Abstract
We consider construction of deletion and substitution correcting codes with low redundancy and efficient encoding/ decoding. First, by simplifying the method of Sima et al . (ISIT 2020), we construct a family of binary single-deletion s -substitution correcting codes with redundancy (s+1) (2s+1)\log _{2} n+o(\log _{2} n) and encoding complexity O(n^{2}) , where n is the blocklength of the code and s\geq 1 . The construction can be viewed as a generalization of Smagloy et al .'s construction (ISIT 2020), and for the special case of s=1 , our construction is a slight improvement in redundancy of the existing works. Further, we modify the syndrome compression technique by combining a precoding process and construct a family of systematic t -deletion s -substitution correcting codes with polynomial time encoding/decoding algorithms for both binary and nonbinary alphabets, where t\geq 1 and s\geq 1 . Specifically, our binary t -deletion s -substitution correcting codes of length n have redundancy (4t+3s)\log _{2}n+o(\log _{2}n) , whereas, for q being a prime power, the redundancy of q -ary t -deletion s -substitution codes is asymptotically \left({4t+4s-1-\lfloor \frac {2s-1}{q}\rfloor }\right)\vphantom {{\lfloor \frac {2s-1}{q}\rfloor }_{j}}\log _{q} n + o(\log _{q}n) as n\to \infty . We also construct a family of binary systematic t -deletion correcting codes (i.e., s=0 ) with redundancy (4t-1)\log _{2} n+o(\log _{2} n) . The proposed constructions improve upon the redundancy of the state-of-the-art constructions.