$Z_4$-codes and their Gray map images as orthogonal arrays

TL;DR

This paper extends Delsarte's classical connection between code strength and dual minimum weight to codes over arbitrary finite rings, especially focusing on $Z_4$ and their Gray map images as orthogonal arrays.

Contribution

It generalizes Delsarte's result to codes over finite rings and establishes the relationship between Gray map images and Lee weights for $Z_4$ codes.

Findings

Delsarte's observation extends to codes over finite rings.

The strength of Gray map images relates to Lee weights.

Results apply to non-linear codes over $Z_4$.

Abstract

A classic result of Delsarte connects the strength (as orthogonal array) of a linear code with the minimum weight of its dual: the former is one less than the latter. We show that Delsarte's observation extends to codes over arbitrary finite rings. Since the paper of Hammons \emph{et al.}, there is a lot of interest in codes over rings, especially in codes over $Z_4$ and their (usually non-linear) binary Gray map images. We show that Delsarte's observation extends to codes over arbitrary finite commutative rings with identity. Also, we show that the strength of the Gray map image of a $Z_4$ code is one less than the minimum Lee weight of its Gray map image.