Systematic Codes Correcting Multiple-Deletion and Multiple-Substitution Errors

Codes Over Absorption Channels

“…The study of codes that correct single-deletion and single-substitution errors was first introduced in the context of DNA-based data storage in [18] and further developed in [19]. More recently, codes that correct multiple-deletion and multiple-substitution errors were proposed in [20]. These results apply to our error model as well, but in this paper we demonstrate that it is possible to use specific absorption properties to achieve higher rates in our codes.…”

“…So we can apply multiple-deletion correcting codes for our setting. There are already a myriad of works on binary multiple-deletion correcting codes (see, for example, [20], [22], [23], [29]- [31]). For t = 2, the best known result was given in [23], where an explicit binary 2-deletion correcting code of length n with redundancy at most 4 log 2 (n) + O(log 2 log 2 (n)) was constructed.…”

“…This code is polynomial-time encodable and decodable. For general t 3, the best known result was contributed in [20], where the authors proved that there is a binary systematic t-deletion correcting code of length n with redundancy at most (4t − 1) log 2 (n) + o(log 2 (n)). The encoding and decoding complexities are O n 2t+1 and O n t+1 respectively.…”

“…In [19], the study of single-deletion single-substitution codes was initiated, and the authors gave a q-ary single-deletion single-substitution correcting code of redundancy at most 10 log 2 (n) + O(1) (measured in bits) [19,Corollary 17]. For more details about this kind of codes, we refer the interested readers to [19] and [20].…”

“…After that, we explain the difference between t-absorption and t-deletion-t-substitution, which justifies our searching for better codes for our setting. Our construction is based on the single-absorption correcting code given in Theorem IV.9 and the syndrome compression technique with precoding developed recently [20].…”