A general family of Plotkin-optimal two-weight codes over $\mathbb{Z}_4$
Hopein Christofen Tang, Djoko Suprijanto
TL;DR
The paper classifies Plotkin-optimal two-weight projective codes over $\mathbb{Z}_4$ by deriving necessary identities and bounds, proving that such codes must have two Lee weights with $w_1=n$ and $w_2=|C|/2$, and providing explicit weight distributions. It then provides a general construction framework to generate an infinite family of these codes from a given two-weight code, and specializes to concrete families, including a $k_2=0$ case and a broad $k_1,k_2,t$ family with exact parameters. The Gray images of these codes correspond to SU1-type binary codes, linking $\mathbb{Z}_4$-code theory to binary strongly regular structures and enabling a transfer of weight-distribution properties. This work generalizes and unifies prior results (notably Shi et al. 2020) and expands the catalog of Plotkin-optimal two-weight codes over $\mathbb{Z}_4$, while also highlighting the existence of non-Plotkin-optimal two-weight projective codes.
Abstract
We obtain all possible parameters of Plotkin-optimal two-Lee weight projective codes over $\mathbb{Z}_4,$ together with their weight distributions. We show the existence of codes with these parameters as well as their weight distributions by constructing an infinite family of two-weight codes. Previously known codes constructed by Shi et al. (\emph{Des Codes Cryptogr.} {\bf 88}(3):1-13, 2020) can be derived as a special case of our results. We also prove that the Gray image of any Plotkin-optimal two-Lee weight projective codes over $\mathbb{Z}_4$ has the same parameters and weight distribution as some two-weight binary projective codes of type SU1 in the sense of Calderbank and Kantor (\emph{Bull. Lond. Math. Soc.} {\bf 18}:97-122, 1986).