We consider locally repairable codes over small fields and propose
constructions of optimal cyclic and linear codes in terms of the
dimension for a given distance and length.
Four new constructions of optimal linear codes over small fields with
locality properties are developed. The first two approaches give binary
cyclic codes with locality two. While the first construction has
availability one, the second binary code is characterized by multiple
available repair sets based on a binary Simplex code.
The third approach extends the first one to q-ary cyclic codes
including (binary) extension fields, where the locality property
is determined by the properties of a shortened first-order Reed–
Muller code. Non-cyclic optimal binary linear codes with locality
greater than two are obtained by the fourth construction.