Binary Complementary Code Structure via a Simple Necessary Condition

Abstract

A complementary set of $K$ binary $\pm 1$ codes of length $N$ has the useful property that the sum of the autocorrelation sequences of the codes has only one nonvanishing element, its size- $KN$ peak. The $N\times K$ matrix whose columns are the codes of a binary complementary set, arranged in order, is the code set's complementary code matrix (CCM). One might ask for which $K$ and $N$ such matrices exist. A simple necessary condition for the existence of an $N\times K$ CCM is introduced, involving the sum of the squares of the imbalance values of the codes in the set (the imbalance of a binary $\pm 1$ code being the difference between the number of 1 and number of $-1$ s in the code). The one-to-one relationship between binary CCMs and binary complementary code sets allows the necessary condition for the $N\times K$ CCM existence to extend to a necessary condition for sets of $K$ complementary code sets of length $N$ . We examine the sum-of-squares condition for the separate cases of $N$ even and $N$ odd. The odd case is especially interesting, since representations in terms of sums of squares of odd integers are equivalent to representations as sums of triangular numbers, a subject of recent research in Number Theory. Furthermore, we exhibit a refinement of the existence condition in terms of the imbalance values of pairs of interlaced subcodes. This refinement is new, to the best of our knowledge. Finally, we investigate whether odd-length complementary code sets exist for $K>2$ , unlike the Golay case of $K=2$ where odd-length sets cannot exist. We easily find, by computer search, a four-code set of length 15, suggesting that these are likely not uncommon.