to classical Error Correcting Codes (ECCs) when the function is bijective. Analogous to this our bound reduces to the Plotkin-bound for classical ECCS for both the b-symbol and symbol-pair read channels \cite{Plotkin-b-symbol, Plotkin-symbol-pair} when we consider linear bijective functions.

Paper Structure

This paper contains 7 sections, 2 theorems, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. FCBSCs for Linear Functions
  5. Plotkin-Bound for FCBSCs for linear functions
  6. Special Cases
  7. Recovering Plotkin-like-Bounds for ECCs

Key Result

Theorem 1

FCBSC For any function $f: \mathbb{F}^k_q \rightarrow Im(f)$ and $\{\boldsymbol{u_1}, \ldots, \boldsymbol{u_{q^k}}\} = \mathbb{F}^k_q$, we have

Figures (1)

  • Figure 1: Illustration of the FCBSC setup. The input to the encoder is a message $\boldsymbol{u}$, which features an attribute $f(\boldsymbol{u})$ that is of special interest to receiver. To guarantee recoverability of this attribute, transmitter encodes the message $\boldsymbol{u}$ to a redundancy vector $\boldsymbol{p(u)}$ and transmits $(\boldsymbol{u},\boldsymbol{p(u)})$. Given an erroneous version $\boldsymbol{y}$ of the codeword $\boldsymbol{c}=(\boldsymbol{u},\boldsymbol{p})$ and the knowledge of the function $f$, the decoder can correctly infer $f(\boldsymbol{u})$.

Theorems & Definitions (6)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Corollary 1
  • proof