In this work, we study linear codes with the folded Hamming distance, or equivalently, codes with the classical Hamming distance that are linear over a subfield. This includes additive codes. We study MDS codes in this setting and define quasi MDS (QMDS) codes and dually QMDS codes, which attain a more relaxed variant of the classical Singleton bound. We provide several general results concerning these codes, including restriction, shortening, weight distributions, existence, density, geometric description and bounds on their lengths relative to their field sizes. We provide explicit examples and a binary construction with optimal lengths relative to their field sizes, which beats any MDS code.

翻译：本文研究具有折叠汉明距离的线性码，或等价地，研究在子域上具有经典汉明距离的线性码。这包括加法码。我们在此框架下研究MDS码，并定义准MDS（QMDS）码及对偶QMDS码，它们达到经典Singleton界的一种更松弛的变体。我们提供了关于这些码的若干一般性结果，包括限制、缩短、重量分布、存在性、密度、几何描述以及其长度相对于域大小的界。我们给出了显式示例和一个相对于域大小具有最优长度的二进制构造，该构造优于任何MDS码。