Block Golay sequences with applications

Golay sequences have been used extensively for constructing base sequences, Yang numbers, T-sequences, Hadamard matrices, SBIBDs and Hadamard matrices with maximum possible sums. The possibility of obtaining new Golay sequences is diminishing and only non-existence results are appearing nowadays. We introduce block Golay sequences. It turns out that every result on Golay sequences could be extended to block Golay sequences. The abundance of such sequences and their applications will be presented. Let A {aI, a2, ... , be a sequence of variables of n. The nonperiodic auto-correlation function of the sequence A is defined by { o , j 2: n. Two sequences A {aI, a2, ... , B = {b I , b2l ... ,bn } are called Golay sequences of length n if all the entries are (1, -I)-and NA(j) + NB(j) 0 for all j 2: 1. Golay sequences exist for orders 2 I026 , a, b, c, non-negative integers. The sequence A = {AI, ... , An}, where are L.A.J.'''''''J.J.,",'''' of order m, is called a Block Barker sequence of length n and block size m if: (i) AiA; AjA~ for all ~,J; (ii) AiA~ = nm1m; { n-j L AiA;+j = 0 for J 1,2, ... ,n-l, (iii) NA(j) i=l 0 j 2: n. Australasian Journal of Combinatorics §. ( 1992 ), pp. 293-303 Two sequences A = {AI, A2 , .•• , An}, B {Bl' B 2 , • .. , Bn}, where AiS and Bis are (1, -I)-matrices of order m, are called block Golay sequences of length n and block size m if: (i) (ii) L~=l (AiA~ + BiBJ) (iii) NA(j) + NB(j) Lemma 1: Let X {A, and block size m. Then Z n -+ e and block size m. Proof: Note that (Nx + Ny)(j) = forward. 0 Lemma 2: for j 1, 2, ... , n 1, o for j 2:: n. be two Barker sequences of lengths n,.e respectively, (A, B), Y (A, -B), is a block Golay sequence of + 2N B (j) 0 for all j 2:: 1. The rest is straight(i) If there is a block Golay sequence of length n and block size m, then there is a block sequence of length rn and block size m, where r is the length of a sequence. (ii) If there are block Golay sequences of then there are block Golay sequences of a, b are non-negative integers. Proof: n, .e and block size m, k respectively, and block size m a kb where (i) Let B be block Golay sequences of length n and block size m and C, D Golay sequences of length r. Using an idea of Turyn, the following are the required block Golay sequences: 1 A x -(C + D) 2 A x ~(C* D*) 2 1 + Bx 2 1 B X 2(C* + D*) , where X* is the sequence whose elements are the reverse of those in X. The proof that the nonperiodic auto-correlation function is zero is similar to that of Turyn and the rest is straightforward.