Maximum distance separable codes for b-symbol read channels

Recently, Yaakobi et al. introduced codes for b-symbol read channels, where the read operation is performed as a consecutive sequence of b>2 symbols. In this paper, we establish a Singleton-type bound for b-symbol codes. Codes meeting the Singleton-type bound are called maximum distance separable (MDS) codes, and they are optimal in the sense they attain the maximal minimum b-distance. We introduce a construction method using projective geometry, and then construct several infinite families of linear MDS b-symbol codes over finite fields. The lengths of these codes have a large range. And in some sense, we completely determine the existence of linear MDS b-symbol codes over finite fields for certain parameters.