Efficient exhaustive search for binary complementary code sets

Abstract

Abstract-Binary complementary code sets offer a possibility that single binary codes cannot -zero aperiodic autocorrelation sidelobe levels. These code sets can be viewed as the columns of so-called complementary code matrices, or CCMs. This matrix formulation is particularly useful in gaining the insight needed for developing an efficient exhaustive search for complementary code sets. An exhaustive search approach is described, designed to find all sets of K complementary binary codes of length N , for specified N and K. Results for several cases are examined.

Group symmetries of complementary code matrices

“…Next, we present the row-correlation function, which gives an equivalent representation of a CCM in terms of its rows [2,6].…”

“…Using this theorem we can see that with one CCM many more can be created. Coxson and Russo created an algorithm for narrowing down the search space for complementary code matrices using these symmetries [2] . This is important due to the fact that the search space for p-phase N × K CCMs is p N K ; as the dimensions increase the search takes exponentially longer to conduct an exhaustive search.…”

“…The following is an explanation of our exhaustive search algorithm for CCMs. Our search algorithm is based on the search algorithm of Coxson and Russo [2]. While their work searches for the binary CCMs, ours focuses on any poly-phase.…”

“…In Section 4, we extend Coxson and Russo's [2] efficient exhaustive search algorithm for binary CCMs to p-phase CCMs. This algorithm was implemented for quad-phase N × 4 CCMs in Section 5 to obtain a classification of all equivalence classes for N = 2, 3, 4, 5, 6 (see Table 1 for a description of the number of equivalence classes and those that are represented by Hadamard matrices).…”

Doppler tolerance, complementary code sets, and generalised Thue–Morse sequences

“…When the code is binary, the set is called a Golay complementary pair, after Marcel Golay who discovered these sets while solving a problem in infrared spectrometry [4]. Complementary code matrices (CCMs) provide a useful matrix formulation for the study of complementary code sets [5–7]. Given a set of K codes of length N , the corresponding N × K CCM has the k th code as its k th column, k = 1, …, K .…”