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A survey on quaternary codes and their binary images

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2009
Özkaya, Derya
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.