1. Systematic Codes Correcting Multiple-Deletion and Multiple-Substitution Errors.
- Author
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Song, Wentu, Polyanskii, Nikita, Cai, Kui, and He, Xuan
- Subjects
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DECODING algorithms , *POLYNOMIAL time algorithms , *CRUSH syndrome , *ERROR-correcting codes , *VIDEO compression - 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. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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