Mathematics Colloquium, CSU
Thursday, May 1, 2008
3PM–4PM in RT 1516
Sequences with good correlation properties
K. T. Arasu, Wright State University
Abstract. An r -dimensional matrix A = a[J1 , . . . , Jr ] with 0 ≤ Ji < Si (1 ≤ i ≤ r )
is called an S1 ×· · ·×Sr array. The array is called perfect if the periodic autocorrelation
coefficients
RA (u1, . . . , ur ) =
SX
1 −1
J1 =0
···
SX
r −1
a[J1 , . . . , Jr ]a[J1 + u1 (mod S1 ), . . . , Jr + ur (mod Sr )]
Jr =0
are zero for all (u1 , . . . , ur ) 6= (0, . . . , 0), for 0 ≤ ui < Si . The array is binary if each
matrix entry is ±1. In the general situation, we allow the entries of the array to be
complex numbers and modify the above definition of autocorrelation coefficients by
using the complex conjugate in the second factor of each term of the r -fold sum. In
other words, it is simply a generalized Hermitian inner product. The case r = 1 will
correspond to “sequences”. In this case, the periodic autocorrelations are, in general,
congruent to the length of the sequence (mod 4). The smallest possible such value
in the range {0, 1, −1, 2, −2} would correspond to what we call “perfect” sequences.
Of particular interest are the p-ary sequences, where p is a prime, and the entries
of the underlying sequence are p-th roots of unity. The ternary case has entries that
are complex third roots of unity. In this case, the prefix “perfect” for the underlying sequence (i.e. 1-dimensional array) refers to the case when all the out-of-phase
autocorrelations are equal to minus one. In this talk, we give some new construction
methods for these interesting combinatorial objects. This is a joint work with John
Dillon and Kevin Player.
Biographical information. Professor Arasu received his Masters degree in 1977 at
Panjab University, Chandigarh, India and Ph.D in 1983 from Ohio State University.
He has been at Wright State University since 1983. He also spent 2 years in Germany
as a Humboldt fellow. He has given lectures at various universities in North America,
Asia, Europe and Australia. His research interests lie in the general area of applicable
algebra. He has published over 80 research articles and is on the editorial board of the
international journal Designs, Codes and Cryptography.