We define new metrics for linear codes over the ring Fq[u]/(ut) via an Fq-module monomorphism on linear codes over Fq. The construction generalizes the Gray map, Gray weight, and Lee weight; and the technique allows us to find some new optimal linear codes and their weight enumerator polynomial.

This work is focused on linear error-correcting codes over chain rings. By a linear code over a chain ring R of length n, we mean a R-submodule of the module Rn . The basic introduction to the theory of finite commutative chain rings and linear codes over them is given. We especially emphasize here their al- gebraic description. Minimal homogenous and Hamming distances of these codes are extensively studied. We explain, how the generalized Gray map can transform linear codes over a chain ring into general non-linear codes over a field. We deal with the construction of linear codes over a chain ring and the construction of generator matrices based on random generation is described. Obtained codes are compared with known results.

Certain nonlinear binary codes having at least twice as many codewords as any known linear binary code can be regarded as the binary images of linear codes over Z4. This vision leads to a new concept in coding theory, called the Z4-linearity of binary codes. This thesis is a survey on the linear quaternary codes and their binary images under the Gray map. The conditions for the binary image of a linear quaternary code to be linear are thoroughly investigated and the Z4-linearity of the Reed-Muller and Hamming codes is discussed. The contribution of this study is a simplification on the testing method of linearity conditions via a few new lemmas and propositions. Moreover, binary images (of length 8) of all linear quaternary codes of length 4 are analyzed and it is shown that all 184 binary codes in the nonlinear subset of these images are worse than the (8, 4) Hamming code. This thesis also includes the Hensel lift and Galois ring which are important tools for the study of quaternary cyclic codes. Accordingly, the quaternary cyclic versions of the well-known nonlinear binary codes such as the Kerdock and Preparata codes and their Z4-linearity are studied in detail.