  In classical coding theory, it is common to construct new codes via
propagation rules. There are various propagation rules to construct classical
block codes. However, propagation rules have not been extensively explored for
constructions of locally repairable codes. In this paper, we introduce a few
propagation rules to construct good locally repairable codes. To our surprise,
these simple propagation rules produce a few interesting results. Firstly, by
concatenating a locally repairable code as an inner code with a classical block
code as an outer code, we obtain quite a few dimension-optimal binary locally
repairable codes. Secondly, from this concatenation, we explicitly build a
family of locally repairable codes that exceeds the Zyablov-type bound.
Thirdly, by a lengthening propagation rule that adds some rows and columns from
a parity-check matrix of a given linear code, we are able to produce a family
of dimension-optimal binary locally repairable codes from the extended Hamming
codes, and to convert a classical maximum distance separable (MDS) code into a
Singleton-optimal locally repairable code. Furthermore, via the lengthening
propagation rule, we greatly simplify the construction of a family of locally
repairable codes in \cite[Theorem 5]{MX20} that breaks the asymptotic
Gilbert-Varshamov bound. In addition, we make use of three other propagation
rules to produce more dimension-optimal binary locally repairable codes.
Finally, one of phenomena that we observe in this paper is that some trivial
propagation rules in classical block codes do not hold anymore for locally
repairable codes.
