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The $r$-th generalized Hamming metric and the $b$-symbol metric are twodifferent generalizations of Hamming metric. The former is used on the wire-tapchannel of Type II, and the latter is motivated by the limitations of thereading process in high-density data storage systems and applied to a readchannel that outputs overlapping symbols. In this paper, we study theconnections among the three metrics (that is, Hamming metric, $b$-symbolmetric, and $r$-th generalized Hamming metric) mentioned above and give aconjecture about the $b$-symbol Griesmer Bound for cyclic codes. %Furthermore,we explore the combinatorial function of the size of the $b$-symbol weight setof a code $C$.

The connections among Hamming metric, $b$-symbol metric, and $r$-th generalized Hamming metric