The Rank of Random Binary Matrices and Distributed Storage Applications

Random binary matrices appear in a variety of signal processing and encoding problems. They play an important role in rateless codes and in distributed storage applications. This paper focuses on block angular matrices, a class of random rectangular binary matrices that are particularly suited to distributed storage applications. We address one of the key issues regarding binary random matrices in general, and block angular matrices in particular: the probability of obtaining a full rank matrix, when drawing uniformly at random from the set of binary matrices with compatible structure. This paper gives a closed-form expression for this probability, as well as some bounds and approximations.