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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 andfundamental problems in coding theory. Inspired by the construction ofVarshamov-Tenengolts (VT for short) codes via check-sums, we provide anexplicit construction of nonlinear codes with lower redundancy than linearcodes under the same length and minimum distance. Similar to the VT codes, ourconstruction works well for small distance (or even constant distance).Furthermore, we design quasi-linear time decoding algorithms for both erasureand adversary errors.

Nonlinear Codes with Low Redundancy