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 quasi-linear time decoding algorithms for both erasure and adversary errors.

翻译：给定长度和最小距离下，确定q元码的最大规模（等价于寻找最低冗余度）是编码理论的核心基础问题之一。受Varshamov-Tenengolts（简称VT）码通过校验和构造方法的启发，我们提出了一种显式构造非线性码的方案，在相同长度和最小距离条件下，该方案具有比线性码更低的冗余度。与VT码类似，我们的构造方法在小距离（甚至恒定距离）下表现良好。此外，我们还设计了针对擦除错误和对抗性错误的拟线性时间译码算法。