Function-Correcting Codes for Locally Bounded Functions
Charul Rajput, B. Sundar Rajan, Ragnar Freij-Hollanti, Camilla Hollanti
TL;DR
This paper generalizes function-correcting codes (FCCs) to locally bounded function classes, introducing the notion of locally $(\rho,\lambda)$-bounded functions under a contiguity assumption on their function balls. It provides a constructive FCC for the subclass $(2t,4)$-bounded with redundancy $r_f(k,t) \le 3t$ and extends the framework to a general bound $r_f(k,t) \le N(\lambda,2t)$ for locally $(2t,\lambda)$-bounded functions, where $N(\lambda,2t)$ is the minimum length of a binary code with $\lambda$ codewords and distance $2t$. The authors also show that any function can be treated as locally bounded for suitable parameters, illustrating this with Hamming weight and weight-distribution functions, and they offer simple FCC constructions in these settings. The work broadens the FCC paradigm beyond bijective or specific-function cases, enabling targeted protection of function outputs with potentially lower redundancy than protecting full messages, and it highlights contiguity as a key structural property for efficient FCC design.
Abstract
In this paper, we introduce a class of functions that assume only a limited number $λ$ of values within a given Hamming $ρ$-ball and call them locally $(ρ, λ)$-bounded functions. We develop function-correcting codes (FCCs) for a subclass of these functions and propose an upper bound on the redundancy of FCCs. The bound is based on the minimum length of an error-correcting code with a given number of codewords and a minimum distance. Furthermore, we provide a sufficient optimality condition for FCCs when $λ= 4$. We also demonstrate that any function can be represented as a locally $(ρ, λ)$-bounded function, illustrating this with a representation of Hamming weight distribution functions. Furthermore, we present another construction of function-correcting codes for Hamming weight distribution functions.