TL;DR
This paper investigates weighted Hamming metrics that allow classical binary Hamming and extended Hamming codes to be perfect, and explores the construction of 2-perfect codes within these metrics.
Contribution
It characterizes weight structures enabling Hamming codes to be perfect and proposes methods to construct 2-perfect codes in weighted Hamming metrics.
Findings
Identifies weight structures for perfect binary Hamming codes.
Provides constructions for 2-perfect codes in weighted Hamming metrics.
Extends understanding of code perfection in non-standard metrics.
Abstract
A weighted Hamming metric is introduced in [4] and it showed that the binary generalized Goppa code is a perfect code in some weighted Hamming metric. In this paper, we study the weight structures which admit the binary Hamming code and the extended binary Hamming code to be perfect codes in the weighted Hamming metric. And, we also give some structures of a 2-perfect code and how to construct a 2-perfect code in some weighted Hamming metric.
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