Stopping Set Analysis of Iterative Row-Column Decoding of Product Codes

Eirik Rosnes

doi:10.1109/tit.2008.917686

Abstract

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 p denote the product code of two binary linear codes C c and C r of minimum distances dc and d r and second generalized Hamming weights d 2 (C c ) and d 2 (C r ), respectively. We show that the size s min of the smallest noncode- word stopping set is at least mm(d r d 2 (C c ),d c d 2 (C r )) > d r d c , where the inequality follows from the Griesmer bound. If there are no codewords in C p 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 (C r ) < 2d r and d 2 (C c ) < 2d c . Finally, as an example, we apply the derived results to the product of two (extended) Hamming codes and two Golay codes.

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