Some nonlinear codes, such as Kerdock and Preparata codes, can be represented as binary images under the Gray map of linear codes over rings. This paper introduces MAP decoding of Kerdock and Preparata codes by working with their quaternary representation (linear codes over Z4 ) with the complexity of O(N2log2N), where N is the code length in Z4. A sub-optimal bitwise APP decoder with good error-correcting performance and complexity of O(Nlog2N) that is constructed using the decoder lifting technique is also introduced. This APP decoder extends upon the original lifting decoder by working with likelihoods instead of hard decisions and is not limited to Kerdock and Preparata code families. Simulations show that our novel decoders significantly outperform several popular decoders in terms of error rate.

翻译：某些非线性码（如Kerdock码和Preparata码）可表示为环上线性码在格雷映射下的二元像。本文通过利用这些码的四元表示（Z4上的线性码），引入复杂度为O(N log₂N)的Kerdock码和Preparata码的MAP译码，其中N为Z4码长。此外，还介绍了一种采用译码器提升技术构建的、具有良好纠错性能且复杂度为O(N log₂N)的次优逐位APP译码器。该APP译码器在原始提升译码器基础上，通过采用似然值而非硬判决进行扩展，且不限于Kerdock和Preparata码族。仿真结果表明，本文提出的新型译码器在误码率方面显著优于多种主流译码器。