Two-weight codes over the integers modulo a prime power
This work addresses theoretical coding theory problems for researchers in finite fields and combinatorics, providing incremental results on code properties and graph constructions.
The paper proves that irreducible cyclic codes of length p^2-1 and dimension 2 over integers modulo p^h have exactly two nonzero Hamming weights, and constructs strongly regular graphs from their duals, with some codes meeting known bounds like the Griesmer bound.
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.