title: Kasami code
text: # Kasami code
  
  Kasami sequences are binary sequences of length $2^N−1$ where N is an even integer. Kasami sequences have good cross-correlation values approaching the Welch lower bound. There are two classes of Kasami sequences—the small set and the large set.
  
  ## Kasami Set
  
  The process of generating a Kasami sequence is initiated by generating a maximum length sequence a(n), where n = $1…2^N−1.$ Maximum length sequences are periodic sequences with a period of exactly $2^N−1.$ Next, a secondary sequence is derived from the initial sequence via cyclic decimation sampling as b(n) = a(q ⋅ n), where q = $2^{N/2}+1.$ Modified sequences are then formed by adding a(n) and cyclically time shifted versions of b(n) using modulo-two arithmetic, which is also termed the exclusive or (xor) operation. Computing modified sequences from all $2^{N/2}$ unique time shifts of b(n) forms the Kasami set of code sequences.
  
  