Irregular MDS Array Codes

In this paper, we extend the concept of maximum-distance separable (MDS) array codes to a larger class of codes, where the array columns contain a variable number of data and parity symbols and the codewords cannot be arranged, in general, in a regular array structure with equal column length. These new codes, named irregular MDS array codes, find applications in problems of distributed data storage with multiple sources of information generating data at unequal rates. We solve the problem of finding optimal parity symbol allocations that achieve minimum redundancy for a given level of protection against block erasures. We provide a classification of irregular MDS array codes according to the parameters of their parity symbol allocation and we show how regular MDS array codes are a special case of this wider class. We derive necessary and sufficient conditions for such irregular array codes to be MDS and extend the concept of the lowest density generator matrix. Finally, we show how a simple constructive method allows to design irregular lowest density MDS array codes with alphabet size independent of the size of the array columns.