Function-Correcting Codes for Linear and Locally Bounded Functions Over a Finite Chain Ring
This work addresses incremental advancements in coding theory for error correction in specific function classes over finite rings, relevant to researchers in coding and information theory.
The paper tackles the problem of extending function-correcting codes for linear and locally bounded functions over finite chain rings, deriving improved bounds like a Plotkin-like bound for irregular homogeneous distance codes over Z4 and providing explicit constructions to assess bound tightness.
In this paper, we further extend the study of function-correcting codes in the homogeneous metric over a chain ring $\mathbb{Z}_{2^s}$ for broader classes of functions, namely, locally bounded functions and linear functions, and for weight functions, modular sum functions. e define locally bounded functions in the homogeneous metric over $\mathbb{Z}_{2^s}^k$ and investigate the locality of weight functions. We derive a Plotkin-like bound for irregular homogeneous distance code over $\mathbb{Z}_4$, which improves the existing bound. Using locality properties of functions, we establish upper and lower bounds on the optimal redundancy. We provide several explicit constructions of function-correcting codes for locally bounded functions, weight functions, and weight distribution functions. Using these constructions, we further discuss the tightness of the derived bound. We explicitly derive a Plotkin-like bound for linear function-correcting codes that reduces to the classical Plotkin bound when the linear function is bijective, we further discuss a construction of function-correcting linear codes over $\mathbb{Z}_{2^s}$.