Cyclic LRC codes, binary LRC codes, and upper bounds on the distance of cyclic codes
Itzhak Tamo, Alexander Barg, Sreechakra Goparaju, Robert Calderbank · International Journal of Information and Coding Theory · 2016
We consider linear cyclic codes with the locality property or locally recoverable codes (LRC codes). A family of LRC codes that generalises the classical construction of Reed-Solomon codes was constructed in a recent paper by Tamo and Barg (IEEE Transactions on Information Theory, No. 8, 2014). In this paper, we focus on distance-optimal cyclic codes that arise from this construction. We give a characterisation of these codes in terms of their zeros and observe that there are many equivalent ways of constructing optimal cyclic LRC codes over a given field. We also study subfield subcodes of cyclic LRC codes (BCH-like LRC codes) and establish several results about their locality and minimum distance. The locality parameter of a cyclic code is related to the dual distance of this code, and we phrase our results in terms of upper bounds on the dual distance.