Described are methods, systems, and apparatus, including computer program products for error correction coding and decoding procedures for data storage or transfer. A plurality of data blocks is received. A plurality of checksum blocks are generated by multiplying the plurality of data blocks by a coding matrix, where the coding matrix comprises values of at least one basic interpolation polynomial and the multiplying is according to a finite field arithmetic for a finite field comprising all possible values of the plurality of data blocks and the plurality of coding blocks. The plurality of data blocks and the plurality of checksum blocks are stored in a data storage medium.