Abstruct-A new family of MDS array codes is presenled. The code arrays contain p information columns and T indelpendent parity columns, each column consisting of p -1 bits, where p is a prime. We extend a previously known construction for 1 he case T = 2 to three and more parity columns. It is shown that when r = 3 such extension is possible for any prime p. For larger values of T , we give necessary and sufficient conditions for our codes to be MDS, and then prove that if p belongs to a certain class of primes these conditions are satisfied up to T 5 8. One of the advantages of the new codes is that encoding and decoding may be accomplished using simple cyclic shifts and XOR operations on the columns of the code array. We devellop efficient decoding procedures for the case of two-and three-column erroirs. This again extends the previously known results for the case of i i singlecolumn error. Another primary advantage of our codes is related to the problem of efficient information updates. We present upper and lower bounds on the average number of parity bits which have to be updated in an MDS code over GF (2"), following an update in a single information bit. This average number is of importance in many storage applications wlhich require frequent updates of information. We show that the upper bound obtained from our codes is close to the lower bound and, most importantly, does not depend on the size of the code syimbols.
A necessary and sufficient condition for the extended BBV code with four parity columns to be an MDS code is given in REF , and some results for no more than eight parity columns are provided.
MDS Array Codes with Independent Parity Symbols
IEEE TRANS. ON INFORMATION THEORY
Computer Science Mathematics