New infinite families of uniformly packed near-MDS codes and multiple coverings, based on the ternary Golay code

We present five new infinite families of linear near-MDS codes uniformly packed in the wide sense (UPWS). These codes are also almost perfect multiple coverings of the deep holes or farthest-off points (APMCF), i.e. the vectors lying at distance R (covering radius) from the code. The families are constructed by m-lifting when one takes a starting code C over the ground Galois field Fq with a parity check matrix H(C) and then considers the codes Cm over Fqm, m ≥ 2, with the same parity check matrix H(C). As starting codes we used the ternary perfect Golay code and codes obtained by its extension and puncturing. To prove the needed combinatorial properties (UPWS and APMCF), we used the m-lifting of the dual codes and features of near-MDS codes. A general theorem on infinite families of UPWS near-MDS codes is proved.