Determining the largest size, or equivalently finding the lowest redundancy, of $q$-ary codes for given length and minimum distance is one of the central and fundamental problems in coding theory. Inspired by the construction of Varshamov-Tenengolts (VT for short) codes via check-sums, we provide an explicit construction of nonlinear codes with lower redundancy than linear codes under the same length and minimum distance. Similar to the VT codes, our construction works well for small distance (or even constant distance). Furthermore, we design $O(n\log^4 n)$ bit operations decoding algorithms for both erasure and adversary errors, where $n$ is the code length.