Isometric embeddings of mathbb{Z}_{p^k} in the Hamming space mathbb{F}_{p}^{N} and mathbb{Z}_{p^k}-linear codes

Isometric embeddings of $mathbb{Z}_{p^n+1}$ into the Hamming space ( $mathbb{F}_{p}^{p^n},w$ ) have played a fundamental role in recent constructions of non-linear codes. The codes thus obtained are very good codes, but their rate is limited by the rate of the first-order generalized Reed–Muller code—hence, when n is not very small, these embeddings lead to the construction of low-rate codes. A natural question is whether there are embeddings with higher rates than the known ones. In this paper, we provide a partial answer to this question by establishing a lower bound on the order of a symmetry of ( $mathbb{F}_{p}^{N},w$ ).