On Function-Correcting Codes
arxiv(2024)
摘要
Function-correcting codes were introduced in the work "Function-Correcting
Codes" by Lenz et al. 2023, which provides a graphical representation for the
problem of constructing function-correcting codes. We use this graph to get a
lower bound on the redundancy required for function correction. 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 of small
distances. We identify a range of paramters 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, we propose the use of this framework to
study systematic classical codes. Further, we study the structure of this graph
for linear functions, which leads to bounds on the redundancy of
linear-function correcting codes. We also show that for linear functions, a
Plotkin-like bound proposed by Lenz et.al 2023 simplifies for linear functions.
We present a version of the sphere packing bound for linear-function correcting
codes. We identify a class of linear functions for which an upper bound
proposed by Lenz et al., is tight. We also characterise a class of functions
for which coset-wise coding is equivalent to a lower dimensional classical
error correction problem.
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