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 $\mathbb{F}_q$-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.

翻译：$b$-符号度量是汉明度量的一种推广。在$b$-符号度量中，线性码已被应用于输出由$b$个连续符号组成的读取信道。当$q$固定且码长足够大时，对于汉明度量下的$\mathbb{F}_q$-线性码，Griesmer界优于Singleton界。这一情形同样适用于$b$-符号度量。Shi、Zhu和Helleseth近期对$b$-符号度量下的循环码提出了一个猜想。本文通过串联线性码与单形码，提出了线性码的$b$-符号Griesmer界。基于循环码和扩展循环码，我们构造了两类关于$b$-符号Griesmer界距离最优的线性码族。