New infinite families of near MDS codes holding t-designs and optimal locally recoverable codes
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Linear codes with parameters [n,k,n-k] are called almost maximum distance separable (AMDS for short) codes. AMDS codes whose duals are also AMDS are said to be near maximum distance separable (NMDS for short). In 1949, Golay discovered the first near MDS code holding 4-designs, i.e. the [11,6,5] ternary Golay code. In the past 70 years after this discovery, only sporadic NMDS codes holding t-designs were found. In 2020, Ding and Tang made a breakthrough in constructing the first two infinite families of NMDS codes holding 2-designs or 3-designs. Up to now, there are only a few known infinite families of NMDS codes which hold t-designs for t=2,3,4 in the literature. The objective of this paper is to construct new infinite families of NMDS codes holding t-designs. To this end, some special matrices over finite fields are used as the generator matrices of the NMDS codes. We then determine the weight enumerators of the NMDS codes and prove that the NMDS codes hold 2-deigns or 3-designs. Compared with known t-designs from NMDS codes, ours have different parameters. Besides, several infinite families of optimal locally recoverable codes are also derived via the NMDS codes.
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