GOLAY SEQUENCES

This list contains the Golay sequences, i.e. four (1,-1) sequences A, B of length n which satisfy the conditions: N_A(s)+N_B(s)=0, s=1,2,...,n-1. More results on Golay sequences can be found in . Open cases and exhaustive search for lengths up to 100 can be found in . All Golay sequences of lengths n=2a10b26c exist for any non negative integers a, b, and c.

Known Cases: (Primitive pairs)

• n=1 :
A=(1), B=(1)
• n=2 :
A=(1,1), B=(1,-1)
• n=10 :
A=(1,1,-1,1,-1,1,-1,-1,1,1), B=(1,1,-1,1,1,1,1,1,-1,-1)
A=(1,1,1,1,1,-1,1,-1,-1,1), B=(1,1,-1,-1,1,1,1,-1,1,-1)
• n=20 :
A=(1,1,1,1,-1,1,-1,-1,-1,1,1,-1,-1,1,1,-1,1,-1,-1,1),
B=(1,1,1,1,-1,1,1,1,1,1,-1,-1,-1,1,-1,1,-1,1,1,-1)
• n=26 :
A=(1,1,1,1,-1,1,1,-1,-1,1,-1,1,-1,1,-1,-1,1,-1,1,1,1,-1,-1,1,1,1),
B=(1,1,1,1,-1,1,1,-1,-1,1,-1,1,1,1,1,1,-1,1,-1,-1,-1,1,1,-1,-1,-1)
In , it is proved that all Golay sequences of length n<100 are derivable from this set of five primitive pairs.

Open cases for lengths n<200 are as follows.
Open cases:   n = 106, 116, 122, 130, 136, 146, 148, 164, 170, 178, 194.

 P.B. Borwein and R.A. Ferguson, A complete description of Golay pairs for lengths up to 100, Mathematics of Computation, 73 (2003), 967-985.

 C. Koukouvinos, Sequences with zero autocorrelation, in CRC Handbook of Combinatorial Designs, C.J. Colbourn and J.H. Dinitz (Eds.), CRC Press, (1996), 452-456.