Griesmer Bound and Constructions of Linear Codes in b-Symbol Metric

The
b
-symbol metric is a generalization of the Hamming metric. Linear codes in the
b
-symbol metric have been used in the read channel whose outputs consist of b consecutive symbols. The Griesmer bound outperforms the Singleton bound for F
_q
-linear codes in the Hamming metric, when
q
is fixed and the length is large enough. This scenario is also applicable in the
b
-symbol metric. Shi, Zhu, and Helleseth recently made a conjecture on cyclic codes in the
b
-symbol metric. In this paper, we present the
b
-symbol Griesmer bound for linear codes by concatenating linear codes and simplex codes. Based on cyclic codes and extended cyclic codes, we propose two families of distance-optimal linear codes with respect to the
b
-symbol Griesmer bound.