Locally maximal recoverable codes and LMR-LCD codes

In this work, we propose two new types of codes with locality, namely, locally maximal recoverable (LMR) codes and \(\lambda \) $\lambda$ -maximally recoverable ( \(\lambda \) $\lambda$ -MR) codes. The LMR codes are a subclass of codes with \((r, \delta )\) $(r,\delta )$ -locality such that they can correct h additional erasures in any one local set, in addition to having \((r, \delta )\) $(r,\delta )$ -locality. These codes are a restricted case of maximally recoverable (MR) codes, which enable recovery from all information-theoretically correctable erasure patterns in a local set. The \(\lambda \) $\lambda$ -MR codes are a subclass of LMR codes which can also handle \(\lambda \) $\lambda$ erasures from any coordinate positions. We give constructions for both of these families of codes. We also study the LMR codes that satisfy the complementary dual property. It is well known that codes with this property are capable of safeguarding communication systems against fault injection attacks. We give a construction of distance-optimal cyclic LMR codes that satisfy the complementary dual property.