We introduce a MinSum-based list decoder for product codes and analyze its performance. We show that it can guarantee successful list decoding for decoding radii that lie above half the code's minimum distance.
Product codes have a poor minimum distance, but an efficient low-complexity decoding algorithm. To measure the performance of a decoder, like Reddy-Robinson's for product codes, which decode error patterns beyond half the minimum distance, we use the notion of equivalent diameter of the decoding region [4]. We produce here a bound for the performance of product code which is, for the considered example, far below the simulated performance but still above the performance of known BCH codes with same parameters.>
Motivated by the significant performance gains which polar codes experience under successive cancellation list decoding, their scaling exponent is studied as a function of the list size. In particular, the error probability is fixed and the trade-off between block length and back-off from capacity is analyzed. A lower bound is provided on the error probability under MAP decoding with list size L for any binary-input memoryless output-symmetric channel and for any class of linear codes such that their minimum distance is unbounded as the block length grows large. Then, it is shown that under MAP decoding, although the introduction of a list can significantly improve the involved constants, the scaling exponent itself, i.e., the speed at which capacity is approached, stays unaffected for any finite list size. In particular, this result applies to polar codes, since their minimum distance tends to infinity as the block length increases. A similar result is proved for genie-aided successive cancellation decoding when transmission takes place over the binary erasure channel, namely, the scaling exponent remains constant for any fixed number of helps from the genie. Note that since genie-aided successive cancellation decoding might be strictly worse than successive cancellation list decoding, the problem of establishing the scaling exponent of the latter remains open.