Two-weight codes over the integers modulo a prime power

TL;DR

This paper studies two-weight irreducible cyclic codes over integers modulo a prime power, showing their properties, bounds, and connections to strongly regular graphs via Galois rings and p-adic lifts.

Contribution

It introduces a new class of two-weight codes over Galois rings, analyzes their parameters, and constructs related strongly regular graphs using p-adic lifts.

Findings

Codes have exactly two nonzero weights.

Codes meet the Griesmer bound when primitive.

Constructed graphs are strongly regular and have a common cover.

Abstract

Let $p$ be a prime number. Irreducible cyclic codes of length $p^2-1$ and dimension $2$ over the integers modulo $p^h$ are shown to have exactly two nonzero Hamming weights. The construction uses the Galois ring of characteristic $p^h$ and order $p^{2h}.$ When the check polynomial is primitive, the code meets the Griesmer bound of (Shiromoto, Storme) (2012). By puncturing some projective codes are constructed. Those in length $p+1$ meet a Singleton-like bound of (Shiromoto , 2000). An infinite family of strongly regular graphs is constructed as coset graphs of the duals of these projective codes. A common cover of all these graphs, for fixed $p$, is provided by considering the Hensel lifting of these cyclic codes over the $p$-adic numbers.