A data storage system stores a set of codewords in memory. The set of codewords are encoded in accordance with a joint nesting matrix specifying multiple layers of integrated interleaved codes, including first, second and third layers of integrated interleaved codes, and the set of codewords stored in the memory include first, second and third layers of parity information corresponding to the first, second and third layers of integrated interleaved codes. When decoding a first codeword and a first subgroup containing the first codeword fail, the system decodes a group of codewords that include two more subgroups of codewords, including the first subgroup of codewords, using the third layer parity information for the group of codewords. The second and third layers of integrated interleaved codes are configured to enable decoding of two codewords, in a subgroup of codewords, having errors beyond the correction capability of the first layer codes.