The VT and Helberg codes, both in binary and non-binary forms, stand as elegant solutions for rectifying insertion and deletion errors. In this paper we consider the quaternary versions of these codes. It is well known that many optimal binary non-linear codes like Kerdock and Prepreta can be depicted as Gray images (isometry) of codes defined over $\mathbb{Z}_4$. Thus a natural question arises: Can we find similar maps between quaternary and binary spaces which gives interesting properties when applied to the VT and Helberg codes. We found several such maps called Naisargik (natural) maps and we study the images of quaternary VT and Helberg codes under these maps. Naisargik and inverse Naisargik images gives interesting error-correcting properties for VT and Helberg codes. If two Naisargik images of VT code generates an intersecting one deletion sphere, then the images holds the same weights. A quaternary Helberg code designed to correct $s$ deletions can effectively rectify $s+1$ deletion errors when considering its Naisargik image, and $s$-deletion correcting binary Helberg code can corrects $\lfloor\frac{s}{2}\rfloor$ errors with inverse Naisargik image.

翻译：VT码与Helberg码（包括二进制与非二进制形式）是纠正插入与删除错误的优雅解决方案。本文研究这两类码的四进制版本。众所周知，许多最优非线性二进制码（如Kerdock码和Prepreta码）可描述为定义在$\mathbb{Z}_4$上的码的Gray像（等距映射）。因此自然产生一个问题：能否在四进制与二进制空间之间找到类似映射，使其应用于VT码与Helberg码时展现有趣性质？我们找到了若干此类映射，称为Naisargik（自然）映射，并研究了四进制VT码与Helberg码在这些映射下的像。Naisargik像与逆Naisargik像为VT码与Helberg码提供了有趣的纠错性质：若VT码的两个Naisargik像生成相交的单位删除球，则这些像具有相同权值；专为纠正$s$次删除而设计的四进制Helberg码，在考虑其Naisargik像时可有效纠正$s+1$次删除错误，而纠正$s$次删除的二进制Helberg码通过逆Naisargik像可纠正$\lfloor\frac{s}{2}\rfloor$次错误。