Fuzzy linear codes based on nested linear codes
Jon-Lark Kim
TL;DR
This work builds a bridge between fuzzy set theory and linear coding by showing that a fuzzy linear code $A$ corresponds to a family of nested linear codes via upper level cuts $A_{\alpha}$, and vice versa. It develops the arithmetic of fuzzy codes, introduces fuzzy self-dual and self-orthogonal notions, and constructs new fuzzy code families including fuzzy Hamming, Golay, and Reed-Muller codes, accompanied by a general syndrome-decoding algorithm that exploits nested structure. The results demonstrate that fuzzy linear codes parameterize a base linear code through a chain of subcodes and duals, enabling efficient decoding and a unified, modular view of classical codes. The approach offers practical decoding advantages for fuzzy codewords in noisy channels and provides a versatile framework for exploring code families via fuzzy self-duality and self-orthogonality.
Abstract
In this paper, we describe a correspondence between a fuzzy linear code and a family of nested linear codes. We also describe the arithmetic of fuzzy linear codes. As a special class of nested linear codes, we consider a family of nested self-orthogonal codes. A linear code is self-orthogonal if it is contained in its dual and self-dual if it is equal to its dual. We introduce a definition of fuzzy self-dual or self-orthogonal codes which include classical self-dual or self-orthogonal codes. As examples, we construct several interesting classes of fuzzy linear codes including fuzzy Hamming codes, fuzzy Golay codes, and fuzzy Reed-Muller codes. We also give a general decoding algorithm for fuzzy linear codes.