Two weight $\mathbb{Z}_{p^k}$-codes, $p$ odd prime

TL;DR

This paper proves the non-existence of certain two-weight codes over the ring _{p^k} for odd primes, using graph-theoretic methods related to strongly regular graphs.

Contribution

It establishes a non-existence result for regular homogeneous two-weight _{p^k}-codes with specific dual distance properties, expanding understanding of code structures over rings.

Findings

No regular homogeneous two-weight _{p^k}-codes exist for odd prime p and kgeq 2 with dual Hamming distance at least four.

The proof connects code properties to the existence of strongly regular graphs built on cosets.

Provides conditions under which such codes cannot be constructed.

Abstract

We show that regular homogeneous two-weight $\mathbb{Z}_{p^k}$-codes where $p$ is odd and $k\geqslant 2$ with dual Hamming distance at least four do not exist. The proof relies on existence conditions for the strongly regular graph built on the cosets of the dual code.