Stopping set analysis of iterative row-column decoding of product codes

In this paper, we introduce stopping sets for iterative row-column decoding of product codes using optimal constituent decoders. When transmitting over the binary erasure channel (BEC), iterative row-column decoding of product codes using optimal constituent decoders will either be successful, or stop in the unique maximum-size stopping set that is contained in the (initial) set of erased positions. Let C,, denote the product code of two binary linear codes C, and C,. of minimum distances d(c). and d(r), and second generalized Hamming weights (12 (C,) and d(2) (G), respectively. We show that the size si,, of the smallest noncode-word stopping set is at least min(d(1)-d(2)(C-c), d(c)d(2)(C-r)) > d(c)d(2) where the inequality follows from the Griesmer bound. If there are no codewords in C,, with support set S, where S is a stopping set, then S is said to be a noncodeword stopping set. An immediate consequence is that the erasure probability after iterative row-column decoding using optimal constituent decoders of (finite-length) product codes on the BEC, approaches the erasure probability after maximum-likelihood decoding as the channel erasure probability decreases. We also give an explicit formula for the number of noncodeword stopping sets of size s(min), which depends only on the first nonzero coefficient of the constituent (row and column) first and second support weight enumerators, for the case when d(2)(G) < 24 and d(2)(C.) < 2d(r). Finally, as an example, we apply the derived results to the product of two (extended) Hamming codes and two Golay codes.