The Connections Among Hamming Metric, b-Symbol Metric, and r-th Generalized Hamming Metric

Abstract

The r -th generalized Hamming metric and the b -symbol metric are two different generalizations of Hamming metric. The former is used on the wire-tap channel of Type II, and the latter is motivated by the limitations of the reading process in high-density data storage systems and applied to a read channel that outputs overlapping symbols. In this paper, we study the connections among the three metrics (that is, Hamming metric, b -symbol metric, and r -th generalized Hamming metric) mentioned above and give a conjecture about the b -symbol Griesmer Bound for cyclic codes.