Abstract – Recently, a new direction in coding theory has been to apply the Gray map to codes that are linear over Z4 to obtain binary nonlinear codes better than comparable binary linear codes. The distance properties of these codes as well as the correlation properties of sequences obtained from Z4-linear codes depend in many cases on exponential sums over Galois rings. We present a survey of recent results on exponential sums over Galois rings and their applications to coding theory and sequence designs.

Keywords – Coding theory, cyclic codes, sequences, exponential sums, Galois rings, Z4-linear codes.

Introduction

In an important paper, Hammons et. al. show how to construct well known binary nonlinear codes like Kerdock codes and Delsarte-Goethals codes by applying the Gray map to linear codes over Z4. Further, they explain an old open problem in coding theory that the weight enumerators of the nonlinear Kerdock codes and Preparata codes satisfy the MacWilliams identities. Nechaev has shown that the Kerdock code punctured in two coordinates, is equivalent to a cyclic (but still nonlinear) code. The coordinate permutation that yields the binary cyclic code is identified by making a connection between the Kerdock code and a Z4-linear code. These discoveries lead to a strong interest in Z4-linear codes, and recently several other binary nonlinear codes which are better than comparable binary linear codes have been found using the Gray map on Z4-linear codes.

Many of the new codes are constructed from extended cyclic codes over Z4.