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
_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.