Some Properties and Constructions of Weakly Self Dual LRCs

A code is called $(n,\ k,\ r,\ t)_{a}$ locally repairable code (LRC) if all the symbols (coordinates) of each codeword can be retrieved from at least t disjoint sets of at most r other symbols called repair sets (recover sets). In this work, we go through the properties of $(n,\ k,\ r,\ t)_{a}$ weakly self dual LRCs. We show that there is no weakly self dual LRC for $t\geq 2$. Moreover, we remark that the existence of $(n,\ k,\ r,\ t)_{a}$ weakly self dual LRC implies that the availability t of each symbol is strictly 1. Further, we furnish some constructions, via parity check matrix, of weakly self dual LRCs over different fields, some of which are also optimal against the Singleton-like bound. Finally, we also provide the existential criteria for $(n,\ k,\ r,\ t,\ x)_{a}$ weakly self dual LRCs having intersecting repair sets and propose a necessary condition for binary $(n,\ k,\ r,\ t,\ x)_{a}$ LRCs to be weakly self dual.