On List Decoding of Insertion and Deletion Errors

Insdel errors occur in communication systems caused by the loss of positional information of the message. Since the work by Guruswami and Wang, there have been some further investigations on the list decoding of insertion codes, deletion codes and insdel codes. However, unlike classical Hamming metric or even rank-metric, there are still many unsolved problems on list decoding of insdel codes. The contributions of this paper mainly consist of two parts. Firstly, we analyze the list decodability of random insdel codes. We show that list decoding of random insdel codes surpasses the Singleton bound when there are more insertion errors than deletion errors and the alphabet size is sufficiently large. Furthermore, our results reveal the existence of an insdel code that can be list decoded against insdel errors beyond its minimum insdel distance while still having polynomial list size. This provides a more complete picture on the list decodability of insdel codes when both insertion and deletion errors happen. Secondly, we construct a family of explicit insdel codes with efficient list decoding algorithm. As a result, we derive a Zyablov-type bound for insdel errors. Recently, after our results appeared, Guruswami et al. provided a complete solution for another open problem on list decoding of insdel codes. In contrast to the problems we considered, they provided a region containing all possible insertion and deletion errors that are still list decodable by some q-ary insdel codes of non-zero rate. More specifically, for a fixed number of insertion and deletion errors, while our paper focuses on maximizing the rate of a code that is list decodable against that amount of insertion and deletion errors, Guruswami et al. focuses on finding out the existence of a code with asymptotically non-zero rate which is list decodable against this amount of insertion and deletion errors.