A theoretical study of complementary binary code sequences and a computer search for new kernels.

Abstract

Complementary binary sequences were invented by Golay in an investigation of infra-red multi-slit spectrometry. This dissertation formalizes the basic results obtained by Golay and develops new concepts and techniques for examining the characteristics of these special binary codes. This work has developed new understanding of the structure and methods for the decomposition of complementary sequences. Complementary sequences have the property of an infinite correlation peak to ambiguity ratio when detected with a matched filter. These binary sequences should find much application as pseudo- random noise modulation signals for both radar and communications systems. A discussion of the need for such sequences is included in the introduction and is followed by a state of the art description. An operations group on the sequences is formulated and the proofs of several theorems concerning the operations group are given in a rigorous manner. One reason for developing the operations group is the application to elimination of redundance in the computer search for new codes. Several invariant properties of complementary codes are proved through the use of the Hamming distance concept. Many more invariant properties of the sequences are demonstrated through the introduction of a Hamming vector. The concept of a Hamming vector is extremely useful as a complementary code decomposition tool. A large number of theorems are proved to enhance its use in this field. Several computer searches for complementary sequences are described and the actual computer programs for the CDC 1604 are included in the Appendix.