On Function-Correcting Codes

TL;DR

This work extends the function-correcting codes (FCC) framework by introducing a function-dependent graph to bound redundancy and by specializing to bijective functions to obtain bounds for systematic ECCs. It derives a Cartesian-product based lower bound on FCC redundancy, showing nontrivial performance in the regime $2t+1 \le k$ and matching the single-error $ZLL$ bound, while remaining competitive with the $BGS$ bound in many parameter regimes. For linear functions, the authors obtain a Plotkin-type bound expressed through the kernel weight distribution, uncover a recursive block-circulant structure in the FCC graph, and employ spectral methods to bound redundancy. They identify function classes where coset-wise coding is optimal or reduces the problem to a lower-dimensional classical ECC, and provide constructive schemes for these cases. The results illuminate when FCC-inspired bounds approach classical ECC limits and lay groundwork for tighter bounds and efficient constructions, particularly for linear function corrections.

Abstract

Function-correcting codes were introduced in the work "Function-Correcting Codes" (FCC) by Lenz et al. 2023, which provides a graphical representation for the problem of constructing function-correcting codes. We use this function dependent graph to get a lower bound on the redundancy required for function correction codes. By considering the function to be a bijection, such an approach leads to a lower bound on the redundancy required for classical systematic error correcting codes (ECCs). We propose a range of parameters for which the bound is tight. For single error correcting codes, we show that this bound is at least as good as a bound proposed by Zinoviev, Litsyn, and Laihonen in 1998. Thus, this framework helps to study systematic classical error correcting codes. Further, we study the structure of this function dependent graph for linear functions, which leads to bounds on the redundancy of linear-function correcting codes. We show that the Plotkin-like bound for function-correcting codes proposed by Lenz et.al 2023 is simplified for linear functions. We identify a class of linear functions for which an upper bound proposed by Lenz et al., is tight and also identify a class of functions for which coset-wise coding is equivalent to a lower dimensional classical error correction problem.

Figures (6)

• Figure 1: The function Correction Setting
• Figure 2: FCC Graph $\mathcal{G}_f(t,k,r)$ for $t=1, k=2$ and $r=2$ and the function $f((u_1u_2))=u_1\lor u_2$. An independent set of size 4 is highlighted in bold.
• Figure 3: Graphs $\mathcal{R}$, $\mathcal{G}_f(t,k,0)$ and $\mathcal{G}'_{f}(t,k,r)$ for $t=1$, $r=1$, $k=3$, and the function $f:\mathbb{F}_{2}^{3}\to \{0,1\}$ defined as $f((u_{1}u_{2}u_3)) = u_{1}\lor u_{2}\lor u_3$.
• Figure 4: Plots of $\Delta r_{BGS} := r_{BGS}-r'$ vs $k$, for different values of field size $q$ and Hamming distance $d$, where $r_{BGS}$ is the lower bound on parity obtained using the BGS bound (Corollary \ref{['corr:sys bound']}) and $r'$ is the lower bound on parity obtained from Corollary \ref{['corr:classical lower bound']}.
• Figure 5: Plots of $\Delta r_{BLB} := r_{BLB}-r'$ vs $k$, for different values of field size $q$ and Hamming distance $d$, where $r_{BLB}$ is the best known lower bound on the parity of linear codes and $r'$ is the lower bound on parity obtained from Corollary \ref{['corr:classical lower bound']}.

Theorems & Definitions (59)

• Theorem 1
• Theorem 2
• Definition 1
• Remark 1
• Definition 2
• Remark 2
• Definition 3
• Definition 4
• Remark 3
• Remark 4